Conversations

0:00 That is one of the things that he certainly had as part of his program. I'm not sure of the answer. Next time I see him I must ask him. I know you said it's certainly a part of the program yeah it seems very natural conjecture that it would because you know it would give much more kind of natural control over the variability of the sets than in the Cantorian setting. And also, it yields this minimal parts geometry, which of course must, well... Yeah, surely implies the continuum hypothesis, or at least possibly. So, yeah. What's very interesting is we're looking at the... My motivation is this, that if you manage to define in mathematics something that defines a macroscopic level on mathematics, I think that's precisely his line of thought as well, his conjecture. He actually has some motivational remarks in the book about that, in the sort of more introductory historical philosophical chapters in the book. The section that's called The Serpent Encantles Paradise, that he deals precisely with that, is clearly part of the motivation. It's a very, very interesting work. He's also an extremely congenial and approachable person. I mean, very, you know, great, great pleasure to be around. Oh, yes, huge. I mean, he's not, I mean, he's somebody who has many interests outside pure mathematics. He has a great, his children, he's a delightful man, he's a delightful man, great man. Well, you must, I mean, you must, actually, I'm afraid I've forgotten. So I don't have a note of your name. Could you give me your coordinates and I'll put you in touch with him and suggest that he emails you. Apart from anything else, he also has, of course, a deep love for ancient Greek mathematics.

2:30 Ah, you have one of his cards. Good. He also has, of course, tremendous interest in the history of ancient Greek mathematics, as is apparent from his book, all that discussion of Arithmos and... We can make something like a two-day seminar or something like that. This would be great. This would be great. He's retired now, of course, so he's done that. Well, no, no, it's on the contrary. It's easier because it means he doesn't have the commitments. He's been teaching ancient history, ancient Greek mathematics, and giving seminars on structuralism and category theory from his own, actually, fairly negative. He recognizes, of course, but from the foundational viewpoint, he does, of course, think that it's not, it's claims to be, in any sense, what he thinks is required, it's claims to satisfy what he thinks is required from the notion of a foundation for mathematics or not. Very strong. He sees it as a fantastically rich conceptual framework for the organization of mathematical knowledge, but not in the strict sense of what he would term foundational. Fundamental, but not foundational. Well, no, because he has this particular line, which is partly of a philosophical position as to what, strictly speaking, is required of a foundation. And, you know, from that point of view, he thinks that set theory, not in the sense of any axiomatic set theory, but... It's something which is quite outside the concerns of most mathematicians, and certainly should not become part of their concerns, it would probably stop a lot of them from doing good mathematics if it did become part of their concerns, but all the same it is necessary just in the same way that the plumbing of a building is necessary, however magnificent the architect. The vision of the architect, unless somebody has installed the, you know, the ventilation ducts and the plumbing and the sewers, there will be problems with that building eventually. That's the metaphor he likes to use. So its function is a very lowly function. It's like that of the sanitary engineer relative to the visionary architect, but without the sanitary engineer, the visionary architect has tended to produce...

5:00 Any way, I think that it would be very interesting to have a discussion with that person. He's very interested in hyperfinite arithmetic, I know, so it would be, but I'll certainly give you, I'll give his card, not only just give him your card, because I've been a very old friend of his now for 20 years. Press him very hard to get in touch with you. He's always, he's a little bit isolated these days in Bristol because sadly at one time Bristol had a tremendous reputation in logic and set there in a model theory. It was a very strong department in logic in the days of people like, well Solovey was there for a time and I'm trying to think of who, but they had a lot of famous names there in the 70s and 80s. But now the department is very, very skewed towards applied mathematics and towards operational analysis and coding theory and things like that, because that's, of course, where the money is for complex, and they have hardly any, they actually have very few pure mathematicians left, they have a few topologists, they have some very good Russian, a couple of Russian number theorists, but... But very few pure mathematicians, and he's the only magician, self-theorist, model theorist that they have left, so, and he does actually feel to be honest that they had a new head of department about a year before he retired, who was rather a pleasant piece of work, I would say. He's an American, very much an applied mathematician, and absolutely no interest in foundational questions at all. He tended to push John out, and there were personality clashes there as well, which is why he was much happier to go off to teach in the philosophy department. But of course his real interest, he still has this excellent group of research students doing the, continuing to develop this program, Euclidean set theory, and as I said, particularly the corollaries of the Euclidean set theory viewpoint for things like geometry and measure theory, and some of the, this mineral parts geometry that his student, who is in fact now his son-in-law, in fact married his daughter.

7:30 Dick and Lush, who had been a pure set theorist and a very good one. He did his PhD under Jensen, which doesn't get much higher than that in set theory. I mean, Jensen, Woodin are the two greatest names in large cardinal. Very, very gifted set theorists. But after doing work in set theory, got very interested, really, for purely conceptual reasons, in Mabry's approach. ...and decided to do a second PhD thesis, which is extraordinary in Bristol, he already had it on, because he was so interested in the subject and he wanted to work out what the consequences of redoing foundations of geometry in the setting of Euclidean set theory would be, and it turned out... They produce a very interesting minimal parts geometry which has many connections with these kind of theories of the stochastic diagonal and the way that, well, for instance, the way that people have Feynman in physics defined. So there are all sorts of interesting connections and motivations. I wish I knew more about it, because there's a very beautiful and elegant sort of lattice theoretic aspect to this minimal pass geometry as well, which I don't pretend to understand at all, which also has lots of interest from the point of view of topology, and he, Lush, is pushing that side of the program together with Pettigrew. Mabry's most recent student was a very, very talented young man of whom I think we will hear a lot more in the future, so it's an interesting time, but of course because they're in Bristol itself, they're really completely isolated, they are always very, very keen to speak to people in other parts of the world, and especially from Greece because John absolutely worships everything to do with Greece. I'm sure he'd love to come. I remember going with him in Italy around the temples at Paestum. One year in 1994, I'm going to do one of the most wonderful Greek attempts that I have in the Gulf of Salerno, and just seeing how ecstatic he was to be standing where Pythagoras and Heraclitus and these people have stood. But I don't think he's ever been to Greece. I don't know, but I don't think he has. I think if he had been, I would know. Because I've known him for 20 years now, I don't think he, very sadly he lost his wife Anita about five years ago now to cancer. She was a lot younger than him actually, so he is now 67, no actually come to think of it he's probably 69 this year, and people are, yeah, no you must be much younger than that. I'm 66, right.

10:00 He's, I was going to say he's 67, but in fact I think I've probably missed out a year because, you know, when you get to our age, people always tend to stay the age they were when you last spoke to them, you know, or when you last, you know, or even when you last met them. No, John is, no, no, no, no, no, actually he's not. No, because Bill Laugher is 71 this year, and John is four years younger than Bill, so he is 67, yes, he's 67, yeah, because he took early retirement, he left the department about two years earlier than he would have needed to instruct, yeah, yes, so he's 67, but I mean, very, absolutely, you know, tremendously. Vigorous and active. But a delightful man to be around. A very good company socially. No, I shall certainly try and get you together. You'd enjoy his company as well as the mathematical discussions that you undoubtedly have. But sadly his wife Anita. This is where I, in fact, Mabry and I both went to Bolzano in 1998 for this meeting at which Gonzalo Reyes also spoke, and Bill Lorvier. It was in 1998. It was a very interesting meeting. It was on the theme of holes and parts. Holes and parts in mathematics, in physics, and in biology. I don't know if the organizer was... Pauli, Pauli. He was one of the organizers. The organizers were Roberto Pauli and this guy Alberto Peruzzi in the University of Florence, who is very close to the program. So now it was all again Pauli who organized the levels of reality. Pauli is the regular organizer. Yes, yes. I would love to have gone to that. I wasn't able to get to that one. And now I have promised him for... There is another one this year, that we should write together a paper about the levels of reality in mathematics. I have written a paper published in Sykes, a volume of... Sika, Sika.

12:30 Oh, Gian Domenico Sika, the guy who has that publishing house, Polymetrica. Yeah, yeah, yeah. Polymetrica di Milano, who published these books on philosophy and mathematics. Very interesting young man. So, in one of these foundations of mathematics or something like that. Yes, yes. Oh, really? I must go. I will certainly read that. Which book was that in? Which of the books was that in? Foundations of mathematics, something like that. So he's published a whole series. They have Foundations of Geometry, Foundations of Topology. Yes, he edits all of them. I mean, he's very much a one-man band. I mean, he does it all on a shoestring. He's got tremendous energy. The only problem is that he does need a bit more editorial resources. Oh, he's very young. He's only about... I actually went down to Milano to meet him in connection because we had a lot of... There's a lot of material in our archive which we had permission to have transcribed and written up, some of which he was going to publish as part of his introductory book on category theory. John Bell introduced me to him. He's also an old friend of mine. He's actually based, not actually in Milano itself, but a little town just north of Milano. I can't think now what it's called. Very charming. Send me an email. Absolutely fast. Just to have you also here. Yeah, sure, sure, sure. We'll do that as soon as I get back, I promise. And I was very impressed. He's only about 25, 26. He actually borrowed the money from his father. Yes, I can imagine he would be. Yes, yes, yes, very young, very young. And right now, I was very, I was impressed. I don't like to be impressed. I would like to see if I have a continuum hypothesis. I completely agree. If I have one of the ways, might be Mabry's, that is using the Euclid, Euclid, of course, he needs to have this, you know, to have geometry, you have to have a level of reality, macroscopic, you know, to have a triangle, otherwise... Yes, otherwise we have no experience of... Triangle, nothing. So, in this way... But the idea that the ultimate reality might be some atomistic, you know, which of course is. Essentially the idea of the minimal parts geometry has been around well at least since at least since democracy.

15:00 Points, structures, and levels of reality. This was your title? My title. Is that on the web? Is there a way I can look that up? On your website? Yes, you can download it. But it is also in Seeker's volume. In Seeker's book, yes. But I don't think I have that particular book, so I... You can download it from my webpage. I think I have it in the card. So, and Pauli saw this paper and wrote me some, some images and invited me to write something for... Axiomathes, which is an interesting journal, which Bale and Peruzzi have both often written for. I said to him, that's very good and so on, but since you have, from a philosophical point of view, deal with left and right, why don't we write it together? And we agree with that. ...have been passed months. Yeah, he has this reputation, I'm afraid, Roberto Pali. I am the guy that... You're the perfectionist, but you know... I should write, and sometimes I must sit down even in this maybe not perfect form to write something. This is the thing. All the independence propositions, actually, they are nothing else but etiquettes. There is no fact of the matter at all about whether the continuum hypothesis holds or not, it holds in certain kinds of, it holds for certain structures both in mathematics itself, for instance it holds in topology, but also in mathematics itself. The physical universe and the world around it, it clearly holds for the kinds of structure which can be represented in a very atomistic way, built up from below by terms of ultimate ingredients.

17:30 But in the case of structures where you have to take account... In direct ways of the kind of cohesion and variation that is involved in the organic integration of the structure, then it's in general falsified, which, of course, connects with the point we were talking about the other day, the way one thinks of the axiom of choice in terms of the non-existence of obstructions to inverses of mass. In other words, we can think of it in terms very intuitively where you have got, for instance, a space in which you have got... Closed and bounded components in a park-disconnected space so that things can be moved around in a completely free way independently of each other, so that you've got a complete lack of cohesion, a co-discrete space. Then, of course, the... In this setting, typically there is some version of the axiom of, some version of the continuum hypothesis ought to hold, and certainly the axiom of choice naturally holds in such a setting, but in a case where you've got things like, well, the situations which are studied in homology, which typically involve obstructions, the existence of inverses, maps, more complicated spaces which are carriers of non-trivial homology and cohomology, it will be falsified. Even all the Boolean values of that, maybe it's better. Look, it's very interesting. Lord Beer has always said to me that he always thought, even at the time, with the Cohen proof and the Scots solid proof, These just naturally fit, these just naturally fill in a corner of a geometrical picture because this is, this is, this is the part of his theory that theory is really just a branch of... This is just a natural filler in of a small corner of a much deeper geometrical picture which takes account of what he terms cohesion and variation. In this model they prove that the action of choice is strictly between 0 and 1. Let's inverse the problem. Suppose that you have a model of theory in which the action of choice takes... Let's move this truth value and then you will have a whole bunch of models in which I imagine that if you go from 1 to 0, the truth value, then the...

20:00 The universe becomes more fuzzy and in zero it's chaos. You don't see anything. So something like that. This is exactly how L'Auveria thinks. He gave a beautiful talk in Cambridge Maths in 1989, which was actually published as a paper in Philosophia Mathematica about five years later in 1994. I must send it to you. This is exactly what he says. But this is very much his point of view and specifically about the case where you have a space which is completely discreet, co-discreet, so the one point... These are obviously metaphors for the kind of degree of cohesion of the background which is which is actually A kind of metaphorical notion that could be made much more exact in terms of Isbell's notion of adequate and co-adequate subcategories of a category, where you've got generating figures in the subcategories and the case where the... Figures are just trivially points, and points are adequate and co-adequate in this technical sense to do with functoriality of constructions, is really just the case where you have the category of sets, but just thought of geometrically, it's a very special, it's the category of discrete and co-described spaces, which is part of what he's got in mind by saying he thinks that set theory is just naturally falling into place within algebraic geometry as a kind of substructure within it as the, okay, zero cohesiveness. Put in a book, then... I'm sure that all these things that they are talking about, this talks, this is a formalist idea. The philosophy of mathematics is garbage. What passes under the name of the philosophy of mathematics in philosophy departments is just the most moribund and, I'm afraid, worthless, pointless subject you could imagine.

22:30 It's pursued by people who don't know any mathematics. I'm sure that this is perceptional principles about mathematics, not axioms, it is just... Basic issues of mathematical conceptualization. And this is what we call foundation in this sense. This is the true foundation. And this is what I mean. Foundations is to do with why universal constructions are indeed universal constructions. Also, the point that Bulbaki made about mother structures. I mean, the mother structures in Bulbaki, the ones which are really clearly the sources of all the deep ideas in mathematics and the ones which constantly kind of cross-fertilize each other to produce new kinds of unification of structure and new branches of subject, are from the point of view of logic, of course. I mean, they're typically not categorical. They're not... Well, it was a phase, a very important phase in the consolidation of mathematics. This is a very nice idea that almost produced all mathematics. Well, no, no, no, but I think that particular focus on what they call the mother structures and the reason why they were criticized, of course, for having ignored logic and being actually ignorant of logic, it has to be said, that the French... The French algebraic geometry school of the 1950s, you know, the people who were in the Boubaki group, like almost all French mathematicians except for Herbrand in the 20th century, really didn't pay any attention to logic. Mathias was right. Yeah, they didn't pay... Oh, you've read this article of Mathias, what he calls the ignorance of Boubaki. Yeah, yeah. Well, Mathias is a very strange man himself, but he's a very... They write... But he's right about that, particularly. In 1950, they saw this mathematics, and they never mentioned... No, no, it was ludicrous. It was ludicrous. I believe somebody has said to me, it may even have been Cartier, that Doudanet didn't even know of Cartier's theorem in 1950 when they wrote that paper, when they wrote that. I think, I mean, I may have said, didn't even know. No, I mean, their ignorance of logic really was pretty profound.

25:00 Well, that's rightly been called the worst book on set theory ever written, at least, well, the worst book on set theory ever written by the bubarki book, the bubarki, well, certainly the worst ever written by a serious man of the texture. No, the bubarki book on set theory is just completely, absolutely, as you say, unusable and unreadable. And yet they did come up, of course, because the history here is very important, which is why I think this archival work is important. They produced this general theory of structures, but just at the time, just before the adjoint functor theorem, and just before it really became clear... I think that category theory was something much more than just an extremely powerful tool. They had that, but of course they never put it to any use. It just turned out to be completely hollow. It had maximal but not the correct kind of generality. Although it's interesting because Mabry has written a very interesting article, I don't think it's been published yet, about the Babacki theory of structures, But I haven't really understood his argument. It's something to do with models of... It's something to do with models, I mean in the sense of tasking model theory, where you're not able to control the way of distinguishing between the occurrence of something as a subset and its occurrence as an element. It's something to do with the way that you understand membership chains in model theory. There is no doubt that no other foundational system has ever really been able to handle this problem, but the Burbanki, the old Burbanki theory of structures, which has never been put to use by anybody for anything, does, in fact, handle this problem. I would like to know the details. I've only just heard this very much. I want to know what's really going on there. But on the other hand, when I spoke to Richard, his student, he said, well, actually, the trouble is... They found that it is complicated and it was really messy. This is why nobody ever made use of their theory.

27:30 But I think that there might be there something. Well, this is what Mabry claims. There is something there that is worth going back to, to examine. Well, I haven't... I mean, you're talking to the wrong guy because I haven't even read the... What about the notion of structure? Well, in the Baubach in 1948, it's... In set theory, they have a definition of structure, and they use, I think, three sets. You can imagine vector space, for example. You have vectors, you have the scalars, that's two sets, and maybe you need another one. On three sets that they start with, they build up a whole, something like a universe. These three sets constitute the structure? Not the basis of the structure, something like that. And this is like taking morphisms and objects? I don't think so. It's more like the thing that we have in non-standard analysis, we have superstructures. Areal numbers. Then you may say, what can you say about real mathematics, in finitely many steps, not more, finitely, so you take the universe over the reals, countably many, then suppose that you take the category of sets as a basis, so you take a set here and another there, so you have a function, now... Suppose that you built up the superstructure over the A and the superstructure over the B. And then you see in a level somewhere here, this is topological space. All structures will occur. All structures will occur somewhere in this. The function here becomes continuous function there. So you have all the categories which are built on sets. And you can have a morphism, what we say monomorphism usually in non-starter superstructures, which goes from one superstructure to the other superstructure and converts one entity here to an entity there.

30:00 But let me just say one very important footnote to this is that these superstructures, or these structures in the Bilbarke setting, are not the cumulative rank structure of the ZF hierarchy. I mean, that actually shows up just as one structure within this setting, and the point that Mabry is making... Something like that. Something superstructure. Superstructure is a more simplified, say, way of seeing Burbanki's... Remember that Burbanki was writing this stuff just before category theory came along. Or at least before the kind of power of category theory became apparent. But they say that they have a... Is this an institution for non-theoretical foundations, like category-theoretical foundations? Yes, it was intended to be, because it wasn't based on an asylum. Well, they did, of course, Eilenberg came along and helped them make it, because he was part of the group. It was also partly a historical accident, because André Wey... ...was very against the project of going back and rewriting the first volume from the point of view of category three, which a number of people in the group wanted very much to do after Eilenberg had been a member of it. And when they saw the adjoint functor theorem and then, of course, Crotenbeek obviously started using category theoretic reasoning in such immense... This would have meant going back to the beginning, and also there was this personality thing which they hated each other, and two, it would have meant going back and restarting the whole project again. If they were as popular then they might have had the argument that it's the same way. Well, this is precisely the argument they did have because they hadn't seen the specificity of adjunctions. They hadn't understood the power of the notion of adjunction. This was the thing which changed their convictions, changed their outlook completely. But it would have meant going back and redoing everything. And, in fact, it was precisely the recognition of the power of the adjoint functor theorem that provoked Maclean to make his famous remark about which was directed exactly at the Burmarki theory of structures, which is that mathematics should be about correct generality and not maximal generality, correct and mathematically fruitful generality is almost always

32:30 Well, it's almost hardly ever maximal generality. But the point about the, as you say, the... And I have also an idea that the non-well-founded sets following Laver's idea is the category of sets. That is, abstract sets. First of all, in the ordinary set theory, the sets are not abstract. What do you mean? If you have an entity here that is structured, as you go up, so when you say abstract set, this is not abstract. In order to get abstract set, then you have sets coming out from graphs, not from trees. Trees give the classical sets. Graphs give the non-well-founded sets. And therefore, this abstract set that you just keep the outside arrows that, for example, three, three, you have three here and zero, one, two here, not the inside arrows. I see, you forget the composition. Yeah, exactly. So you have only... These are members of three, nothing else. This is by just forgetting the structures, the arrows, the inside arrows, keeping only the outside, which means that zero belongs to three, one belongs to nothing else, nothing between zero and one, for example. So this is an abstract set. And this is the set in category of set theory. And suggested, of course, precisely by an adjoint construction. Even in classical set theory, if you take trees, in order to get, again... Abstract sets. You have to forget some inside arrows.

35:00 Of course, of course. Which is all that additional structure which really doesn't do any work. But, you know, people arguing about whether, well, in the von Neumann ordinals, whether zero is really a member of... Let's X be a set. If you say, let's be in category of sets, that's okay. But if you say, let's X be a set, and now let's define a structure of group on this set. There's no meaning of that. We've gone very radical now. Yeah. Let X be a set is not meaning proof. Yes, exactly. At least if you take the von Neumann universe, if these are the sets, then what does it... And then I will... The obsession was, of course, because they thought... I mean, we'll have to remember the whole point about... These things are supposed to be elements of collections in extension. Therefore, there has to be an answer to the question of identity conditions. Whether a thing is a member of the set or not, which is the whole basis, obviously, of the epsilon way of thinking of sets. That is, before they were thinking about sets like abstract sets. Yes, well, that's what you understood by the meaning of sets as trees, I take it. After that, the Kuratovsky definition of order person, then for Neumann universe. Well, this is the... The universe of sets? Well, what are these sets? These are everything. I mean, if you go to the 25th level, then you have linear spaces. Everything is there. So, I mean, in a few words, if you examine a little bit the things. You will discover this. I see. It's supposed to be structuralized but you cannot dispense with it because you have all this hierarchy. It is the practice of real mathematicians. It's not coherent with the foundation... And that was always L'Ovire's objection to it, of course, that the foundation based on membership... If you take category theory, then it's not... More strictly, the view based on the idea that membership was absolute and global, and fixed global absolute membership. I mean, it's not that the category sets dispenses with the notion of membership, it represents it very well, but it's clear that membership...

37:30 The notion of membership is something local and relative, but not global and absolute, as in the Rhone-Irish. I think this is a particular case of a more general paradigm shift, where reductionism is not so, which was very big in earlier centuries. And also, I think, that Laverie has the right ideas and conceptions, but when he writes something, It's like trying to swim through wet sand, I think somebody wants it. It's like reading Heidegger is like trying to swim through wet sand, I'm afraid. Any simple word. He says a lot of deep things. But this is not the way that... He's never been the most brilliant expulsor of his own ideas, I'm afraid. Because if it was otherwise, then I think right now he should travel until... Okay, we're going to... But many, many, many mathematicians, they don't... He's a theory structure... Most mathematicians are just machines for turning coffee into theorems, as somebody once described them, and they're just not interested in conceptual issues, and a number of mathematicians have said to me when I've made claims to them, or people much smarter, more articulate, and mathematically better informed than I have made the case to them for the conceptual importance of their work. The answer to that is, what did Descartes ever prove? And how important for the history of mathematics do you think Descartes was? And it's a really philistine and stupid reaction, but I'm afraid it is one that you get quite commonly from, you know. Well, I think, well, I mean, it's not nice. It's also not entirely fair because he has proved some quite important things. If you express suspicion like that in a quick way, it's a little...

40:00 Well, it just, it exposes you as a philistine, you know. But I think it's all right because... Oh, yeah, sure. ...because they're different. I don't want to engage with the ideas, but I'm talking about the kind of person who just... ...from 1931 until 1948 visiting Scholar. No, they wouldn't even give him a permanent position, let alone a chair. No, no, they wouldn't, because von Neumann made the famous remark. Which of us... von Neumann refused... Thanks again. Five papers. Five papers and some philosophical... And this is not enough. Well, in fact, von Neumann in the end kind of more or less blackmailed him into making... because he said that he would refuse to accept. But then it was terrible in some of the things he said about Goethe, when they moved Goethe out. Not in the logic. Well, no, exactly. Yeah, sure, sure. Yeah, I'll come with you, thanks. Yes, come on. You guys coming? Okay. Okay. We're all going to the same place anyway, so. Look at this. Splendid. Splendid car. Good muddy car like old cars should be. A working car. It's a rain today from North Africa or something. Serious working car. Good. Please, I'm going to go and see if you can put the car here. Oh, I see. OK, OK. Can I just stick this in the back? Yeah, of course. Is that OK? Just takes up less room. Don't let me forget it, though, because... Yeah, that's great. Thanks very much.

42:30 Well, I did still think the last time he was talking to anybody about mathematics, still think that logic would just absolutely drop out as a tiny byproduct of some general theory about classifying rings. Because it deals with the subalgebra of objects in a topos, and this is naturally the subject of a more general theory of classifying rings, which would unify the general structure of subalgebras of different theories. Cohen proved his theorems and results. They were giving a seminar in Princeton about these things, forcing and things like that. The algebraic geometries and algebraic topologies, they were just passing without giving just a glance. No, no, that doesn't surprise me at all. Absolutely doesn't surprise me at all. At least Lorvier has the grace to say that, you know, he thinks that he thinks that it will all drop into place, you know, as a fragment, you know, as a fragment of a bigger geometric picture. But, you know, he does think it's really important to understand how and why and, you know, it's crazy. No, it's very, yes, it's very hard to cope with. Are there many non-geometers that have tried to follow the proof of the Poincare? Of course not, no, of course not, of course not. And it's a great thing. Yes, but they should have been just a little bit more humble. Oh, I see. Yeah, I agree, I mean, about their own... Yeah, I mean, yeah, everything about the attitude is wrong. Yeah, everything about the attitude towards logic, that it was a... I'm afraid it's not confined to the French. There's a very famous story about... British mathematician and an analyst and number theorist, in fact you probably know Swinerton Dyer, who was said about a colleague who was a logician, and in fact who did some of the most important work in forcing, beat a man gamma subject. Arrogance doesn't come much more gross than that. I mean, Scott suffered from this terribly, you know, with, uh... It's surprising that he said it sounds quite, I don't know, philosophical when he talks.

45:00 Swinerton Dyer. Really? Oh, that's interesting. I've heard him talk a few times, but only on very technical subjects. My impression was that he had very little in the way of conceptual philosophical interest, but that just could be complete misunderstanding. Swinerton Dyer. But I've only listened to him now that he's like very old. Yeah, and famous and just gives... No. Maybe they're more keen on looking at the big picture. Well, of course you are. That's when you're supposed to go off and look at the big picture. That's why people like Atiyah and Penrose spend their time doing it. And good for them. I mean, there's a lot to be said for looking at the big picture. The view that you only do it when you're too old to do real mathematics is, well, sad. It's a widespread prejudice amongst mathematicians. Yes, I obviously can tell just from your talk that you are very much a Grotendiekian in formation, which of course I thoroughly... I don't know the logical... You're probably less interested in the logical aspects of the effective topos, the recursion, let alone the computer. I mean, the fascinating thing about topos is it's such an... Hence of course the title of Peter's book. But it's such an incredibly multifaceted topic that it connects clearly like all the... I mean Atiyah sometime at one point said the importance of any idea in mathematics, the importance of any... A body of concepts in mathematics is the test, the test of the importance of a mathematical idea is the degree of its connection to other parts of mathematics. Well, Proposter certainly meets that test to a very, very high degree, an almost unprecedented degree, which is why I realized right at the beginning how important it would be. Yeah. I'm using quotation by Mario Patil. To his students, saying, don't ever read books. It will only make you depressed. If you want to know something, just ask me. Just ask me. Who is that, Atiyah? Atiyah. I hope your colleagues will be here.

47:30 We'll ring them and I tell you what, we'll give them, may we give them a call and allow you to speak to them. Can you tell them that yourself? We'll give them a ring now. I'm so sorry you've been put in this position. No, no, no, I'm sorry I have to interrupt you. I think it's fair to say that I discovered a lot of category theory when I started as a PhD student. Was that here in Greece or was that in...? No, I did my degree in Cambridge and then I started my PhD in Austria. So obviously that's how you know Peter and Martin, all these guys. But in Cambridge, yeah, I mean, finishing part three, I was thinking all sorts of possibilities for research. There were many channels and stuff, and even analysis, functional analysis. At the end, I picked, I chose the Paul Aguilar Geographic College, and I think that was right for me. And then I started working with my supervisor on things related to his work. Who is your supervisor? What's your supervisor? Yeah, I know the name, but I don't know... So, we worked together on cobordism categories and the gematopathy of cobordism categories and so it's... There is category theory as a tool, but it's also a geometric method. From how I started, I discovered all these modern categories then. Well, thanks to you and to Andrei's talk on the first day, I now at least know what a model structure is. I was really hesitant about giving this talk because it's not about sheaves or logic necessarily, but... Hardly any of the talks at PSSLs I've been to ever happened. This is about the sixth or seventh time I've been to a peripatetic. And I can't, I think, recall a single talk that was actually about sheaves. One or two that would work. No, no, no, this is fine. In fact, if anything, this particular session has been rather more, you know, traditionally categorial, you know, focused in foundational aspects in category theory than the last couple I went to. No, no, no. No, no, people give, I mean, in fact, our young friend, you know, Richard Krasinski, the guy from Walsall, he could probably well have given a talk on quantum gravity and got over these days.

50:00 You know, that's what a lot of people give talks out about at PSSL. It's very, it's become very freewheeling. Anybody who's good and has got something important to say about their work, it's always been a very open culture, isn't it? So yeah, I split my time in between. I got into algebraic k-theory reasonably lately, and I think this may be the point of unification of a lot of things in my mind. Yeah, well this was what we were making apparent in your closing remarks, which I found very interesting indeed. Subtitles by the Amara.org community I never know what... It's difficult to cram in that much into a half an hour talk. It's tough. Do you think I should try this? Yes, I do. I think it would have helped. I mean, I only speak for myself. And, you know, obviously my level of understanding is much lower than that of anybody else in the room. Totally worthless, but certainly not up to the standard of other people in the audience. It would have helped me, I think, if you had had overheads or slides. I hope you don't take this the wrong way, it's not meant to be personal, but your handwriting is not the most legible in the world. So that was a little bit of a problem, but that's, as I say, minor point. All the best mathematicians have terrible handwriting, it's a standard joke. No, no, no, no, please, I'm sorry, I hope I haven't upset you, I didn't mean to make it as any kind of major criticism. For me it's something I can improve, so I take it conservatively.

52:30 In fact, one of the striking things about this particular session was that... So many people, such a high percentage of people did give chalk and talk talks, well, privative and talk talks, and not use, in fact, apart from Eugenia, I don't think anybody used a PowerPoint, which is, these days, is very unusual. You know, they say all PowerPoints are PowerPoints. The last two peripatetics I've been to, in fact, almost everybody used PowerPoint except for Peter Johnston and maybe one other person. But even Martin uses PowerPoint today often when he gives talks. I'm afraid they've all been corrupted. I think there is something just much more natural to a mathematician about the order being, giving the exposition of the ideas as you are writing down the theorems, and as it does seem to be something that you... Well, what you're saying, when you use a fiber tip or a chalk rather than a projector. Yes, I agree. Well, it simply forces you to expose things in the right order. The trouble is that if people have everything on a PowerPoint and they just call up the screen, unless they're very disciplined in the way they go through each projection, they tend to become... And therefore, more difficult to focus on the exposition in a really carefully controlled way, so you tend to get a bit hand-wavy. You mean in Oxford? In Cambridge, I think. In Cambridge, is there? Of course, I've been out of touch for 30 years. My impression was that in the philosophy department, I didn't think they had anybody apart from Tim Smiley. He must be retired, long retired now. But, of course, I could be quite wrong. I always have the impression that the emphasis is on analytic philosophy and logic. I mean, there, of course, is a theory in philosophy. Well, no, but in Oxford. Oxford has always had a much stronger... Oxford has always had the biggest philosophy department in the country. But it's more comprehensive, right? But Oxford...

55:00 The good thing about Philosophy at Oxford is that it contains easily the biggest concentration of really good class philosophers of physics at the moment that are writing anywhere in the UK, or indeed arguably anywhere, well actually there's also a very good group of Princeton, a very good group of Pittsburgh, but they're certainly one of the three most impressive groups in philosophy and physics in the English speaking world. And the thing is that they tend to know, philosophers of physics have given the standards in their field, which are now very high, I mean they are expected to be able to write, to co-author books with theoretical physicists and to write joint papers with those theoretical physicists. Have you ever heard of a philosopher of mathematics who was invited to co-author a paper with a leading mathematician? I don't think so. The standards in philosophy of physics have been quite revolutionized in the last 30 years, and as a result of that they tend to know far more mathematics than any other group of people in philosophy departments. Yes, exactly, and not to mention things like quantum gravity and string theory. They do have to know, and they tend to be much more interested in... Conceptual issues to do with the... And it's interesting, I think that they are probably the group through whom, it's already happening in France, they are probably the group through whom category theory will begin to get into the broader intellectual community, at least as far as the philosophers are concerned. Yes, I know Chris Isham very well. He's a very interesting guy. I'm not particularly impressed with the, you know, those papers on topos theory, they're pretty, they're what I call spray-on application of topos theory. I think, I don't think they cut very deep to be honest, but I think that was the case of Chris. I do like Chris Eichmann-Normansky, he's a brilliant man. And I think he's one of the best theoretical physicists around, and certainly one of the boldest and the most... Yes, he's very, very interested with the right conceptual questions. But the only criticism, my criticism I make, is that he does sometimes tend to go overboard for whatever is, you know. For him, the latest, shiniest, exciting new bit of mathematical toolkit. Like a little kid who's just been given a new chemistry set to play with.

57:30 And, you know, I've seen him go into libraries and kind of, you know, look along the shelves and, you know, pull down three or four mathematics books and sort of say, Oh, this is new, so I didn't know anything about this. And then, you know, two months later, he'll be giving seminars on this subject and on... All the wonderful applications that he's found for it in physics, and right now topos theory is his favorite new shiny bit of mathematical tool kit, but I don't think he's actually asked himself the deep questions about it. You know, the topos-theoretical, category-theoretical perspective in mathematics itself that, you know, should really perhaps have been asking first. He's, and I think he's slightly oversold the claims with the relevance of topos-theory to physics at the moment, but he's still a very good thing, and I like him enormously, and he's the kind of person that, and of course he's precisely representative of those people who, in physics departments, that now work with philosophy of physics people in areas like quantum gravity. And I think produce lots of very interesting work as a result. This young guy from Poland, this guy Richard Pozinski, I've been tremendously impressed with. He does seem to be genuine. I thought at first he might be just a little, it might just be a cocky little bullshitter, but actually after talking to him for a couple of days, I'm really impressed. He seems to really, really know his stuff. He knows a great deal of mathematics. And not only knows it, but doesn't just simply have a desire to show off. He has thought. He knows deeply about the issue. He knows it because he is genuinely searching for the right conceptual frameworks for the solution to the problems that he sees as important. I think on the whole, from what I've heard, he's absolutely right. He's feeling important. I think he's going to be, yeah, he's just finishing a thesis, but in a physics department. He works on quantum gravity. Yes, he's not a mathematician, but he knows more mathematics than most of the, well, the hell of a lot of mathematicians I've met. Well, he talks with great confidence to Anders about his work on ordinary differential equations, and he certainly seems to know a huge amount of algebraic geometry. I'm not so surprised he knows about n-categories because that's become quite a big bandwagon in quantum gravity in the last few years, but he seems to have a very, very broad and deep, especially for one so young, mathematical culture and feel for deep conceptual issues.

1:00:00 I suspect we're going to hear a lot of him. He also seems to be generally an engaging guy. I think of all of the people I've met at this conference, he's the one I'm... Well, apart from the old friends that I came to meet anyway, like Anders and Gonzalo and Andrei, yeah, he's been, I think, the most interesting, for me, interesting new face. I think we're going to hear a lot from him. Christ, he's submitting this year. He looks as if he's about 16. It's ridiculous. It's absurd. Yeah, he's probably very young. He must be, what, about 22, 23? Well, that's what I'm thinking. Unless he's starting extremely young. He looks terribly young. When you get to my age, everybody under 40 starts to look young, you know. It really does hit you when you realize, it's not when policemen start to usually look young, it's when you realize that you're older than the Prime Minister or the Archbishop of Canterbury. That's when you really start feeling old. And then if you look at that, it's a few years ago, maybe two or three years ago, and if you look at three years ago, it's one year ago. And then we have this paper on motion in special relativity, the law of motion in special relativity. Have you seen that? Yes, I have. It's in the LA archive, isn't it? Yes, it's in the LA archive. Michael, you seem to see a lot of things. Well, that's my job. I'm supposed to keep in touch with all of this and get it all in the archive and make sure that it's all kind of cross-referenced. You told me about it when we were in Boston, anyway, that's why I knew. And the most interesting thing is what we did afterwards, as I told you, that we obtained by just analyzing these things, you know, the Maxwell equation. And then these, I mean... But not all four, but two. I think we obtained maybe two of them, yes. Thank you. Thanks a lot. Thank you. Bye-bye. And thank you for looking after us before they arrived. That was very kind of you. Thank you very much for your time, and I hope to see you again soon.

1:02:30 Well, you should both come to my church. I'd love to. What a wonderful invitation. And there, you see, I mean, you just have to pay your trip. Because then you are welcome to stay in our house. Well, you're very kind. I would love to. And of course the same goes whenever you are Maria in France, come and stay with me. Oh, thank you. But I think Richard wants to ask you a question about this work on Maxwell equations. Oh, you know, I wanted, I think that in any case we will talk about this because for me, you know, these things are just important, but I just didn't want to forget to mention several things. Yes, we can return later. Okay, okay. The question of course is in modern physics the fundamental one. Actually, all people who were really deeply trying to approach the foundations of physics, they have tried to focus on exactly one. The first was Max, the second was Einstein, who actually to some extent, well, he succeeded and he failed at the same time. And, of course, the latter approach was Feynman, yes? You're right. Absolutely. Maybe the first one was Peirce. No, the first one was Peirce. Yeah, because Peirce has this, you know, this thing. Yes, Mach was a little bit later. Well, and the motivation with Mach is much more fully worked out. Feynman, he wanted to deduce the laws of relativity. Einstein from just a graviton, and also changes in the graviton, and he said this is cake, you know, to do that. And then he had a seminar and he worked for one year and he was getting mad, mad at his time, and he never did that. But it was published in fact. But no, no, no, it was not the gravity, it was, I mean, this was published, what is published is about the electrodynamics, yes? He tried to do this with electrodynamics. No, but have you seen, no, but have you seen the title, Gravitational Equations? Oh, yes, I know what you mean, yes. He published whatever he did, but he didn't finish. But the actual, you know, the question of... I think that one of the biggest jokes is that one of the main contributions to Feynman's approach in the last, well, probably half a century, no, half a century now, but 25 years, was actually made by you.

1:05:00 You have shown how to precisely write the functional integration, which is totally, I mean... You mean in our book with Nicky, yeah. You know, this is actually the only formulation which is not obscure, which is, at least from the categorical and also differential perspective, precise. I'm happy, because I never met anybody who said anything about that, and I like that. In my present work, I am using this very concretely, and there is a problem. Still about relation with the, I mean with the embedding, if I can go into the test, for me it's still the question about the embedding of continuous vector spaces calculus into the topology, because you have shown only the embedding into Cayet, and Cayet has to know, has, we can structure. Embedding of what? I mean, this embedding, I mean, it has some properties, but I still do not know if those properties are... Present, when we move from Cayet to GES, I mean, there is an embedding, but I don't know if the properties of embedding, of this embedding, really hold all the structures of convenient virtual space. Is that the idea? That's the plan, is it? Okay, fine, good. But you know that I, with embedding, I mean... Yeah, we're walking, we're walking, that's fine. We'll take the bike. Well, that's okay, that's very kind. You sure? That's fine, thanks a lot. Is it far? No, I don't think so. This is made together with Anders, yes? No, no, no. Another paper, written by me alone, which I showed that manifolds with boundaries can be embedded. Oh, yes, this is quite a great paper. You have seen that?

1:07:30 Yes, this is one of the pamphlets. This is a very recent one that appeared in the... Archives? No, no, no, in the Cahiers. Oh, so I haven't read it. I have some paper which was about embedding the manifolds with Bandler, which was present something about a year ago on your website. Oh, yeah, well, that's a thing. That's a thing, because I... It's a piece of good work, actually. I mean... But you see, what I didn't like about that is, again, that all what I could do was only to get foreclosed ideas. So then... Yes, exactly, exactly. Once we have the topos F, sorry, sorry. That's right, but the topos F and other topos which are similar but they have to have the idea to be close. And only for those I was able to prove the result, you see, of embedding. So my impression is that unless I'm mistaken and I did something, I didn't see what I should have done, but my impression is that... This depends very, very strongly on the nation of the topos. I don't know whether, for instance, we could do something better than what we did with Anert, you know, embedding in the higher topos or not, you see? This is one of the most important questions for me, actually. Yeah, and this is the same thing that happened with this existence of... So, this is the uniqueness of the differential equation because we apparently, if this is correct what we did, I mean we ran it the other day, but for the others that's not the idea. Yes, because for others, I mean, you know, my analysis I have done and I have tried to catch the point about the uniqueness. I think that jam representability is sort of crucial. It would be very nice if you tried to work the mathematics out. That would be very interesting and unexpected. You see that you could do something in a big topos that you could not do in a smaller topos. But you think that, oh, you consider the F as bigger. Yes, well, it is bigger, but... No, the barrel topos which have this, as you say, you have...

1:10:00 What is the name of this? Invertible. Invertible. This is what you were talking to me about this morning, isn't it? Well, yes. I think it was this morning or yesterday. I don't remember. You were talking about it with Anders yesterday, but then this morning I asked you about the question about the invertibility and the vial algebra. And this is what you were telling me about. And then... We have the activity, and this is a very big topic, and very few things can be true about it. You mean about buzzers? Yeah, and the buzzers. Although it seems to me that we shall not be mistaken, we have the theorem, the chancellor's theorem. Yes, yes, of course. Well, it's not a theorem, it's a chapter. It's more than a theorem. Yeah, but it was what? I mean, you know, I don't remember very well the thing, but whatever you could see... In the stronger analysis, I mean stronger arithmetic, you can translate it into weaker, with respect, yes? Yeah, it was more coherent. Coherent propositions, which are true. In the G-couples, there they are also true. Is that even the one in the backpack? No, no, no. I think that there was some more assumption, I mean... But actually I have to really say that the chapter about transfer principle was most hard for me always. I mean, I have never gained, you know, the real insight, intuitive insight, which sentences, you know, are translatable or not. I mean, that the... But probably it's because I haven't, you know, really done any original work. It's a nice view, isn't it? You know that you have to say that you had a lot of trouble and reading that book. Yes. For me, it has been horrible. Yeah. Of course, because I haven't done any mathematical work, as I said before, in ten years. And then I have to start looking at that horrible book, you know, which is not written for, you know, for...

1:12:30 In a simple way which everybody could understand, I was written, you know, for you. So which book are we talking about? Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Murdoch. Who wrote this stuff? Can't he make himself clear to anybody? There is too much stuff in that book. There is an awful lot. It is very heavy. It's heavy going, I agree. In some way we're not fine at all, you know. Because the trouble with that book is that... I think this is too good. Yeah, I mean, it's too brilliant. So then, you would have all these ideas, you know, and then, you know, financial skills. They are divided into two brackets, trivial and obvious. True, true. Let me explain. To take an example, you see that financials is a baffled topic. It was probable because, you know, it is a site. Well, the site is not difficult to understand. But the difficult things to understand were the embeddings. Because the embeddings...

1:15:00 In principle, you have to invest only in finite things. But then, in fact, it invests things which are infinite, but the topology needs to be finite, and then he uses the theorem of Ostrand on dimension theory. So why don't you publish this topos, whatever it has to do with this topos, on your own. And so then we will just refer to that. And then he did so. Really? You mean the... There is a filter called... On the topos, we've been working on the Buster topos, and so on. You can see it in our book. Yes, yes, it is. It is there, you see? Actually, you have also published, and I was always wondering why you have not included the second part of rings of small functions and localizations into the book. I mean, you know, when you consider the spectral theory of infinity rings in all... Probably we decided that that was too specialized and maybe this as well and this anyway can be found. And I always, well I didn't regret that, but I regretted that there was not a single, you know, one place where the two papers could be together. Because, and even I thought at one point, just playing with the idea of publishing, you know, just questions of localization in a small book, but then, you know, as Makai said, you should come. Or students. As long as you don't ask them to learn synthetic differential geometry from that book, then you may not have so many students. Well, I have almost... Certainly one or two very, very good ones indeed, but... Are you right? Yeah, you are one of the very few who have learned that.

1:17:30 Well, talking about Icke and Wurteich, it used to be said that the only person who had ever learned topos theory from reading Peter Johnston's first book. The only person who has ever learned synthetic differential geometry from reading Murdoch and Reyes is Richard. Is this where we're going? Or a little bit further on? No, it's further on, apparently. A very close group, which is, you know, making some esoteric or mysticism. You have to passage through the rites, yes? The Eleusinian Mysteries. Like Pythagoras. Yes, exactly, like Pythagoras or the Eleusinian Mysteries. What is really striking is that Marek once said to me that, you know, Reading and understanding one chapter of the first book of Johnston is better than understanding the old book of Goldblatt. Oh, yes, yes, agreed. And when somebody, you know, passes the old book of Johnston, he is something like, you know, an insider. Yeah. Well, Corley MacClarty has put it very well. This is the book that you have to read and understand if you want to claim to know Topos. This is the gold standard. Magna Mara put it up in a very nice way. She said, I would say, you know, John, here is a paper. And then she said, is that for Christians or for lions? Is that for the Christians or for the lions? Yes. And so when, you know, when he finally learned, you know, to take a joint and so on, Marie gave him the lion. Ah, brilliant. I wish I'd met him, I wish I'd known him, but I'm not going to let you go without giving me just a little bit of time on that work that you did with him about the analytic metaphysics, the business about entity, the notion of an entity. Very good. Oh, can we go? Yeah, well, except we want Marie to be with us. I'm speaking about the problem of force and the problem of movement, which is, of course, part and aspect of the problem of motion we were talking about last night, yes.

1:20:00 So, the problem is that after the virus grows, differentiation is not related in any case to movement. So, actually, those two issues became... Sorry. I'm sorry, get that out of the way. I was just keeping that for you. We cannot speak about force, but we cannot speak about movement too, but it's too much. I've tried to join several nonlinear lines of thought in class, but I get confused. I think that there are two different approaches to science as a resolution of the question of course. First, the building theory is... From the regimes where there is no absolute space, and they consider just relative oceans as a basic concept. The Machian program of GRS, you know, danced by Barber and people like that. The basic structure is appropriate. Yes, yes. The relative oceans are complete. Oh, sorry, do we need more space? We can compress them, please. Sure! Yes, we can compress. No, they're going to add a table at the end, I think. Yes, and it is obviously represented today by a position of people like Barbara and Bertotti. The position of what? Barbara and Bertotti. There might be another type of resolution which is based on the idea that making a difference between existence and being and becoming.

1:22:30 When moving from laws of calculations to mathematical physical realisation, I would say that on the level of death, we have the substance... It has partial differential equations in its, you know, existence outside of the world, yes, and it is something like, it is the core of all possible solutions, yes. But when we consider and grade them, yes, we move to the wider space of jam, I mean, the interdisciplinarity of jam, and then the space moves to something that is only proposed on their potentiality, in the very sense of Aristotle, becomes actuality, becomes actualized. I'd love something. Thank you very much for your attention and I look forward to seeing you in the next lecture. We can understand it far better, yes? So, I would say that... In context of topos models of physics and geometry and analysis, germ representability introduces very flavor of observability and is very highly, in my opinion, connected with the question of if properties can be observed.

1:25:00 To some extent... When we deal with things which are only infinitesimal, and when I say infinitesimal, I mean new potentials, not only invertible. When we come only with new potentials, we are still on the level of potential. And only when we are potentially on the other side of what is called in Aristotle's terminology, a killer. Oily, Oily, Oily, Oily, it's pronounced Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, Oily, O It really has the connotation of matter in its pure potentiality as a receptacle of form, but without having any determinate form to begin with, just in its pure potentiality. It seems to share starters and main courses for everybody to share. Correct. I mean, that's one option. The other option is for everyone, it's not me. Means we'll be here till midnight. No, no, no. Share, share, share, share. The other option in this party, I would say, let's share. Eventually we'll be here till midnight. I mean, it's really crazy.

1:27:30 Anyway, why don't you finish that, what you are saying, and then we... Yes, I would just say one thought. It's very connected with the question of localization of couples in terms of setting of which two categories. I mean... Sorry, certain... Localization of the topo. Localization. Yes, localization. Because it's somehow that localization, especially using the subcanonical topologies, as you showed me, the subcanonical topologies, they are better to deal with, yes? So did you say it brings what to the purely algebraic part of... I just didn't quite catch what you were saying. Something like a concrete appearance. Okay, right. Sorry, I just didn't catch the words, that was all. I mean, in my opinion, it's something that there are, you know, before the act of observation, there are certain geometrical structures which are only potential, yes? But in the act of observation, they become localized in respect to something like, you know, existential appearance. Yes, but... Prior to this, they have some kind of potential being, is the broad picture. This is very suggestive. I mean, this seems to me to connect with some very, very general level of connection of suggestive ideas, with some of the ideas that are there in Bowman-Hiley in their book.

1:30:00 Do you know about the undivided universe? Well, they talk quite a lot about potential... Who is that? Bohm, David Bohm, the physicist and highly his collaborator, the algebraist in London. I mean, in a very general level of, you know, kind of convergent lines of structural intelligibility of the underlying picture. This is the way it moves, yes, and this idea for me was not correct. Oh, yeah, no, but that was in Bohm's very early 1952 theory. He abandoned all of that for a much more, I think, much more sophisticated picture. Oh, yes, yes, it's astonishing. One of the things that David Bohm always used to say was, I wish people would not call my theory Bohmian mechanics. You know, it makes me so angry when they call my theory Bohmian mechanics, even the original 1952 theory. ...which he had long abandoned as being kind of hopelessly naïve. The whole point about it was that it was intended to be an anti-mechanist picture. It was absolutely not based on the idea that mechanical principles could be fundamental. There was always this strong dialectical idea in the background. The red wine we had yesterday, very well, I'm quite happy to stick with it. I think we've got red wine on. Me too, me too. How about Anders? I think we're all reds. We're all good reds at this table. Both in the drinking and in the political sense. Both as drinkers and politically speaking, we're all good reds. There are no whites at this table. No, no, well... But I mean the invertible one, yes? Yeah, but the invertible one is bigger. But even in the smaller one, you see that something is flat. You see, in an infinitesimal, then it should be flat in even a smaller one, right? So then I show that even in the smaller one that it falls.

1:32:30 But I found that one formulation of the principle that came to me that has no problem whatsoever is that... At a given point, not at any moment, but at a point, the question is correct. And then I was able to prove something that appeared in the book of, what is the name of this guy, Weinberg, that is completely wrong. Thank you for your attention. Famously, he wrote the book on relativity from a totally anti-geometrical viewpoint. But the book is based on... Relativity is a gauge theory. ...and leads to false results. I'm glad to hear it. The theory of gravity as a gauge theory, which is based on a mythosic background, is... But look, in the case of Feynman... But which is the equation that you were talking about that you showed that only holds exactly at a point in SPG? Sorry, I wasn't hearing what you were saying earlier. He wants to obtain, I mean, what is his name? He wants to show, you see, that this equation that I just mentioned, this populate number two, which says, in my lecture, that says that the free-falling part... Now he proves that from the principle of equivalence. The principle of equivalence is incorrect because it corresponds to say that a ball... You know, locally, it's a plane. And you know that it falls not even locally, not even in a small thing, you know, it's a plane. But then I found one way in which you can make that, but instead of looking at...

1:35:00 I mean, locally, you say that at one point, it satisfies the equation of a point. And from there, I will say, and this seems to me a completely unobjectionable, and from that, I deduce the result that he wanted to deduce. But you see, since he wanted to write something completely out of the geometry, he said, you know, I will try to eliminate geometry, because this is the only theory that uses geometry. And then put it apart from all the other theories on physics. I took that seriously. I said that is very interesting. And then I see how he defines, for instance, the different notions. And then I discovered that he has one more variable than what you would need. They are four, but you have to show that this is independent of that variable, you see? Yeah, which he doesn't. He doesn't do it. He doesn't do it, yeah. And then I try to do that. This is in his book on gravitation. The book on theory of gravitation, yes. And he cannot do that, and I tried to see whether I could do it, and I realized that I could do it only if I introduced geometry. Yes, yes, yes. So then, you know, starting then from this area. From this parallel transport, then I was able to eliminate, to show that what he had done was independent of that product. Of that variable, yes. But then he believes that he was able to get rid... Of geometry completely, yes, I know. Well, that was the motivation of the book. You see, in the discussion after our talk, L'Oreal said there is not too much difference between a metric and a transport. Because if we define the metric at a given point, and then if we have a transport parallel, then we can transport the metric. This is of course completely correct. But then the point is that Weinberg defines the metric at a given point, and without the transport, he just pushes the metric all over here. You see? And then this is crazy! Because there is nothing that changes all that's really there. It's crazy! Talking about introducing new variables and stuff, and he loses sight of the whole of the original motivation of gauge theory, too.

1:37:30 Right, he believes that he can get rid of the geometry, and that is crazy. It's wrong that you can get rid of the geometry when you start doing weird things like in quantum mechanics, in which you start interacting things in the process of... Yes, but, you know, all this approach is based on misunderstanding. I mean, those terms which they use, yes, they know where to find mathematical meaning, that this is the problem. I mean, they use something like a mental... Yeah, but one of them has mathematical meaning that will be found in some... Yes, on the initial variables, yeah. Yes, on the initial day it was a logger, yeah. But as they interviewed this guy, you know, it was really an incredible surprise that somebody like Weinberg, you know, Well, maybe people, I guess, I have no explanation to offer. He does, as you say, have a huge reputation as being a... An extraordinary reputation of being able to take the place of... You know, in the quantum physics community, I think there should be... It seems that after the 60s, this community became really more and more somehow closed. The expression we're looking for is up its own arse, to put it bluntly. They just became fixed to certain calculation techniques, which drove them into the so-called success. That's what is striking. I've been listening to, I don't know, half a year ago or something like this, one of the Nobel Prizes of the future. He came to Poland and he was giving a talk.

1:40:00 And actually, he said everything what was written in the talk of Weinberg, which I have read, and this talk was given 30 years ago. So, I mean, they just have something like a mental glitch. You know, particle physics, explain physics, and we have particles, and whatever. But when you, for example, read, you see that this is completely awkward. You have manipulated the mystical symbols, which have no precise meaning, in order to obtain, you know, the results. Wow! I thought we finished! I have only just begun to fight! No, no, we have our own. Do you want to pass that down? Impression as an outsider, just sociologically.

1:42:30 Particle physics. And of course it was also bound up with the fact that it was the biggest of big physics and therefore of course receiving huge amounts of, relative to any other area of intellectual inquiry, a disproportionate amount of funding and resources, which I think is something which always tends to drive down standards and to make the cleats. The sad thing is that people who are really deep thinkers in physics, like Chris Isham, are of course never going to be in the running for a Nobel. Weinberg is a strange piece of work. I think cosmology in many ways is in the same shape. Yes, yes, yes. You know, to divide the things which are dependent on observers, and to divide the things which are dealing properties on the observers. Absolutely. And that's why, when you observe something on the sky, especially radiation, you never really forget it. When you just nail the trick it, that you can recover... You mean, what the thing was? I wish you the pleasure of working with me.

1:45:00 Sure. Absolutely. Interesting. Statesville. Statesville has written quite a good deal about this, you know, who you met in Boston. Mmm, that's a good one. What about Anders? Come on, you give it to Anders first. No, I'm sorry. How about you guys? It's a little bit tender than splintering. You see, because they say the principle of equivalence says this thing that it's completely wrong, you know, so then, for instance, in... When you talk about the principle of equivalence, in what formulation? In the... In the formulation of the... Of the kind of falling elevator frame? The falling elevator. The falling elevator, OK. The elevator and the formulation of that, which is Einstein and so on, and it says that essentially that, you know, that locally...

1:47:30 Equivalence of inertial frames, yeah. And so then, for instance, you know, this guy, what is his name, Van Gogh, then he says, well, because of that, let's take something that is gone, you know, without really falling. It should satisfy, you know, that during a small time, at least, the equation of the second derivative of the acceleration is zero, but of course then this cannot be realized, because even in a small time, you will never get a plane, I mean, you always have a problem. Of course, of course. Only in the, well not even in the infinitesimal, not even in the first neighborhood, only at a point itself, yeah. Precisely. So then what I do is that I say in this point itself. So we actually have to take back one thing that Grotendieck said, sometimes points are merely points, if they have to be, in fact sometimes points have to be just merely points, otherwise you wouldn't know. Indistinguishability, indistinguishability. Well, it is my native language.

1:50:00 They've already got one up there. No, they seem to be okay. We were just wondering if you wanted any more of our, because they bought us this huge, yes sir, we're never even going to make an impression on it, I'm afraid, okay. This is one of the different questions of mathematical physics. So this is a question to what extent we can... Say that again? Sorry. I mean, this is the question. To what extent we can think about particles and so on as non-linear solutions? Ah, yes. Which of course was Einstein's program.

1:52:30 Well, that was the whole program of Einstein's final unified field, the thing he worked on in the last 30 years. It's quite interesting that those disciplines of knowledge fits directly to the training we see. Yes, yes. Our perspective is just a... We need a general restability in order to cope with it. It's somehow intimately related with the production of a symplectic structure. That's what I want to understand, what the connection with symplectic... Well, it has, of course, been in quantum physics. It has been in... No, no, no, not in SGD. It has, of course, become very important in Spanish. Because without understanding things like symplectic capacity, we've not got there.

1:55:00 I'm sorry, can I just ask one question? How would the development of an account of Simplecto... I'm sure there's a very I'm sure there's a very deep answer to it I just love to try and understand it better okay well we can maybe talk a bit more when we get back The work is on the right thing, so I'm chomping at all this stuff, so... But there is a whole set of people about it. People of the old, it always surprises me, sociologically, that nobody has gone back and looked at that work, that late work of Einstein on, you know, the late work of Einstein on nonlinear solutions of, you know, trying to develop a unified field theory, but directly with... But he was operating with the tools of differential geometry, you know, of the 19... Right, that was the non-symmetric, that was the string, the later work, that was, you know, a few months before his death. Yes, yes, exactly. It was a revival to some extent of what Hilbert and von Lauer and Mies and people had done in the 19th at the same time. He was working on, because they were trying to develop a unified field theory of electric magnetism and gravitation. Oh, Cartan, yeah. Yeah, because apparently when Einstein went to France in 21, 22 years ago, he had the long correspondence with Cartan.

1:57:30 Yeah, at the point he had a discussion with Cartan, you know, and he didn't understand a single word of what Cartan was saying. And then Cartan said, you know... Very few mathematicians would have done that, let alone physicists. Absolutely not. I don't think I... Einstein can be blamed too much for that. Even, you know, the famous Hermann Weyl. Yeah, even Weyl found difficulty in understanding Cartan. Cartan certainly is the greatest geometer of our days, but, I mean, it's very difficult. Yes, exactly. You know, I've read a book, I did a collection of works of Cartan, actually it was edited by one of my teachers, Krautmann. And in his collection of the works, for example, he analyzes the notion of Huffing space and Huffing connection, and it's, you know, it's just, you know, when you read his works, I mean, this is the same what appears in some moments in the Andres book, yeah, you just can take a citation and write, you know, the delineation notions, I mean. His description of Atiyah's space and Atiyah's connections is exactly this, you know, but using the strong differences for relation and definition of all of them. Yeah, the trouble is that he never said exactly what, you know, what he's doing, because it's obvious, it's obvious to people. Yes, because sometimes it's genius, it's genius. It's the same problem with Cartan, of course. So then we, at the beginning in our book, you know, we do an exercise like that. We try to put exactly the words of Cartan for what he's doing and what he's doing, you know, synthetic differential geometry. He has a lot of work. For instance, he's the first person, I think, to present Newton's theory as a gravitation, right, as a relativistic gravitational theory.

2:00:00 And it's easy, it's easy to play, at least the one I play. So from that perspective, as you say, it's applied by SDG. It's an absolutely amazing work. You're the man to do it, of course. You should be doing that, because I have also another project to try to understand what Gödel did. What, the Gödel solutions, you mean? Gödel solutions, yeah. What you have said now, because I haven't heard the last sentence. No, I mean this question of trying to understand what Platon and Einstein did. And this is something that seems to me that is the next step of what I was doing now and maybe, and this is something that, for instance, something like you could look at us and try to understand what he was doing and then, you know, put that into this general context of getting a picture of, of, of, you know, like the way, you know, of people. But what was the connection with the Gödel solutions? Oh, no, it's just that I'm very interested in getting a solution. And then I say, well, I don't understand. But at the time being, how to tie these two things, what I did and the general solution. Well, is there a connection, are you thinking that there's, I'm sorry, it's a pretty crude idea, but are you thinking there's some possible connection with the Einstein-Cartan theory because of the torsion, you know, the torsion might be an explanation of the rotation. No, I just wanted to learn it, and then I think that I have now the tools. What is interesting in the non-symmetric matrix is that it actually encodes, you can use it in order to encode the notion of Machian relationality. I mean, if you have a non-symmetric matrix, you can encode in it the information about all relative velocities. And this is the difference. I mean, the problem with the original Atiyah formulation was why it has not realized the Machian idea, yes?

2:02:30 Because the metric is symmetric, so you cannot use metric to encode in gravitation all relative motions. Because the relative velocities are in totally most cases are anisotropic, yes? And doing it via a generalization of the generalization of the connection in the Einstein-Cartan setting would I think conceptually be a much more Potentially much, much more interesting, deeper way of doing it than the way that people have tried to take account of anisotropy, which is why, for instance, Finsler geometry. Well, I mean, there's this group of people in Russia that have these conferences every year on Finsler geometry, together with the Romanian school that they invite me to. On the other hand, the idea of generalizing the metric from a quadratic to a general polynomial form does give you an incredible richness of structure and obviously automatically gives you anisotropic models and topologically interesting models as well. It would be really very interesting, and actually it was one of the things that I wanted to do somehow, to take care about this, to think about physics and geometry from the synthetic pressure perspective. That would be very interesting. Because when you can write in decimals, yes, you can deal with this... I mean, you can think to some extent, well, infinitesimally, while the ordinary physical geometry is somehow, you know, very merged with the topological issues, and they have to cope, you know, with some problems which seem not essential from a geometrical perspective. That's right, they seem to be very much more to do with, well, actually with, well... This is the number field that you're using actually.