Am discussions: FW Lawvere, C McLarty, L Corry, A MacIntyre, JL Bell

0:00 In this case, it was Birkhofer, yes. I mean, and the prices are really not as outrageous as Kluwer. No, but Kluwer is everything. He's Springer... That's precisely why the prices are so outrageous. It's called Monopoly capitalism already. That's the whole point. I have a friend, an economist in Tel Aviv, he has a book with glue and in his website he says, don't buy this book. I put my glue on my webpage. Good, good. The price is just illegal. It's quite a good kind of strategy they're using. They know that they don't get overwhelmed by people publishing those electronically very soon, they just want to make as much, and that they can rely on institutional budgets, they don't give a damn about people out there who might actually want to read these books. Learn this stuff. Because there's no doubt at all that the same field can be published for a fraction of the cost. I mean, look at what Polytechnica are doing. Look at what KCL are doing. They're publishing the books which involve just the same demands in terms of typesetting and proofreading. They're not much better done on the whole. ...for literally one-tenth of the price. They even wanted that, but I said, no, you don't want that. Okay, the quarters are now running, so let's turn to business. It's the 15th of June, 2005, and this is the AM session of our discussion. We've just been having a very freewheeling exchange over breakfast. There are quite a variety of topics, but all those topics could be broadly said to fit within the framework of a discussion of what form should a general account of structure take and how does it connect together with the idea of a foundation for mathematics and what a foundation or perhaps better foundations for mathematics should.

2:30 One topic which was raised just at the end of yesterday's discussion was whether we are closer to a unified conceptual framework of mathematics than we were approximately a century ago in the time of Hilbert. And I think it would perhaps be useful to discuss that question and particularly some remarks that Colin made about the role of algebraic geometry and homological algebra in providing tools for general theory of structure and to look at the philosophical ramifications of that situation by comparison with the one a hundred years ago when philosophically mighty mathematicians We saw set theory perhaps as the principal point of connection between the concerns of mathematics and those of philosophers, to the extent that set theory was seen as providing a foundation of mathematics or a framework, its significance to the philosophers seemed to be in terms of providing ontology for mathematical structures, and today the connection between the conceptual organisation of mathematics and the concern of the philosophers is perhaps much less clear. Perhaps draw closer towards that, towards the end of the discussion, and particularly perhaps more from Bill on the neutrality of variation of structure, a later increased awareness of the... Okay, there's all a big problem. You'd like to kick off, Colin, on taking up where you left off yesterday? Well, actually, I... On that theme, I was thinking, one of the things, maybe it's useful to approach it, so one of the things that you understated in your description of your book, just over here, was that you have this interplay through the whole thing between uses of structural ideas, I forget your exact terminology, and images of structure, between what's going on and theories of what's going on. Yeah, exactly. Body and images. Yeah, yeah, and could you say more about that? Yeah. There was a thing, is...

5:00 There is this possibility of looking at something that happens in mathematical theories and provide a formal theory of that. That's more or less what happens with category theory. It's a theory about mathematical theories, also model theory in some senses. Let's say foundations. Foundations sometimes have been confused, or perhaps not confused, put together with the question of all kinds of reflexive mathematical theories. Notice that this is something that you cannot do, for example, in physics. There is no physical theory of physical theories, right? You have a theory of heat, you have a theory of matter or energy or whatever, but you don't have a theory of what is a physical theory. That would be outside the scope of physical science. Whereas in mathematics, you have, many times, in many ways, mathematics dealing with what is a mathematical theory, or parts of it, for example, truth theory, recursion theory, or whatever, deals with aspects of mathematics. And therefore, sometimes, a philosophy of mathematics, or foundations of mathematics, or any attempt to say something serious about mathematics... The only way to do that has been taken to be via a theory of mathematics or mathematical theory, which not always should be the case, I think, and I think that in looking at what is a theory, if we remember all the things we mentioned, what Stuart Shapiro says and with other people, try to give some kind of theory of what is a mathematical theory, and what I said about Van der Werden, for example, was not... I was saying that... Van der Werden, in his book, presents a certain picture of mathematics or a mathematical theory. That's what I call an image of mathematics, like saying, look, if you want to... To know what is algebra, and perhaps even more than that, this is what it is. It's not just that I'm telling you there is a theorem that says that every subgroup of a group, the order of the subgroup divides the group, but also look for this kind of theorems.

7:30 Look for this presentation of algebra where, as I said before, you start with the idea of sets and you start adding things to this set and you start... Looking for building blocks and interconnections between building blocks, etc, etc. And when I said that Bourbaki broadens this view for other places or other mathematical domains, I meant At the level of the image, not at the level of the content, in the sense of saying, this is how you did algebra, so let's do something similar for topology. We are not using the same concepts, perhaps by chance, yes, we are not using yet, for example, a theory or a category or something like that, but we want to use the same view. Start from a set, add to it certain things, and then you have a topological space. And then within the topological space you can look at the At the subspaces what property for example if you have a separation property in the space when this separation is inherited by the subspace or if you take the space and you build something on the space for example a compactification or what properties and how the properties go so this when you do this in topology you are in a sense following the paradigm or the image. That Van der Werden consolidated for algebra. So I, of course, I'm not saying this is not an invention of Van der Werden. He just put together many threads that start from Dedeckin, from other people, and then Emanator does it very strongly. But by being a textbook, this is again the thing. Sometimes you take a textbook as a... You can see in textbooks things that you don't see in articles, because the textbook is... Presenting mature knowledge and things that already you have had some time to put together to organize and then you see a certain view, that's what I call an image, a certain view of the domain and then you can move this more or less this image to other domains, perhaps have a general image of how the entire discipline of mathematics should look like. For example, if you look at Burbaki That you don't put inside applied mathematics. Applied mathematics is not part of this. Why?

10:00 Because it doesn't really feel... How do... What do you do in order to to take applied mathematics in some way and put it as part of this structural view? I mean, I think it was part of the ideology to make some kind of separation. So Weil didn't care for physics very much? Weil. He didn't? I mean, there are many reasons why this is so, but it is a clear image. Noel, for example, who studied with Burbach, Dr. Noel in the 1950s, he almost immediately took up precisely the program of… And there is a famous book by Liknerovic on general relativity. This book is incredible because it's very hard to see where the relativity is. It's written in the Burlake style and, as you say, trying to extend. But then what I said, and that's the point where we stopped, or we didn't develop it enough, is here you have this image of mathematics, and now some mathematician can come and say, okay, can we now formulate a theory that's... Tells us exactly what is this in mathematical terms. And this is one of the ways I saw that there was room for a theory at the beginning, like category theory, in the sense that gives... I'm speaking about the early papers of Eilenberg and McLean. And I'm not even speaking about a relation of causality. Here you have this and here you have this. But I'm speaking about... A certain atmosphere where this thing could pop up. For example, how do you move between one domain and the other? You start to see that if you have certain questions in algebra that you are not able to answer them, then you move to topology via homotopy or homology. So is there any... More general framework, mathematical framework, where you can say something about how these relations between theories work. And category theory, I think, at the beginning had this ability to say something about this, at least to formulate in precise terms what was going on. And therefore, this was my surprise when I started to look at the Koumbaki series. And I ask myself, how come these people did not immediately say, this is what we need?

12:30 You know, here, this is, if we haven't written the first chapter and we wanted to write the first chapter here, we have it. But it was a little bit late because they had already written the... The chapter on structures. And so, as far as I know, started some internal discussion in Burbaki saying we should take it, and then people like Serre and Grotendieck start to use it in their own work. So it's a very strange situation because in the collective work they don't adopt the view or the language and even try to develop it, but in their own individual work they are using it and they are using it in a very strong way. Perhaps not developing category theory as such, but at least using very strongly the language to develop their own ideas. I don't know where to put the borderline between the two things. Cartier described an actual treaty whereby it was agreed that Rotendijk and friends would write EGA and on the other hand Bourbaki would continue with the... And so on and so forth. Grotebeek's writings, of course, did use category theory quite systematically, whereas theirs did not. I think this is a mistake, that is to say, to call the global and local the main distinction underlying, you know, categories and no categories at this point. At least at the time, this was a definite, not a debate, but it became a very definite position. Well, what you're talking about, that division starts to break down precisely over flatness. Because flatness is a functorial property, which is in Commutative Algebra, in Borbach. They did do flatness. They didn't do other localization, but they did flatness. No, Commutative Algebra is the most interesting book of Borbach in this sense because it was written relatively late. And you would expect the categorical language to be used there, and it is not. So what do they do? They use all kinds of projections of modules, you know, in an older way, and they work very hard. They spend a lot of effort saying things that could be said very quickly.

15:00 When I was looking at that, that was... And then they have the homological algebra, which comes out in 82's images. Really late. No categories. So you are doing homological algebra in the 80's with no categories. It's almost perverse, you know. Exactly, exactly. So that's the point where you see that there is an ideological clash and people, you know, they are committed to a certain view. They have written about this view all the time. And they cannot say, sorry, we were wrong, and they... But it may not only be that. I mean, to be fair, I think the effort of actually rewriting, going back to stuff that... Maybe they agree, well, you know, it isn't... It's maybe they've already wrecked the building. I mean, then actually they wouldn't have to have done a lot of work, right? Maybe the publisher wouldn't want to do the whole back... Let's have, you know, the whole... Let's have all the mathematics... But then, you know... It would be so obese. Then I would expect someone to come forward and say, look, the book is there, it is good for whatever it is worth, it can be used as a textbook for algebra, for whatever. There is a problem. I mean, we came down to write homological algebra. You write homological algebra without serious, without exact nuances. Maybe they didn't want to admit it. I think they didn't want to admit it. That's what my impression, I think. I know only one mathematician who actually had a beautiful house, probably, you know, cost a couple of million as it was. His wife didn't like it. So they tore it down totally to the ground, moved across the street, and had a completely new one. This particular man is reputed to be worth 200 million dollars, because he was one of the founders of Sun Microsystems. Completely to the ground. But you need resources for that. Exactly. There's a material aspect to it. Exactly. If you don't have those resources... No, but look, I think, again, if I go back to that, you know, you may like or dislike, let's say, Volvaki's books on topology. But a lot of people use it. And if you take it stand-alone, you know, it's a textbook that you can use.

17:30 Even if you come and say, look, forget the entire program, forget the architecture, we wrote these programmatic things, the books are there and you can use it if you like it. That would be fair enough in my view. But there was this thing of the structure, and I think that it has also to do, you know, I have tried to understand all these philosophers, like Michael Resnick, and I mean, perhaps if I tried harder, I could understand them, I don't know. Don't bother. Don't bother, perhaps. Because, you know, I see so clearly the idea of structure. This is a way of doing mathematics even you don't know yet what is a category and you can again you can even remain ignorant of that and you have here a way of presenting algebra a way of doing mathematics with advantages and possibly I think with limitations because today algebraists are doing other things etc. But this is what I see as the structural view of mathematics and you don't need for this. Any particular philosophy or ontology. You have one important thing I think is that numbers They all become less important. They are derivatives. They are not, let's say, foundational. They are not the main thing you have to know in order to build algebra. You know it's an ordered field, you know, an ordered real field. Okay, that's a good definition that starts from a set and has some properties. And then you know what the real numbers are, at least from the algebraic point of view. So I think that's good enough, I don't know, of a way of doing mathematics. Between 1930 and 1980 algebraists did mathematics, I mean the mainstream of algebra did algebra in that way. I mean thinking of that kind of questions, for example if you look at the simple group theorem. I mean, it's a little bit ugly in the sense that it's not really architecture. You would like something more beautiful, you know. It's a little bit ad hoc. Either this or that or that. But here you are investigating the structure in that sense of all these things. You want to know... whether if you go in this direction you get groups of this kind whether it's a it's

20:00 It's really the structure of the category of structure. Well, now if you use the... No, I mean really you're looking at the... The pipe structure I would say in this case, but it's a way of doing things and now I think I'm not sure about this But I think from the 80s or so the algebra is like set. I know this in algebraic geometry happened Okay, we know enough of this stuff. We now really understand how the rings are built and what is an etherean ring and an artesian ring and how these separation properties are formed. Now let's do some calculations again that we haven't done and now we have computers. I think that some algebras consider that phase of mathematics to be a little bit... Behind it, in some sense, in the practical sense, I'm not talking in the foundational sense of trying to understand what is that, but we did algebra structurally. So you're thinking of like David Eisenberg. We're really going to compute Grobner matrices. Yes, like Doron Zylberg. He said, I know enough of these polynomial properties. I even myself proved many important theorems of structure of polynomials of this and that kind. And now it's time to move to use this. I mean, we are not going to throw this. On the contrary, this stays as a support for everything we're going to do. And I also heard this guy, this Indian guy was a student of Zaritsky. What's his name? Abhyanka. Abhyanka. But he, of course, is very specific. Right. He's doing calculations. He has always advocated. I mean, he's always been countering. And so on and so forth. In the end of these, when he published them, he was a major contributor to the resolution of singularities, but essentially by bare-hands methods, however, and he strongly advocated this from the very beginning, that these... Which are special cases, right? Not general theories. No, well, I mean, he did the... one has not ever gone much beyond what he did in characteristic P, in low dimension. Exactly. He's never gone anywhere. I mean, he advocates just very clever use of...

22:30 Direct, almost, I think he probably uses the phrase, high school algebraic. Exactly, exactly, that's what I mean. And heavy use of, of course, also something I was going to, I mean, things like the various preparation theorems and formal power theories, I mean, these are quite crucial in the complex analytic picture, but they're also very fundamental to commutative algebra. I mean, the commutative algebra is a very special case. I mean, in some sense, you're saying probably, I think, correctly, that there isn't much In the method of presentation between Wobbeke and van der Vaarden, really, in those things. But the subject did evolve a considerable amount in the time, in the very short time between van der Vaarden's books and the main... I completely agree. I mean, one did understand a great deal more about these... Completion processes for which flatness, I mean, flatness was an essential item there in the case of magnetic variance of this kind, and I think also Warbecki knew much more than they ever intended to write in this particular treatise on that matter, and I mean, it was, of course, Serre, Cartier alluded to this, Serre had produced all this very refined local theory involving homological algebra and books of art and so on, which, I mean, was One knew was going to be a fundamental ingredient for algebraic geometry, perhaps in a slightly remote future. I mean, Gabber and so on use these same methods now in getting correct definitions of intersectional multiplicities in very general cases. I mean, I think it isn't really clear to me that it would be a sensible strategy for them to redo... That body of results in commutative algebra, that one would be more categorical, unless they were going to do algebraic geometry. I mean, of course, it's indispensable for algebraic geometry, but even subsequent people, people like Mumford. Mumford doesn't really do the commutative algebra, say, in anything but classical language, but he mixes it, of course, with the scheme language. I think that making the treatment of tensor products and those flat is a little more functorial. I agree. The flat is different. I agree. It's different. It's definitely different. Yeah, yeah. It's true. It's a mixture, I think. It's a specific case. A great deal was discovered between Van der Weyden and... Absolutely. But the approach was followed thoroughly, I think. I mean, it's... Therefore, I think that the importance of... Let's say, if you take Van der Weyden, you have two important things. The first thing is that you have to be able to say, you have to be able to say, you have to be able to say, you have to be able to say, you have to be able to say, you have to be able

25:00 First of all, it codified, like many textbooks. This is the discipline if you want to learn it, and when McLean and Birkhoff wrote their book, they essentially did it. But there is more than that. That is the point. It's not just that I'm telling you. You know, it's like the elements of Euclid. It's not that he put together the results. He also said, if you want now to continue, this is the way to do it. You know, you have to use diagrams in this way and you have to derive. So the same Van der Werden, I don't even think that at some point Van der Werden thought that this is what had happened with the book. I mean, he was aware of what he was working with Netter and he was strongly in, but he was young. And he wasn't completely aware, I think, of the great difference. You think he was aware? No, I think Leder was so strident and insistent and everyone knew this was an image of all mathematics. More than all mathematics. All mathematics should be done this way. Okay, okay, okay. But in any case, here is the book. Now, you know, I'll put it in this way. When I studied, you know, I studied in Venezuela. Basically, the study program was burbaki. We had topology, functional analysis, algebra, and we even used the books. And for me, this was mathematics. There was no other thing. You have a set. Well, okay, we also knew some, of course, calculus and all the techniques and everything, but then you move to the real mathematics in the, you know, the second year of your training, and you have always, you start with a set, you define on that set something, and then you start, you know, looking for this and that theorem moving here, moving there. This is, you know, I think that as a student... You get a certain picture of what is mathematics, and it doesn't include statistics, of course, I mean, it's outside, or probability, things like that, or continuum physics, I mean, it's completely out of the picture, and since that was in the early 70s, I think that was, I look at that as a good example of... Perhaps the high point of the Burbaki influence in that sense, specifically in a place where there was no real autochthonous mathematics, at least not at that time, and it was not part of a tradition.

27:30 So the view was taken from some other place, and here you have a complete view, very appealing, sometimes it's easy to teach. Because, you know, it's very structured. You know exactly what you have to teach, how to build a program, etc., unless you need less to know, to have a big view of mathematics, like we were talking yesterday about people like Bernstein and all, that they say, look, there is a strong problem here, let's solve it. Algebra, you learn topology, you learn... By the way, even in Venezuela they changed it very soon after that, because in the 80s they said let's put, you know, statistics and computer science and things like that, but this view I think was quite dominant from the 30s to the 80s, not in every place, if you look in the United States. The Courant Institute, possibly. They took their own view about it. But if you look at Chicago and Harvard, I haven't really seen the programs, but from what I have seen, it's very strong in this way. Follow this view of mathematics. But I don't think... No, that's definitely right. And, you know, there were these... I remember in Oxford there were these... Well, there's still what we might call traditionalists who would get against this idea of doing that, because it's true, it did affect the kind of split from applied mathematics, from mathematical physics, that perhaps hadn't been so glaringly, you know, so definite before, but actually, yes, it provided a kind of model of what modern mathematics, some kind of... The second thing was that you notice very quickly, for those of us at any rate who went into later interlocking, and I didn't start as, I started in physics as so many of us did, and I started in physics as so many of us did, and I started in physics as so many of us did, and I started in physics as so many of us did, and I

30:00 You know, a lot of it because of reading Kelly's book and then Bourbaki gave it to me, yes, well now I really know what's going on, I mean I have some understanding of rigor in the objects of attention or somehow it's only fully presented, but the interesting thing is that, and I later worked in set theory, was you notice rather quickly, if you're reading Bourbaki, And I've read a lot. I mean, I've got all the exercises, which I always thought was the main point. Yes, that's true. The exercises are superb. The topology general and, you know, in algebra and so on. And, you know, the space vectorial, for example. A topologique, I mean, that after the initial definitions, they use sets for what you're really looking for are common patterns, and you get always this repetition, solution of universal problems, which is clear, of course, it's actuates, I mean, they're free, you know, free objects, and quotients always again, and frankly, the set theoretic part of it just falls away, you're not looking for that at all, I mean, but you use sets, you do, you do use them, you can't, But you notice that in some way they don't play in a sense in the formulation of the general patterns that emerge, the common patterns, if you like. Yes, there's a set sort of underneath it all, it's like the bricks, the buildings. What you actually see is the sort of general architecture and that's why that was the idea, I think. When you said, I wanted to say another thing. And the quotients, I mean, of course, the set theory gave one a standard way to realize the quotients, which is in a way quite bizarre. I mean, integers modulo three is not three points, it's three huge sets. Yes, exactly. Equivalence classes. And so there's, you know, the equivalence classes as such are the only way to realize quotients. Well, I think that was, yes, but I think that was, again, because, yeah, exactly, but as I said yesterday, I do think that was some... It was a kind of ontological thing. I mean, there really was a sort of vagueness in the, at least in the notion of an ideal number, except that he did provide, clumsy, the answer except that he provided may have been clumsy, right, with these huge things, but it was regarded as concrete in some way, you know, as definite, even though it was clumsy.

32:30 For a while, but I mean, I think... For a while, for a while, this is not a good idea. I was talking to Bill this morning. No, no, I agree. He's starting to push it now in terms of elements. Of course, that's, I mean, it drives... And so on and so on, but in fact it tends to block a lot of things. Although I do think that was the initial... Well, you know, using equivalence classes is some... It's a universal technique. Once you use it, you use it everywhere, not merely in mathematics. It's better to change the identity relation. There's this whole trend of papers on category theory around the concept of the exact completion. It was first promoted by Aurelio Carboni, I guess, but it's already in bars. But then applying this, it's funny because this is somehow a very simple, naïve approach, you see, I mean you basically, you start with a category that may not have quotients and you consider the internal groupoids. But then up to some homotopy, so to speak, instead of not just the obvious homomorphisms or group lois, but something much coarser, but nonetheless definable within that framework. And you get a new category where you have these formal quotients. But the striking thing is that they're not formal. That is to say, in all kinds of concrete examples, those quotients are isomorphic to the ones that you... You wanted them, you know, as you might think from the way I described it, that you'd be getting these sort of formal quotients which would be different from the set theoretic ones, but it's quite striking, I mean, there's an implicit, a whole viewpoint on that very question that still has to be, you know, sort of summing up the essence of all these papers, I don't think it's still been done, and there's still, you know, you can even find the exact completion of the homotopy category and all sorts of things like this which... From the point of view of equivalence classes, you wouldn't be able to. Does that make sense? Yeah, yeah. I'm guessing, the sense it's making to me is, just take the integers mod 3. You knew these were remainders on division by 3. They're 0, 1, and 2. So you knew this going in.

35:00 So that often these quotients, they have some, you've got another simpler description of them already there. Yeah, in other words, in that case, what is the group void? I mean, the domain and co-domain maps are, let's say, a trivial one, and multiplying by three, in other words, literally, what it's about is multiplying by three, and so you make that into an object. But then you treat it only up to that point. And the Netter School is concerned about this in just that case, and Saunders even has a footnote about it in 1950. In 1950, he wants to compose projections onto quotients. Well, he wants a quotient of a quotient to be a quotient. And on the equivalence class set, it's not. An element here is a set of equivalence classes, not a set of things, although of isomorphism it is. And he mentions there an issue that runs all through the Netter School that I first saw in your book and always puzzled me. He says that instead of doing equivalence classes, another approach would be to say that you describe a group, for example, by giving a set of elements and an equality relation. And we'll regard passing to a quotient as changing the equality of the world. That's what I meant. By the way, in the 19th century, you know, kind of... The view is now adopted, of course, quite routinely by people working in constructive type theory. Quite routinely. And completion is essentially that idea. There are other... You mentioned universals and quotients. There is, for example, the idea of a closure. Algebraic closure. You have a property and then you look for the closure of that property. This is a kind of structural idea in that sense. You see it applied here to the algebraic closure and then you try in topology or in a functional space or whatever. So there are many such ideas that move from one place to another. Of course, later on you can formalize it or put it into a category, in a theory like category. But you can do it without the theory. That was my point, that you see something that works fine over here, you try to see how it will work. Sometimes it works and sometimes it doesn't. But many times there were ideas that moved from one place to another.

37:30 From one site to another. Yeah, by the way, it started inside algebra. One of the points that I think for me was one of the points that was startling when I started this was that People defined in the late 19th century groups, rings, and fields separately. And they treated them separately. Okay, at some point it was clear that you can take a set and have one or two operations and study some things, but the idea that these are three species of the same idea, which is an algebraic structure, this idea comes very late. In fact, it comes around the time of Van der Werf. For example, Dedekind himself, he uses fields and let's say rings, or ideas, not rings. In two completely different ways. The kind of questions that you ask about fields is not the kind of question that you ask about ideas, because ideas are a tool to study fields or the algebraic numbers. So only in Van der Werden you see, okay, now I have, you know, he has this tree, or a light fountain, he calls that. And then he says, the same question that I ask here, I will ask here. The same thing that appears here. Here it is true, here it may not be true. But the question is the same question. And it takes quite a lot of time for the algebras to realize that they are dealing with the same thing. So Gerrit Wirkhoff's work at Cambridge was… Yeah, in lattices. No, Universal Algebra. This was just shortly after the publication of Andrew Barry. Exactly. Three or four years he came up with it. With Hall. He was a guy called Hall in Cambridge. Oh, yeah, yeah. Philip Hall. Philip Hall. I guess they were contemporaries. Yeah, he was something a little bit... Philip Hall certainly did his... I mean, I went to Philip Hall's lectures in Cambridge. No, he was dead by the time he was there. I mean, they were very much, for group TV, they were very much in Dunbar and Stein. He got very far in them. I mean, they were tremendously organized in that spirit. I mean, all was better in Cambridge doing hardcore.

40:00 Very combinatorial work, but he did organize his lectures exactly in that style. Although, you know, this linear expansion, because what Birkhoff did, by summing up and by saying these are all instances of algebraic structure. Which, by the way, McLean says that's what his master's thesis was related to that as well. But, no, not the general idea, but simply another example, namely exponential re-existence. Yeah, that's right. But this was all foreseen by Grossman back in 1884. Yeah, of course. He also has the general notion of an algebraic structure. But look, I want to say something that I forget, and I wrote in my book. Everyone forgot that. There is this case of Øystein-Ore, who came up with this theory. He thought that by looking at the lattices of some structure, you can say a lot of things. In the end, it didn't turn out very well. But structurally, let's call it, it's very similar to category theory, because it says, here we have these things, let's look at them. I'll tell you a very specific thing about this because Vandenberg lived a long time after that, in Zurich actually, so when I went to Zurich as a postdoctoral fellow, I stayed there for several years. So, you know, I could see that Van der Weyden wanted to know who are all these people coming here, so he summoned me, sort of on top of a mountain, it's not in Bergen, and I went up there, okay, so the pretext was he wanted me to help him translate something from German into English, just one word, and so he asked me about this. This was the pretext. But then it developed, you see, that he... He wanted to tell me that, you see, this whole business about the category of sets as a foundation was bound to fail just like the lattice theory had failed, so in other words, he didn't grasp the fact that by going from the lattice of sub-objects to the morphisms that relate objects, you suddenly gain, in some sense, the full power of mathematics, just a fragment of it. This he didn't get, but he explicitly recognized, you see, that the lattice theory idea, you know, lattices of subgroups and so forth, that this was never, this was a huge piece of the information, but it never quite, and certainly not as a foundation of mathematics. So this is what he really wanted to tell me. He must have been quite old then. He lived longer.

42:30 But then also I know what I wanted to say when you spoke about physics. This is what I am so surprised about this matter of the Hamiltonian physics in your story, because it comes, as it were, from outside. I'm speaking about this algebra and topology and so on, and well, perhaps you can connect it with logic, but how come physics enters here from the side and plays an important role in developing this idea? I mean, it's foreign to the story that goes with the mainstream of the structural ideas because Hamiltonian physics was not structured in this sense, in the sense that I've been saying. At least not apparently so. Well, there are two distinct things. I mean, Hamiltonian mechanics, which is usually a finite dimensional system with a finite number of particles. Although, of course, in quantum mechanics, formalism is extended to infinity. But a completely different thing, you know, the physics that engineers actually use in designing rocket engines or interplanetary orbits or, you know, whatever, you know, you look at the vibration of the moon, all these kind of things, it's another thing completely from quantum mechanics or even from Hamiltonian mechanics. It's this continuum mechanics which Truesdell, you know, was like to emphasize Euler developed. Continuum mechanics and Cauchy. So earlier in Cauchy were the big heroes of Truesdell, who was my first teacher. I mean, I wanted to understand electromagnetism, went to Indiana University to study. There I met Truesdell, who opened my eyes quite a lot. Unfortunately, I still don't understand electromagnetism. I have a son who's working on that. But anyway, so I mean, so Truesdell saw... Euler and Cauchy as the big heroes, and they developed something quite different from this analytical mechanics of bare particles because... But still in a variational approach. Deformable bodies.

45:00 Yeah. Deformable bodies, you see. Flexible all strength, you see. How can it move under the influence of force? You know, for example... There is this theory of so-called rigid bodies, which is one thing, but solid bodies is something totally different. A solid body is not rigid. It can move. When it moves, it responds to outside forces in quite a different way than a liquid or a gas would. So to be solid is a constitutive relation of motion, not of rigidity. So this kind of idea, which is quite foreign really to most physicists, actually, but this was the point of view that I learned, and in particular his student, Walter Noll, as I said, already in the 1950s, who had been a student of Hamel, the same one as the... Who was a student of Hilbert. Yes. Georg Hamel was a student of Hilbert. And as you point out yourself... Hilbert was teaching courses on continuum mechanics all the time, so this Hamel, who did this crazy abstract thing about linear spaces over the rationals and all this, his main work was in continuum physics, continuum mechanics, and of course thermodynamics grew out of this as well, so thermodynamics is again basically a continuum. Theoretical theories about heating bodies, how they expand and how they might deform, and how the thermal effects might interact with the mechanical effects and so forth and so on, all of which is under the rubric of continuum physics. So, Noel had studied that sort of thing with Hamel in Berlin. In Brno he was, no? How many, how many was he? Yes, he was in, that I don't understand. I know that. I know that he was in Brno, but he was in, he was in Brno in the early part of the century. Yeah, perhaps. That's right. People in Brno don't even know that. But yeah, that was, that was the early part of the century. So somehow after the war, he must have been, he was one of the few, you know, mathematicians who survived the war. He's, he's mentioned in that there is a... There is a book, I think, or an issue of a journal called Kollegen aus einem dunkler Zeit,

47:30 something like this, which is describing how various German mathematicians managed to survive those awful times. So Hamel is one of them. So Hamel was still teaching up until 1950 or so. But no more at Bruneau. I think just at, is there, there's a university at Charlottenburg? Yeah, yeah, a technical institute. Right, that was where Noah studied. But then Noah came to Paris and studied with Bourbaki. Ah, okay. That's very interesting. I mean, in the sense that he followed the lectures, you know, of the group that we've been talking about. And so he absorbed enough of this point of view to develop really a passionate desire to clarify the continuum of physics using precisely these structures. Considered to be, by many, a solution of Hilbert's problem. One of Hilbert's problems was to axiomatize physics, which again, the narrow-minded physicists who don't understand all this think is somehow just a matter of making rigorous particle mechanics, and indeed there was a big debate between soup aces. And Truesdell, in the very first volume of Truesdell's journal, when it was still in Indiana, when it was thrown out of Indiana, where Suppes and Sugar claimed to have solved Hilbert's problem. Here it is. The Foundations of Classical Mechanics in Light of Recent Advances in Continuum Mechanics. This is Noll's paper. That's Noll's paper, right. So the one of Suppes, Truesdell agreed to publish the paper of Suppes with a long footnote explaining what they've really done here is they've not solved the problem of Hilbert, they've done something else, which is, you know. So, I mean, there's a long structure. So, the way that you saw this... So, Noe became a student of Truesdell, you see. In fact, his thesis was about the fundamental question of how the statistical mechanics of many, many particles can be used as a model for, not necessarily a replacement for, but a model for the continuum structures there.

50:00 Euler, Cauchy, and so on. Maxwell was one of the main important contributors to this. So besides Euler and Cauchy, the big heroes included Maxwell in Truesdell's courses. So anyway, Noah had made a great advance on formalizing the general way that you can deduce from the laws of physics, plus the laws of physics and particles, large numbers of particles, plus the laws of probability. The general balance laws of continuum mechanics. But the balance laws have to be distinguished from the constitutive relations that particular bodies enjoy, and so this was still not really the... Anyway, so he reached this, the paper you're talking about, and this was basically defining a body as a certain type of structure, it's a topological space, it's a given family of sets, and a given family of transformations, and so on. It was completely in the language of Bourbaki. So it was structural, a structural reading of… Completely in the language of Bourbaki. And it was so striking, you see, that… Large numbers of Trusdell's followers, loosely speaking, have formed into a society known as the Society for Natural Philosophy, of which I'm happy to be a member. But this work of Snowell was so striking that large numbers of these people have just accepted this as dogma. They use this language every time they write a paper. Still today? Yeah, still today. Meanwhile, Noe himself, it's like Grotendieck, you know? Meanwhile, Noe himself improved drastically, qualitatively, this drawing work in successive papers in, I think, 72, maybe, or 73. So, this is the... But now, I, observing all this, you see, this was my original reason for going into category theory. Of course by Truesdell in functional analysis, simultaneously of course by Truesdell in fluid mechanics, where it seemed to me that these physical ideas about bodies and forces and motions and constitutive relations and all that had a meaning, had a possible rigorous expression independent or at least largely independent of the detailed mathematical ontology.

52:30 Which mathematics provided, you see, because, you know, according to the mathematics as it stood at that time, now it's changed. You can't talk about a manifold without talking about charts. You can't talk about a manifold without first co-ordinatizing, you know, in Euclidean space. You can't talk about real numbers without having Cauchy sequences or something. You can't have blah, blah, blah. So that the ontological foundation for expressing such physical ideas is so incredibly complicated. I mean, you have to be a to think simultaneously to advance your thinking while dragging along all this baggage. So this was the basic aim was to find a framework sufficiently general so that one could still reason rigorously about these continuing physical matters without committing oneself to a particular category of functions or particular ontology. If you talk about the numbers are no longer basic, well, after all, the real numbers are just an abstraction from all kinds of measuring processes. Measuring processes involve physical dimensions, like mass and time are different in length and force and action and so on. So to really deal properly with the physics, one needs a deeper kind of quantity, really. Because the real numbers are really only the pure quantities within that framework, the dimensionless quantities. So the proper general framework should include a variable notion in some sense of what are the real numbers even, adapted to the physical needs. So this is how it is. It wasn't all from the side. But how do you go there to categories I don't really see? I mean, you could... Well, because it's about the motions are mathematics and bodies are objects. You have studied what occurs in mathematics that involve categories? I mean, your story starts from physics. And then you have a tool that comes from mathematics. I start from physics and two things happen. I meet Truesdell, a very persuasive, charismatic figure.

55:00 And also, at the same time, I'm meeting a barrier in learning physics, because none of it makes really sense, because they're not rigorous mathematically. I mean, it doesn't really make sense for several reasons, but at least at the time, at the young age of 19 or whatever, I could say, well, part of the problem here is mathematics. So therefore I switched my concentration in the university, saying, well, first I'll straighten out the mathematics, and then I can bring that back and straighten out the physics. More than one person has made that move, that dream, right? Just an incredible number of people. If I look at my friends, you know, Gonzalo Reyes, Jack Duskin, you know, just any number of people that started in physics and switched to math, more or less. Cartier as well was saying he started with general relativity. Yeah, so when I learned about the Grotopos algebraic geometry in Saint-Gabriel, I saw this as a paradigm. It's a specific example, but it's a very nice paradigm for the later work and actualizing of continuum physics. This paper here. Which just gets to the infinitesimal body. Even though the infinitesimal bodies are much bigger than the bare particles, you see. And then, if I look at this and then I ask myself, okay, how does it affect back physics? I mean, has physics somehow become aware of these things and tried to use it? He's very, you know, he's very, he's enlightened, you know, he really is interested in all kinds of things. He wrote a book, as you may know, called Mathematical Physics, which is, which is, well, it's lectures in which he tries to give, of course, he doesn't do very much physics. It's, it's really a kind of introduction to category theory at a, not at a deep level. I, I, I did tell him, you know, I didn't mean it defensively, I said...

57:30 To him, well, you know, you've given this account, and he does do definitions, and you can see he introduces the kind of structures, mathematical structures, that are used in physics, and various things, and functional analysis, and basic analysis. And he states a lot of these things, and he does emphasize the idea of a map. But essentially, he doesn't really use it. It's a sort of cosmetic. I told him this. I said, well, this is very nice, but do you really use it? And he admitted no. I mean, he was aware that there was something important going on here. But of course, he didn't really see how to use it. In terms of the actual problems that he was dealing with, you know, in physics. Another example, you see, specifically they often use fiber bundles, bundles of one sort or another in that discussion, in the physical discussions. But again, these concepts of bundles exist in many different categories, some of which, you know, might be equally good candidates for a basis for a particular branch of engineering or physics, so. So one needed a series of bundles which didn't first assume, well, you've got an analytic manifold. Oh, no, you've got a C-infinity manifold. Oh, no, you've got a continuous manifold. Can I say something very quick on that point? There was a very interesting meeting at the Newton Institute last December on quantum gravity, and I heard a couple of the talks there, and one which impressed me deeply was a young German mathematician who is a general relativist. And we did talk about the applications of ideas from synthetic differential geometry in GR, particularly he was working on what they termed sectional curvature as a preliminary to trying a particular approach to general arturgy. ...but very much starting from the general relativity structure rather than assuming that was something which had to be broken down and built up again from the quantum level. And he made exactly this point about, you know, the many different categories in which the notion of fungal-fiber bundle... And I got talking to him afterwards and he was certainly aware of full work and wanted to read. And I found this very exciting to read. I was going to wait until the appropriate moment to tell you. And his wife, who is Spanish, is also a mathematician and she works in fluid dynamics.

1:00:00 He is aware, just how few people out there have yet been prepared to learn this language, but he tells me that the situation amongst the younger general relativists is changing, and that particularly because of the role that sectional poetry plays in some of these approaches, people are now beginning to become much more aware of synthetic geometry, so it may be coming. Marrying women who work in fluid dynamics helps a lot of work. That! Especially attractive South American women. Okay, I just mentioned a few parts. The situation is perhaps not quite as... There are a few papers, if you start to think of it, in this archive. You know, the web thing. I can't remember, you know, where the Russian... Well, of course, the Russian... Russians had some... Physicists did know, had some notion, like, of Topper's theory, maybe a bit... Well, I don't know, I mean, perhaps a bit muddled sometimes. Well, you remember, 20 years with Aksharina and these people, and there were, and you see, yeah, yeah, I know, but there were... No, but nevertheless, there's always been a, there's a certain adventurous... But there is a certain adventurousness, of course, among cosmologists. Under the old regime, you could more or less think and say what you liked in that department, and so I think there has been. I noticed a couple of Russian... I can't remember their names now... Who have attempted to use synthetic differential geometry as a kind of framework for developing general relativity. There are a few in Russia, but unfortunately I can't remember the names now. It needed a certain openness of mind, you know, to consider this, and it's interesting that it's more common in Russia than it is in the West. Atiyah had an open mind, the problem was he hadn't filled it with anything except a few buzzwords about topos, which is the same thing he had, you know, corresponded with Geroch, but it was pretty obvious that he didn't really, like Chrisley of the great Siberian woolly topos of Ormsk. I mean, I don't think synthetic differential geometry has had a very big impact on people working in differential geometry as such.

1:02:30 So far. This was, right, I mean, of course, there were people who were geometers, you know, who actually, or who primary interest was there and got into that and developed that more or less exclusively, but there don't seem to have been very many people in mainstream, you know, differential geometry who've actually, or even, I don't know whether they're even aware of it. I think it only becomes striking when you get beyond the... You know, marveling at elements with square zero instead of doing, say, the functional analysis, which is possible in the topology. Their work on the weight equation, for example, is quite, I think, you know, quite striking in the sense that it puts a different light on this very classical subject. You refer to Walter Knoll's axiom of equation in what connection? Yes, because I'll tell you why. The book, you know, the book is about the sixth problem, because the sixth problem speaks about the thematization of physics, and it was usually considered, and if you look at the books that were written later, 50 years after, 80 years after, where are we standing with the problems? And always there is one page for the 6th problem. Let's say 70 problems, Langlands program, you have for every problem, or they were solved, or 10th problem, you know, it's the O'Fantin equations, algorithms, etc. So the book is about, first of all to say... For Hilbert this was a real problem and he devoted a lot, a lot of effort of himself to do it. Second, some of his students followed him. For example, Hamel wrote several articles about the axiomatization of the, let's call it the parallelogram of force theorem, which is elementary, but it's in the way of... To what extent he influenced some people. So no, let me tell you, it's something that has to do with thermodynamics. If I am not wrong, it is because...

1:05:00 In 1909, Hamlet published an article on the principle of mechanics that contained philosophical and critical remarks concerning the issues discussed in his own earlier article, the one on vector addition. It was about vector addition. In particular, he discussed the concept of absolute space, absolute time and force as a priori concept of mechanics. Except I mention it because of a brief passage where Hamel discusses the significance of the Hilbert axiomatic method. More importantly, it also contains an account of a new system of axiom for mechanics. It is 79 and then, where is null here? There has to be a connection because it's null. According to Clifford Truesdell, I say here, This article of Hamel, together with the much later Noll, 1959, are, and I quote, the only significant attempt to solve the part of Hilbert's sixth problem that concerns mechanics that have been published. But I say this is not true. I mean, there are many other things. For example, there are, Canna Teodori published interesting articles on thermodynamics. As part of his attempt to deal with the automatization of physics. Born, who was very close to Hilbert, published many things. But all of this has been, like, diluted. Well, I mean, the point is true. I think Truesdell was aware of all that. In other words, he had looked at Kara Theodore, he had looked at Moore, he had looked at Sufis, and he decided that these were not serious attacks on the problem in its proper generality, or for various reasons. It's not that he was not aware of these things. Well, it would be interesting, yeah. And then there is another article by Hamel in 1927 that appeared in the Handbuch der Physik about the axiomatization of mechanics. Ah, I didn't know that one. Handbuch der Physik, 1927. I can send you later the reference. No, I think that there was a lot of work on it. In fact, what Hilbert does on general relativity is part of his work. It derives from the program for the axiomatization of physics, because he takes what has been done for mechanics in general and he changes a little bit the kind of Hamiltonian function that you put in the variational principle in order to get the... But going back to your point then...

1:07:30 At some point it connects with logic, all this work that you start doing with categorical algebra, topos and so on. Or that's a part of... Of course it does. I mean, again, I... So you start with physics, move into... I draw the historical parallel, actually, in my paper in the Peer volume, the second Peer volume, on the comments on the development of topos. Because he, as a student at Brown University, interested in foundations of continuum mechanics, you know, he got the idea that, well, foundations are being done by logicians. I should study logic in order to know how to go about developing foundations. So he actually moved to Princeton. He was even going to change his whole, as I change mine, but he's going to change his major and everything. And he took the notes for Alonzo Church's book, Mathematical Logic. It says in the preface that this is based on notes provided by Truesdell, and Truesdell participated also in revising them. But he realized that this was not really the kind of foundations that he wanted in one way or another, and he went back and dropped it entirely. So, in the same way, when I was studying category theory with Eilenberg in Colombia, I had courses from Max Dorn, for example, before, and so I knew a little bit about these things, but I decided in order to deal with these feared big categories. I needed to know more logic and so I moved from New York to Berkeley and as an unregistered student listened to Tarski and also Dana Scott and Robert Vaught and Neil Craig and various logicians at the time to try to learn as much as I could in order to apply this logic to the foundations of category theory, which of course I did. I mean I took away a certain amount of...

1:10:00 Now is there, which has always been useful. Wait, for example, I can talk intelligently with Angus. But I hope it's intelligent. I wish more logicians had gone to Berkeley. Yeah, no, period. You're right. I think you're right. It wasn't even about the logicians. They don't know everything that was going on there. But again, you see, I soon realized, well, there's something missing here. This is not really the foundations. This is not the kind of foundation. There are a lot of things that I need for category theory for continuum mechanics. It could be an important component, but it can't be the thing, so... But, for example, with a paper like this, you seem to have completed the original question. Well, it's a big step, but I don't deal seriously with constitutive relations there. From the point of view of the people who work in the foundations of continuum physics, that's the main thing, and this is a framework within which constitutive relations may be discussed in a later paper. It's a step forward, I mean. Essentially, I talked there about spaces of states. I formalized the notion of a second order equation. You know, 99% of what's done in mathematics articles under the slogan of dynamics, dynamical systems, is really not, it's really only about first-order equations, so that the fundamental discoveries of Galileo and even of Fibonacci before him, that the laws of motion are laws which work on the, are laws of becoming and not laws of being, really, has not really been taken into account by most of those. But here I do achieve really a different kind of category of dynamical systems, but of second-order dynamical systems, or states of becoming rather than states of being. But to really get into continuum mechanics, the spaces of states have to be analyzed again as function spaces, embeddings of a body into ordinary space.

1:12:30 These are the configurations and the states may be tangent bundles or histories of those things. So this analysis of the space of states into Given body, so to speak, ordinary, or some containing space, and the space of maps between them, is not here. So the constitutive relations deal with the body independently of the embedding, by quantifying over the embeddings, you see. So how is this body going to respond if it's heated on this side and hammered on that side? Making a sword by tempering the sword, heating and hammering and all this kind of thing. How the sword is going to react is a property of the sword, not of how it's momentarily embedded. So to really analyze these properties of the material itself, one needs to go a further step beyond this, a further step of analysis. Nonetheless, I do think, yes, it is an advance toward providing a proper categorical framework for such discussions. In terms of providing a proper categorical framework for discussion of logical structures, this might be an appropriate place to ask Bill, in connection with the way that you see logical kind of structures and that kind of logic is fitting within the program. Any re-description of structure in general suggested by the developments in chronological algebra and algebraic geometry made a remark, I think in your 1989 Cambridge lectures, that logic could be seen as having, for its subject matter, the study of the roots and supports of intensive quantity. And that obviously connects with the roots of intensive and the supports of extensive quantity, and that of course naturally brings in this whole question of how a general theory of structure, which takes proper account of covariance and contraherence, the thing which is of course completely missing from this defective theory of structure we talked about over breakfast, the one that just thinks it's all about making maths and physics and ontology for its own, how this connects with... Well, a deeper understanding also of real-world variation and cohesion. But I wanted to ask Bill if he could expand on that remark and perhaps say a little bit about the whole relation of structures of intensives as categories of intensive and extensive quantity and how they put together the understanding of categories of space as logic connects.

1:15:00 It's a huge, huge subject, I know, but... You probably had this already since you were... Yeah. Ah, that's Sebastian. Yeah. Yeah. You mean that kind of material? Yeah, I was just thinking that might very much be something I would like to hear about, too. Yeah. So, Frege has given a lot of credit for having a lot of quantifiers, but I... I don't like Fragon, might as well admit it. So I see various flaws in his character and his mathematics and his philosophy and so on, so if you want to accuse me of this, okay, but I tend to discount at least the absoluteness of this idea of inventing quantifiers because he was a student in the milieu, you know, late 19th century Germany where countless of variations was a major... Subject of study and one of the two that actually made his degree in calculus and variations. But in any case, the basic object of studying calculus and variations is, okay, you can say optimization, but optimization of what? Of functionals. In other words, of maps from r to the power x into r. The real-valued functionals of real-valued functions or complex or vectors, basically. So, it seems, given the previous work of Google and so forth, it wasn't a big jump to go into two-valued or, say, truth-valued functionals of truth-valued functions. And that's all, you know, that's what a quantifier is, right? In fact, Frege himself uses the analogy to explain bound variables, you know. These are analogous, you see, so he sees them as functionals taking directly the typical one from calculus of variations as an example.

1:17:30 Now, of course, there's a very important further step, namely, why is universal and existential quantification? You know, really singled out in the case of truth values, whereas it seems that there's a whole vast variety, the steepest descent, the steepest that, the most this, and maximum profit there, whatever, in calculus of variations, there doesn't seem to be kind of a preferred functional. Well, of course there is. I mean, it depends on the fact that this object R is an ordered set. And of course, there's always the inclusion of constant quantities, constant functions into general functions, and more specifically, you can ask for averaging processes, which are functionals, which compose the identity, in other words, the functional applied to a variable that happens to be constant gives back the same constant, that one condition, so... If you're exploiting the fact that R is an ordered object, if it's even a complete ordered object, then of course you have the sup and the inf of a variable quantity, which gives you constants. And these are indeed both averaging in this very general sense, but they are distinguished by being left and right adjoint to this trivial inclusion of constants, the very definition of inf and sup. You know, by Bolzano, whoever actually did it, is really nothing but an instance of the notion of adjointness between constant and variable quantities. So, if R is conceived as truth values, then indeed you have, you know, adjointness there results in existential and universal quantification. Maybe it's worth explaining how they're adjoints. I'm not sure if this is something that's familiar to you, because it sounds like we've gone from the infant soup into, oh, what is it, Chapter 5 of Maclean's Categories for the Working Mathematician. Yeah, well... But in fact we've just noticed what the infant soup say. Right. Right. I mean, the supremum of this function is less than... The rules of entrance for universal and existential.

1:20:00 Quantification, as I learned them from Max Zorn in Kleene's book on metamathematics, for example, if you just look at these rules of entrance, which are really in the, you would call it the Genshin-style terms, right, they exactly express agonists. The rules of entrance are just agonists. I was the first to point out that they're adjoints, but the fact that they're functionals was an easy, I wouldn't even say generalization, but specialization from the real case. And nevertheless, no one formulated it that way before Frege. Not that I know of. No, before Frege. Okay, but mathematicians didn't look at that. I mean, it's not, I'm not telling you the chronological... No, no, I said that mathematicians did not take up the concept and use it by looking at Peirce, but by looking at Frege, right? Well, I don't think that didn't take up logic for a long time after that. Yeah, okay, okay, but great graduates, no, no, I didn't see the specializing down from this, you know, it's already there in the case of analysis. Mackay made a very nice comment about, meaning he said, Volvier's contribution to logic was to point out that substitution is the basic concept and everything else is defined in terms of that. So, viewing the rules of inference as adjoints, it tells you that they're defined, they're just, knowing what substitution means in a given system. There's a unique possible meaning for the quantifiers. And in turn, well, that's slightly different. Why substitution is basic, but also... Composition. Well, substitution is composition, of course, but, I mean, this idea of what you call the image and so on, I mean, there was already this, I guess, philosophical concept of...

1:22:30 The general concept comes in two dialectically opposed aspects, the abstract general and the concrete general, you see, so that, now that the concrete general is typically a category because, no, of course, Urbaki's formalization of structures left open exactly how you're going to deal with suborphism, but he doesn't. He does allow for the possibilities. You have a notion of structure and a notion of morphism semi-compatible with it if you consider that. So if you can consider that as an abstract general, then corresponding to it is the category of realizations, the concrete general. Somehow this is inevitably a category. But it's a further step to realize that the abstract general itself is a category. The abstract general itself is a category. Where the composition is typically called substitution, but it's not the same category at all, it might in special cases be dual to a subcategory and so forth, and that depends on a particular doctrine, particular situation. So, in other words, Mackay's comment could be put slightly differently by saying that around 1960 we realized that the abstract general is also a category. I wasn't the only one who did that. John Isbell did something like that. Claude Chevalet and his famous Lost Manuscript. That's what we don't know exactly, but he must have been doing something. Like that, but philosophically it amounts to recognizing that, you know, whatever particular doctrine you're talking about, the abstract general should be a category, then it'll have an additional structure, it'll have different properties, and so forth, but first of all it's a category because the substitution there will be dual by naturality to the notion of morphism in the concrete general. Now, is it maybe worth looking at quantifiers this way? Because I remember, I mean, at a time when I hadn't studied a lot of category theory, I'm trying to think what I might have. I mean, it's just, when does a proposition with a variable x entail another one, which we think of having a variable, but it doesn't depend on it. So we've got this property of a general x, when does it entail a tail psi? Well, just in case for some x pi.

1:25:00 All of these things entail that side. We'll get rid of the free variable. So this general thing implies that just in case. And when does this thing that, you know, we think of as having a null dependence, when does it imply a statement about x? Well, just when it implied the universal quantification. This is just the jointness. This substituting has on one hand the adjoint existential quantification, on the other hand the existent adjoint. So, again, yeah, just to emphasize that the point is a simple one in itself, and it's a case of a well-known, you know, a pattern of some points itself. But that is to say, if these are functors, what kind of functors are they? Ah, they're adjoints to substitution. And that's the point that this is all about substitution. Can you say a little about how it connected with what Isbill was doing? Is this the... Kind of like a recognition of the adequacy and co-adequacy conditions. Yeah, in that same period, I think. Right. I'm just, sorry, I'd like to understand better how that is, where is a further case, a further instance of this. No, not that as such. I mean, more or less, that is similar. You know, the idea that algebraic structures are functors on a small category which plays the role of the theory. He had a paper that he revised considerably when he received a copy of my thesis. Saying that, you know, a large part of what I was trying to do has been done here and so forth. It was independent at first. Quickly became. And Chevrolet, you know... What is this paper? I don't know. I mean, lost paper? Manuscript. Manuscript, yes. Chevrolet was, again, I think he was probably commissioned by Bourbaki, you know, to systematically write up... Everything about, you know, categories and counters and limits and co-limits and algebraic structures and, you know, the theorem that filtered co-limits commute with finite limits and sets, you know, this kind of thing, which would be an essential framework, you know, whether they were going to use it.

1:27:30 There's another question about the rigidity. They were certainly preparing. It's like contingency plans. We're prepared to go to Moscow even if we never do it. It's a contingency plan, at least that. Well, since they do see themselves as the general staff of mathematics, of course, I guess that's exactly what you're doing. Actually, I think Cartier did specifically say when we were in conversation with him that the Chevrolet... In fact, Chevalier was among all the people there, the first generation, the one who was interested in said theory to begin with, that the others were not. So then from said theory you go to other foundational things, then you go to Chevalier and ask him to do that. Yeah, it's not clear that he wrote such a paper. I mean, perhaps rather than a paper... No, no, he wrote it anyway. It was lost by the shipping company. Ah, okay, okay. MacLean mentions it in categories for the working method. Conker gave details while he was sitting where you're sitting. Okay, okay. I didn't know that. A couple of years ago when I called a fairly humorous category theorist for some historical questions, he's telling me various things. I've got this manuscript by Chevrolet, you should look it up. But this was not necessarily, you know, foundations in the ontological sense. The foundations were actually doing algebra and topology class. But probably, no doubt, still phrased in terms of, you know, the tau symbol and all this, if any foundation was involved. I'm just speculating. I don't know if you've seen it. I think I can see it. Foundations according to... Do you have another one? It's duality in foundation? What's the name of that? Adjoinments. No, but here this is specifically citing Truesdell and Eilenberg for their use of the term foundation, which I think is probably... And I think you also... It's an explicit guide for developing a certain subject.

1:30:00 And you also of course quoted a paper with Eilenberg and Steenrod as a very suggestive source for that. Yeah, I cited two of those two in that book in the first part. So Chevalier presumably is going to be a foundation in that sense but not in the extreme and ontological sense. And in fact, the connection with the distinction, with that purely ontological sense, you know, it only got a little way, actually, I think, in setting out for Leo this point about logic as the study of the roots of intensively the supports of extensive quantity, which has got as far as the substitution of the quantifiers being agitized to substitution. So if you want to carry the explanation... Right, well, I mean, if you've seen this in the San Sebastian paper, you know, intensive and extensive, the dualization, the pairing between them and all that, but I point out that, you know, in this special case of tooth value functionals, there's been relatively little development of the extensive aspect, I mean, in other words, the quantification itself is an example of an extensive part. Like integration. But there are lots more, you see. In other words, just to take an example, the idea of, what's his name, the German guy at, and so did I, and also in Berkeley, no, it's a Swiss guy, Gerhard, yes, yes, yes, the set theorist. Later, Reinhardt, no, no, that was Bill Reinhardt. He was a student at Berkeley. Yes, yes, I know. It was Gerhardt, I think. What was it? His, his, his... You know the study was there they're at least uncountably. Yes, yes, generally. Oh, we had a question though, right, right. It's in Alan Slomson. I can't remember. Yeah, yeah, it's part of Alan Slomson's thing. He's a nice guy. Yeah, yeah, I never met him. I met him years after your birthday. We are all around this table with the exception of Leo, past the age in which nominal or phase, you know, I think we'll set aside the search for the name. Just can you just tell us what he did?

1:32:30 And how it relates to this point about the lack of study of the supports of extensive quantity aspect as an objective. So, you know, specifying the size of the set of things that's required to exist instead of just saying that something exists, at least in some combinations of that idea, are, you know, like having a different probability distribution as opposed to the uniform ones. You know, he did some of the idea of generalized quantifiers quite system... Firkin, Gerhard Firkin. Firkin, yeah, that's good. You won the race. Chug, chug, chug, chug, chug. Chug, chug, chug, chug, chug, chug, chug. But I think in the book, Rostovsky actually really did try to consider, you know, generalized in, you know, quite systematically and tried to develop a kind of general framework for, well, I mean, a framework for dealing with generalized quantifiers. He mentions it in... Thirty years of foundational studies, yeah. Yes, but let's not get away from the thought that I love conceptual expositions of this vision of logic, vis-à-vis algebraic geometry and the structures. So the essentially extensive, in other words, my basic claim is that mostly what's been done in narrow sense logic is an intentional nature. The fact that we take the inverse image, i.e. the substitution of, you know, credit... Propositional or quantificational combination, we look how to substitute and so forth, which is encoded into Toko's logic or even bar-exact category logic or whatever the fact that you have sub-objects and you can take unions and intersections of sub-objects. And you can map another object into your object and take the pullback, which will certainly preserve the intersection and in many cases also the union. So this pullback is essentially looking at the solution set of a function, for example. You have a function and you have a subset of positive values, you take the inverse and you just have a positivity set, or you just equal, you take the inverse and you just have a point along the map, and then you get the usual equation. So the typical kind of thing that's done is essentially intensive, in the sense that it's contravariant and multiplicative.

1:35:00 Contravariant and multiplicative, whereas the extensive quantity, distributions, measures, and so on, are typically covariant and not multiplicative, but merely additive. So that, you know, I'm just saying that there should be a parallel development of extensive logic, and these, at least some of these, I'm sure... Systematizing it by a logician, you get all sorts of alternations of things that only about a fourth of that would be fitting into the narrow idea of a covariant functor. But still, you know, it's interesting to know. Yes, it was the broader point that I wanted to bring out. There's something that is being done now, namely all these students at... The Carnegie Mellon of Steve Aldi are following a very narrow line, which he claims stems from McLean. I don't think that's even true. But the basic thing that they keep considering is of such a nature. It's a kind of extensive quantity. I mean, they start off with something that's not even a topos, but then they embed it in the topos. And once you've embedded it in the topos, you've got the ordinary power set, so to speak, there, which is both intensive and extensive. That's the idea. The quantifier is being defined on all of the propositional functors, as opposed to... So it's the same power set. Now, the power set functor is both covariant and contravariant, the total one. But you have interesting subfunctors in the power set functor, or the covariant power set functor. The sub-functors of the covariant power set. So, things defined by cardinality restrictions are sort of the most obvious things, like, well, you could take finite subsets... So there exists a finite subset, this kind of thing. Major theoretic, almost all? Almost all, right. Yeah, there are a number of examples. All but a finite number. Again, to check some of those negations of other things. But, yeah, or even up to, right, but covariately, you see.

1:37:30 I think there's probably much that could be done in that area. And this algebraic set theory program is a very, very partial, not a good way, a very partial anticipation of the kind of structures that need to be investigated. That doesn't even consider it as such. They talk about the power set when it's not the power set, it's not the representer for all sub-objects in the category, it's just this other parameter. So what I'm saying is, well, it's quite natural you have many such parameters, it's not just one. Well, and their achievement was to find something that it could do short of parametrizing all the power sets. Because it parametrized all the power sets, it broke out of the framework, so it found something that it could do without breaking out of the framework, which was close enough to being a power set. But there are many such functions which in some sense should be taken at the same time, instead of one of them being arbitrarily singled out and called the. Well, I mean, in the historical order, it wasn't arbitrarily, I mean, from hindsight, but it wasn't, they didn't arbitrarily single out, they had trouble finding them, having found one. No, they give, their theory presumes that one is given. It's arbitrary in the sense that you could have given a lot of others, but it's not singled out by a unique faculty board. Is this a reflection or connected in any way with the fact that the singleton in the category set is a free object, and therefore... The singleton, right. So the singleton map is essentially the Dirac distribution, you see. In fact, many notions of extensive quantity are actually, you know, monadic, so they have a... It's the one that says, yes, just once. Well, again, it's important. This is about extensive and intensive. It's important for physicists, for Dirac, and ultimately for Laurence Schwartz, because distributions are not generalized functions. This is a fundamental mistake in terminology conceptualization, which is pointed out by many people. And Courant, and they all pointed out that, well, you know, we don't mind generalizing, but these things don't behave like functions. We can't substitute for exactly that. You see, we cannot substitute. That means what? They're not a contravariant functor. So, you know, I mean, the distributions have to be considered sort of alongside and operating on the functions, but the variance is opposite.

1:40:00 And the fact that you have this operating on means that the distributions form a module over the functions. So if you choose a particular distribution, you can consider all the possible multiples of it by functions, which gives more distributions. So that's the embedding of functions into distributions, the sense in which it's, quote, generalized, but along the arbitrary chosen. Starting point. Which, of course, I have in mind to use Lebesgue measure for that if you're in Euclidean space, but often Lebesgue measure is not relevant to relativity or continuum mechanics, it's not an invariant, and so there really is no preferred embedding of functions into distributions, and hence to call the latter generalized functions is a fundamental conceptual error which has confused a lot of people. Even though they didn't do anything wrong mathematically that I know of. The terminology was disastrous. It just leads people down the wrong path again and again. This is attempting to be, you know, pointed out, rectified by the work of Koch and Reyes on the wave equation, for example, where they say, well, the solutions of the wave equation were from the beginning supposed to be extensive quantities, not intensive ones. Because you see, in particular, if you realize that, then the so-called discontinuous is often really a very, very nice distribution. Like the Dirac distribution, if you recognize it as such, it's the most simple thing. It corresponds to the function that's evaluated at this point. It's only when you try to force it to be, to have a density with respect to a measure or something that you start getting into contradictions. But so in particular, so I always call this, you know, the Dirac map in all sorts of different contexts where you have something corresponding to this Dirac map. From the domain space, let's say x, into the space of all distributions on x, you have this natural transformation which gives you the distribution which is concentrated on the given point.

1:42:30 Each point in the domain space has a distribution that only says yes to n. And then typically, you see, then the typical, I mean, Riemann integration is essentially the approximation of one functional by another kind of functionals, where the kind of functionals are just linear combinations of these deltas, particular points that you can test. Assuming that the extensive quantities form a linear space, you can then take linear combinations of these. And so it's by having a sequence of those are supposed to approximate the Riemann integral, for example. And coming back to the set theory then, the covariant power set function is the singleton map which plays the role of the Dirac. Does that take care of what you were saying? Absolutely. Yeah, in a completely masterly way, thank you. But I think it was very useful to get it, as it were, onto the machine, so that was very useful. I'm going to stick my neck right out and risk your wrath, Bill, by saying I think that, tactically, the use of intensive and extensive may have been efficient, because what I tend to encounter when... I tried to explain that sort of to people the same way is uh this reaction of intensive extensive what the hell are you talking about ah yes this is the philosophers no philosophy oh yes isn't there something about that in hayden intense and extensive well yes and not just in hayden it's also taken up by maxwell It's important in the 19th century. Yeah, exactly. And they say, oh, that stuff, that was all the stuff which was really important to those guys in the 19th century, who we know, of course, were completely confused and had no serious ideas about mathematics, philosophy of mathematics. Because they were just hopelessly confused. It's all been shown because anybody, you know, if it's in Hegel, it must be confused because Russell showed that Hegel didn't know anything about mathematics.

1:45:00 I think you're right. In Spivak's five volumes on differential geometry, he wants to distinguish things you integrate over from things that you don't. Absolutely, absolutely. And of course, it's there. It integrals some derivatives. I'm making purely a kind of pedagogical, tactical point. If you say that this is all simply to do with the covariance and contravariant functoriality of the maps and into and out of the respective domains and the moment that you start speaking about covariance, contravariance, functoriality. Doesn't have only to do with that. No, I understand it has much more to do with that but just this has just been my experience that if you if you talk to them about if you introduce the idea to them through This general issue of, you know, how much has been missed and how to completely defective any general theory of structures that doesn't take account of conoverian, contraverian, factorial. Then they tend to listen. But if you start talking straight away about intensive and extensive, it may just be because they dismiss me as obviously an outsider and incompetent anyway. They just tend to switch off because that's some crazy stuff that's in the back of the 90s. It's not your fault, Mike. It happens to me. Well, maybe for the same reason. I've had the same experience. But I just find there's this huge distance. You can just see them. You can just see the shutter coming down, the vine switching off, intensive, extensive. Hegel, Maxwell, I mean, this must be allowed to query an interest. Hegel was wrong on it anyway. Because Hegel said they were indistinguishable. Sure. Don't mention any philosophers. That's the best. Oh, I wasn't mentioning any. Let me pick up. The essence of science is implacable hostility to ignorance. Not to the ignorant, of course, because it's our own ignorance which we first want to attack, diminish, and eliminate. But ignorance on the part of physicists, ignorance on the part of mathematicians. We have to simultaneously help to bring out enlightenment in both, you see. So, I mean, these concepts are still used in thermodynamics and engineering thermodynamics, the engineering of... Heating a house is based on the thermodynamics which heating engineers learned in the language of energy, enthalpy, and so forth, but all those within the framework of intensive and extensive.

1:47:30 So they're very, very special. I absolutely agree. First convince people that this exists, and then tell them that it has, if you want to tell them, it has to do with the two kinds of heat hunters. Because it exists because... As Newton gave a basic example, mass and volume, if you divide one by the other in some sense, what is the sense? Well, you get density. Density is an intensive quantity. This is what one should start with. Intensive quantities are ratios of extents as well. Then, of course, you have to tell a whole story about what ratio really means, which most people have never been able to do. It's calculation or something like that. But basically, you multiply density times one extensive quantity, you get another extensive quantity. So in thermodynamics you have pressure, which is intensive. This could be, for example, a ratio, depends on which constitutive relation is correct, but in some contexts it's energy divided by volume, where energy is extensive. You don't have the energy of a point, you have the energy in this room and the energy in the next room taken together makes a bigger energy and the dividing. Energy divided by volume might be pressure, for example. Temperature is also intensive rather than extensive. So there's a whole list of familiar quantities, or more or less familiar quantities, some of which are of an extensive character, some of which are of an intensive character, some of which are clearly of the character of ratios. And so then you can start thinking, well, there's a functorality because whenever I talk about these quantities, there's a certain domain of variation. The intensive ones are varying one kind of way, the extensive ones another kind of way within that object. And now, in fact, I'm changing the object when I look at a part, for example. A part is a map from a smaller thing. Or when I observe something, I'm applying a function to this domain space, so when I start worrying about transforming these quantities as they are existing, say, on the state space of the, you know, the gas in the box or whatever it might be, along either an observable function or a parameterization of a part, I will see that they have this covariant and covariant aspect.

1:50:00 I think traditionally, although extensive and intensive magnitude began, the notion was sort of named. Well, it's early. It goes back a long way. It goes back to scholastic, you know, to the scholastic development of the Middle Ages. But I think as it emerged, I mean, everybody knew during sort of basic physics in sort of high school that... You have quantities that are like density and temperature and so on. At least the way you sort of saw it was that they are defined, if you like, at points. And they don't add because there's no way of adding them up at different points. It doesn't mean anything. You take two temperatures and try to stick them together. And then on the other hand, one has... Extant quantities like mass, bond, which of course are properties of regions. Now the connection between the two is affected by the differential and integral calculus. And everybody knew in some sense that's what you pick up when you learn basic physics. Even perhaps though, and I think the mystery about it was part of the fact, well I think the fact, that the actual numbers, the pure numbers, right, Real numbers, say, or rational numbers that you actually get from measurement, are the same in both cases. I mean, just as numbers. In other words, when you blur the distinction between intensive and extensive, you end up with a temperature of 80 degrees, let's say, and a mass of 80 grams, which is really quite, these are two, the numbers are the same, but the actual quantities they represent in that case really are different, and that's why I think why... But the point is you could take two times either one. Exactly, that's right, you could take two times either one. Or pi times either one. That's right, that's right, you can multiply them. But I think that was part of the, you know, those distinctions are things you sort of pick up, and they're very natural, and you do it more or less automatically in sort of basic physics, mechanics, and so on, and then later on these distinctions get sort of lost, I mean, in some way, in the development of pure mathematics.

1:52:30 There is bias in favor of intensive quantity, at least in actual mathematical history, so that one continually tries to replace the extensive by the intensive, by some device or other, and this can This can distort the conceptual view of it. When I think of specific philosophers and where do they lose it, philosophers in mathematics, where do they lose it, one way to concentrate it, they think you can integrate a function over an interval. They don't know what a form is. They just take the standard measure for granted. They don't think it's entering into the problem. You take the integral of a function. But you can convince them that you can't take the integral of a function. You take the integral of a form. And forms vary differently than functions. The contra-variantly and the co-variantly. And you could show them that if they happen to be interested in it. Yeah, no, and actually, I was just wondering whether, in fact, because it's possibly just the order in which I read your paper's bill, but it also happens to reflect the order in which they were written. This is an extremely central and fundamental notion of distinction between intensive and extensive quantity and the way that they act respectively on categories of spaces and domains. In your San Sebastian paper, which I guess I read about ten years or more after reading the earlier The paper in the island of Festschrift on Variable Structures of Variable Quantity in Topology, and on there you make a very remarkable remark, which was one of the first things which I'm sure I could slander, which may relate to about the two different ways in which one can analyze the domain of variation of a variable and how they still tend to think of as the way of doing it, which is to see the values as the concentrated points. Can can just be seen as a special restricted case of the second where you're looking at kind of lattice of organisms from

1:55:00 parts of the quantity into parts of the domain. And I was wondering if you say a little bit more about that and also how it relates to the whole issue of the way one sees inclusion is more fundamental than the membership in the analysis. Well, you know, remember in Bolzano you were talking in the context of the whole part relations about the relationship between... Mariology precedes membership. Yes, mariology is where it precedes membership. I'm expressing it very badly, but yes, the whole issue does include and precede membership. You're a cloud of steam, right? Certainly mariology is more important than membership. Well, I'm not trying to put words in anybody's mouth, I'm just trying to get clear on, no, I don't think Bill would say that, but it would still be very interesting to have a clarification of how he does in fact see the relationship between meritology and membership, and that would actually connect with this other issue about the relationship between the two ways of analyzing variable quantity and its domains. Never recognized a distinction. They treated membership as a special case of belonging to... Of inclusion. Inclusion. Yeah. One of the extreme examples of the influence of dogmatic ideology is that somebody, not perhaps very important, but retranslated and published Dedican's works on Watson and Vassourian and so forth. Changing all the symbols. Shane there was a footnote saying, now we know that we have to distinguish epsilon from inclusion. So the whole thing is completely... Well, he's correcting Dedekinds. Where's that? What's his name? Pogorowski. Is this the one published in Maine? In Maine, yeah. About four or five years ago now? Yeah, he has more than one. Well, he actually changes his notation!

1:57:30 Changes it completely and explains it in footnotes. Now we know better, you see. This was the naive Dedekind. I realize, of course, this whole business about Singleton basically as being a sort of change of type rather than just whatever was basically introduced by Peano and Frege praised him for having done this and so this is a conspiracy which I see is the root of some of the difficulties we have. So if you think that that's a later clarification, then you correct Dedekind. You don't understand Dedekind. Everything Dedekind said probably was correct. And Russell makes this really fundamental. For Russell, this is a fundamental insight. This is what suggests type theory. This is what lets you get away from the paradoxes. Russell, in his relative position of power, not only promoted Wittgenstein, way beyond any reasonable proportion, but also became a fan of Peano at the International Congress in 1900, and was a very active copycat of this sort of thing. But also Bonnock. Bonnock, in his book on linear operators, that I studied, also with Truesdell, by the way. Where he introduces, you know, Frechet spaces and Banach spaces and various important theorems of functional analysis. He also does not use membership as distinct from inclusion. So in my book with Rosebrook, I use the word belonging, which is basically just factorization. You can consider it in sort of complete generality in a category. It might be possible to, with the same co-domain, it might be possible to express one as a composition of something with the other one, and that something would be viewed as a proof that one belongs to the other. If one of them is a monomorphism, then the proof is unique to the most one proof, so it becomes a property. But then the other, the other, the role of the other map, there are indeed sort of two special roles to the other map, one is if it's also a monomorphism, and you're just talking about inclusion, I think everyone would agree, at least everyone who understands that sub-objects are maps, not objects, would agree, would agree to that, the inclusion. But then the other thing is, the other, the other version of the, of the first map is,

2:00:00 Typically, in a category, we could go into, but typically there are a few preferred objects, like figure types. In the extreme case, it's just the one point and mass from it are elements in the narrow sense. But typically, one wants to pay special attention to figures of those preferred shapes. Usually, there's not just one. There's some small category. These might be curves in topology. You pay special attention to curves. Yeah, even though you're in a world where you have surfaces and everything else. So, in fact, I use the epsilon symbol. It's a special case of inclusion, but not a special case where the first thing is not necessarily a monomorphism. That, by the way, would be called a non-singular figure. The term non-singular is just a question whether the figure itself is a sub-object or not. But if the domain is one of these preferred figure types and you have a proof that it belongs to, for example, to a part, I use membership as a special case of a general inclusion for which the usual inclusion is a different special case. It's really just factorization. I mean, you have a notion... Bourbaki on page one could understand this. You have a multiplication, i.e. composition, hence you also have a division problem. It's not a determined operation, but since the multiplication is non-commutative, there are two division problems. And so I claim that equally important with inclusion and membership is the opposite thing. Namely, if you have two maps with the same domain... The existence of a commutative triangle is the proof that one depends on the other one. So like when you have observed quantities of statistics, for example, you often say one statistic is sufficient for another statistic. It means that given the values of one, there's some way of computing the values of the other without going back to the domain. And again, there's a special case where these things are epimorphisms, where both are epimorphisms, or where one is an epimorphism and the other one is a function as opposed to a figure, which means the codomain now is special, maybe real numbers or complex numbers.

2:02:30 Duality plays a role here too, because these supposedly fundamental notions really are real notions in mathematical practice, just as important. We're always worried about what depends on what. Of course, if you apply a function space operation, then dependency transforms into belonging. But it's more basic. You don't have to have function spaces to discuss these things. It's sort of like in any category. This is a mistake that Goldblatt makes, for example. When Goldblatt wants to discuss membership, he somehow thinks you have to have the power set countries. But no, it's much more elementary. Belonging and depending. It's just division. Two kinds of division. And in the case of the analysis of variable quantities and their domains, again one sees the... Oh, that's okay, sorry. It's a special case. Yeah, so that particular realm of ideas... Okay, what can I say? Okay, we're talking about some category of domains on which there may be defined contravariant functions of a sort of intensive, co-variant ones of a sort of extensive, relations between them, and so forth. Well... You see, I mean, after all, we didn't say these objects had points. No, no. They may have a few points. They may not have, quote, enough points. They may not even have enough figures if you've chosen too small a set of figure types. So to explain what are the points, or more generally the figures, of an object, an alternative way to perceive, or you could even say in some sense an experimental way to perceive in some highly abstract sense, is You look at it by double dualization. You look first at the intensively variable quantities, then you dualize that, and it turns into extensively variable, and there's the analog of the Dirac map, the singleton map, from one to the other.

2:05:00 Now, the idea would be that if you have a sufficiently powerful notion of extensive and intensive quantities operating in that case, then this Dirac map... Could become actually an isomorphism, could actually become an isomorphism, namely, because notice, I forgot to mention a crucial thing, since the intensive quantities are usually multiplicative, if you look at variable intensive quantities over x, say, and constant ones, then, well, a general extensive quantity is linear functional. Linear functional might be required also to preserve multiplication. Again, from intuitively using statistics, the idea of an averaging process that preserves products, what does it mean? In general, with respect to an averaging process over a domain where the domain code may have multiplication, you would say that the average of f times g It's not equal to the average of f times the average of g, because averaging is nearly linear, plus being a retraction onto the constants, as I said. And so you would actually say that these particular quantities, f and g, are independent if this equation holds. And even if most aren't, in other words, the averaging process doesn't preserve multiplication. In particular, when is a quantity independent of itself? Namely, that the average of the square is the square of the average. Well, that means the standard deviation is zero, doesn't it? Because the standard deviation is just the difference between those two, or the square root of two, however you explain it. But to say that those are equal means the standard deviation is zero. Doesn't that mean that the distribution is concentrated at a point? You see, so this intuitive reasoning is a kind of... Conjecture in a concrete context. Maybe a multiplicative linear function is actually just a point. In any case, this would be a definition of point. I can take all the multiplicative linear functions and declare those to be points. This may or may not then agree with some previously given notion of point or figure.

2:07:30 Banach raised this question actually, and he and Ulam and Tarski worked on it in a particular context, and so the basic realization was that, well, yeah, that sort of thing is really true, unless you're dealing with too large a space. You're dealing with a space so large that it'll never come up. You know, it isn't just a matter of McLean doesn't know about L for omega. The point is that in functional analysis, partial differential equations, and all the domains where in mathematics we never reach those sets that are so big that this intuitive principle about the standard deviation fails. So this is something that I've been trying to calculate lately, and I think there's a relatively simple... Categorical lemma, which would point out that the subcategory of those spaces for which such an equation holds is in fact closed under operations of exponentiation. So in that sense, it's a model of ordinary mathematics, set theory. And so now, this is... I mean, this precise context where this is the case is where we were talking about countable additivity. When I say linearity, this can mean many things. If it just means preserving unions, then it's the definition of ultraculture, you see, so that typically you would expect only finite sets. In other words, if you take one of these... One of these ghost points, or non-finite... When can you describe the elements of a set by the ultrafilters on it? Well, as a finite set, every ultrafilter is principled, so if you have an ultrafilter, you have a point. If it's in between the ultrafilters, there are principles in ultrafilters. Well, not a point. It's also a point of compactification. It is a point of compactification. It's not a point of compactification. Which, of course, brings us back to the point... But the precise meaning of compactification sort of depends on what... And what is your exact definition of extensive quantity? Exactly.

2:10:00 Also intensive. Yeah. I mean, you see, so... Yeah. Catalog additivity goes way, way, way, way, way beyond finite additivity. Yeah. In the sense that you will have this equation for sets, you know, way beyond anything ordinary. Until you get to the very big set, the so-called measurable cardinals. So anything less than a measurable cardinal, you can describe as points by countably additive ultrafilters. Exactly. That's right, they're countably additive and multiplicative, and those are just points. Again, I have great trouble with the word measurable, you see, because I would rather use it just in the opposite way. The origin of it was this, that such a functional, which is both countably additive and multiplicative, which is not coming from a point, they call the measure, in other words, the existence of such a thing, a non-representable by point functional, is a measure. Of course, to call it a measure is already misleading in another way because it's not really many. I mean, the whole point is it's multiplicative as well. That's the only way you're going to conceivably cut down from sets to points. The classical picture was what measures do we have, say, on the reals that measure every subset. Well, the principal ones, the ones that say something has measure zero if it contains its point, measure one if it contains it, measure zero otherwise. But can we imagine something so big that we've got a measure on all its subsets that isn't concentrated in a point? And that becomes a measurable cardinal. But what Bill is saying is those are the ones where the measures don't tell you as much, where the measures betray the actual elements of the set. They should have been called non-measurable cardinals. Immeasurable. Immeasurable. Measurability means you measure this. That's making it bigger rather than smaller. Yeah, that's right. Anyway, further clarification of this, I think, was given by Hisbell in 1960, where he said, OK, well now, instead of considering intensive quantities where the constants are just 2, like 2 and false, we can consider those which are some countable set, like the natural numbers.

2:12:30 So in other words, we can look at, for any space x, we can look at maps from x into the natural numbers. These are our intensive quantities. And so in particular, the constant intensive quantities are just natural numbers in an averaging functional. will be something that assigns to every map from natural to natural numbers, a particular natural number, which by the retraction is an averaging condition, is one of these points, and so you think of it as, well, consider any possible countable partition of my space, an oracle tells me which of these my point is in. But it does this in a completely natural way. Instead of talking about that kind of relativity and multiplicativity, it suffices just to consider the unary operations of any possible map from natural numbers to itself, you see, because a map's a natural, you know, if A is an algebra, then A to the X is also an algebra, you see. So a functional from A to the X into A could be required to be a homomorphism with respect to whatever operations you had. So let's take all operations on A. You want the strongest possible naturality conditions to try to force this oracle to be in the real world. And so he shows that in fact that particular way of considering, namely... Taking a bigger space of constants, but then correspondingly, instead of taking countable arities on operations, you can take only unary operations, or binary operations, no, just the unary ones, on the bigger, that will have the same effect. This is entirely the same thing as the Banach-Gulam-Tarski concept. So there you clearly are measuring. You're measuring the set by making all possible tries and then, you know, it's sort of abstractly analogous to real measuring processes. You're confronting the variable with the constant in all possible ways. So I think there's, you know, there's an interest for mathematics in all this. I mean, if mathematicians looked at all the stuff on measurable cardinals, they'd say, well, okay, you guys are... You guys are okay, bye-bye, you know. But in fact, there's an important way that this keeps coming up in mathematics, whether it be in category theory and analysis and topology or whatever, namely that this style of... Actually, there's a contravariant pair of functions, you know, the intensive quantity type.

2:15:00 And then the homomorphic extensive, part of the extensive, these are actually an adjoint pair of functors, and sort of intuitive duality about topology, monological vector spaces, and all sorts of other things, which suggests that, well, really, these adjoints should be pretty close to being inverse to each other, so these two categories are in duality. But then there's always this exception, that the spaces are too big, it's not true. So one very nice example, which is directly relevant to algebraic geometry, because algebraic geometry, in the hands of Grothendieck, is really taking up from von Neumann, Gell-Thon, and so forth, this paradigm from functional analysis, whereby the points are maps that preserve just addition and multiplication, or maybe scalar multiplication as well. So, you know, there's not a huge number of operations and not a huge arity, just two binary operations, but on, you know, a suitable ring like the real numbers, so the continuous functions, and this is something that we all studied back in the 50s, rings are continuous functions, so the point was, of course, if you take all continuous real valued functions on a space, It enjoys a much, much, much richer algebra than just rings because any map from R cubed into R will act as a ternary operation and so forth. But somehow you don't have to look at those. It's enough to look at addition and multiplication. And you get this and you get this duality. So this is kind of a third example. Instead of two with countable area operations, or instead of natural numbers with unary operations, just take the real numbers with addition and multiplication, and exactly the same pattern, and exactly the same result, namely that the mathematically reasonable duality is true if and only if there's none of these Bonac, Lula, and Cardinals. For me, it's a kind of a boundary to ordinary mathematics, and I think it's, it's, it's, you might, it goes into the possible, possible ways of actionitizing the category of categories, you see, that, of course, we want the category of finite sets to be an existing category. It's an object, not just a subcategory of the world, right? But then, the mathematical practice, by categories, anyway, the Grosvenor d, everybody, there's always this

2:17:30 This idea that, well, there's something called U, which is the universe of sets, which really is an object in the category of categories, so to be honest, object, and at the same time, arbitrary exponentiation applies to it. Usually U makes sense and so forth. So the world is a Cartesian closed category, but there is a fixed object that we'd like to dualize into once again, you see, because the structures are actually functors into the background on various kinds of domains. It's double dualization and all this. So, again, if we restrict the kind of categories that we are considering to those which are sort of commensurable with this U, we tend to get nice dualities. But again, only if there are no magical cardinals in you. So it's quite, it's quite, this is, you know, the Czech school has a whole series of results of that play. Yeah, Vopenka and so on. Vopenka, Trinkova, Adamek and so on. So quite interesting results. So it seems to me, you know, so I want to have an object in the category of categories to which I can use as a dualizing object for the, you know, not just for the real quantities, but for the categorical quantities. I don't really care if there are magical cardinals outside of that or not. They might or might not exist in the category of categories. But I definitely don't want them to be in you. That's also something to do with paradox, the avoidance of... I wonder if there is a measurable cardinal or something like that or to begin with you. Naive descriptions should be correct. A lot of naive descriptions of say spaces by dualizing in some way are correct if there is no measurable cardinal. But if there is a measurable cardinal you could violate this naive intuition. So the idea is we'll let the solids work in a universe where there isn't one, so there are naive ones. The question that you raised, I think, was one of the things you most wanted to bring out in discussion with Bill on your list of topics, I seem to recall, and this was one of the things, or at least I'm taking it, that this is the explicit content of that... I wonder if there's any connection, I mean, one of the most interesting formulations of measurable cardinals was in terms of elementary embeddings, you know, the universe into... Now I'm wondering...

2:20:00 Right? I mean, it's very beautiful. I don't know if it's Solovey or... I mean, it became folklore rather quickly, but it's very beautiful, you know, that you have a measurable cardinal. Measurable cardinal is essentially fixed points, right, of elementary... of functions which are elementary embeddings of the universe into some transitive... You know, into some transitive parts, sub-model. Now, I'm wondering if there's any connection, I mean, some way of, you know, because if you look at it in you, as you describe it, what you're saying, the non-measurable cognitive, it's really telling you something about the non-existence of certain kinds of... Can we describe a set by knowing... What embeddings do to it. That's right. Well, if there are measurable cardinals, no, because embedding can change it. If there are no measurable cardinals, of course we know a set by knowing what every embedding does to it, because it just leaves it alone. Can we assume that if you've done what amounts to some embedding, you still know what your set was? Well, yes, if there's no measurable cardinals. And no, if there are. It should be possible to make that explicit. That's just the embedding theorem. Well, they wouldn't even have to be elementary embeddings. Well, right, right. They just have to preserve, they have to be closed frontlets. They have to preserve exponentiation. Yeah, your project is to look for something simpler than elementary embedding in that statement. But if you mean elementary embedding, that is the theorem. Right. But now, so the question is, can we come up with something simpler than elementary embedding and still say that if you could do this to a set, you'd still describe the set? If and only if there are no measurable cardinals. Yeah, preserving exponentiation. This alternation of quantifiers, you know, is arbitrary alternation of quantifiers. Well, that's my mathematical practice. No, I mean, I haven't looked into the, you know, it's a good question to look into what, well, you could isolate what's needed in order to prove, yeah, I mean, what properties have to be preserved.

2:22:30 What do you think is the most important function under the embedding in order to get this result? What Colin says, I mean it because the formulation of the double duality just uses the Cartesian closed structure. So if a functor that preserves that, certainly an elementary embedding would in particular be a functor that preserves equanimation. It sort of made that, just parenthetically, I mean, that result of Koonin's, you know, that there are no, you know, which really pushed that whole program of looking at fixed points, you know, of elementary embeddings, you know, getting large cars, and Koonin, I forget who suggested that that's the thing, the fixed point of an elementary embedding of an whole universe into itself. Koonin, of course, showed that it didn't exist. I mean, there's some absolute, you know, barrier to... You know, to this kind of, can I put it, distortion, if you like, of the universe of sets. And I've often wondered, you know, what the categorical content, if you like, is. It's a beautiful theorem. It's a beautiful theorem. I tried to formulate it in second order logic. But it's telling you something categorical. I couldn't quite... I mean, I don't know what it is. I was wondering about that, too, but I'm just reminding you. So what we need is a grad student to do it, because I don't think I'm probably going to learn it. No, but basically, clearly an elementary embedding is going to give you a functor, preserve composition, and preserve the definition of the function set, so the categorical closed category. So this category U should have an endomorphism that... Yes, in that case, of course, the Kuhnian theorem would then... You know, the possible existence of measurable cardinals throws some kind of monkey wrench, if you like, into the program that you were describing, whereas whatever the equivalent would be in the case—there are no Kuhnian cardinals in this set of things, which are fixed points under—I don't know what the corresponding—if you formulate categorically, there would be presumably something similar to the situation that's presented in the case of measurable cardinals, but which is not— But for which, you know, there isn't any obstruction. I mean, the fact that there's some general fact thing that is, that principle that actually will hold, which is, right, which is going to be weaker than in the case, of course, of the other thing, corresponding thing that measurable cardinals prevents.

2:25:00 But which in the case of coding, since they don't exist, there's going to be some general feature which has not yet been identified, but which holds, right, in the case of the universe, or in the connection between analysis of quantity. You do that or the opposite. You're right. Yeah, but you see what I mean. So there may be some unknown young man or woman whose career has been made this morning. Sorry, McIntyre's been very quiet. No, I was just trying to think about the role of the ordinals. The role of the ordinals is very crucial. Well, you do need all but... Well, you could look at... Well, I'm sure that a lot of that could be formulated in terms of the, you know, Mordach and Joyal's, you know, characterization of ordinals, you know, categorical ordinals there. That's basically beside the point. Well, I guess the issue of the Kuhn thing and the actual choice is also a bit delicate. Well, I mean, there ought to be some nice way of putting it. But clearly... Yeah, no, it does happen that the... The statement itself was nothing to do with order. Yeah. No, no, that's quite true. But the... This proves that it's very much... Yeah. It goes on and on and then it collapses. Most obviously it collapses and all the rest of it. I mean... No, it's certainly very well what we're looking at. Well, the thing is, there has to be a first one. Look, the problem there is that in the case of characterizing the measurable cardinals as fixed, it's the first one. It would be interesting to see if there's someone avoiding all that and just talking in terms of sets. I mean, in other words, avoiding ordinals altogether, but I really don't know. Because you have to talk about the first one and then there's some argument, you know. As far as I remember, yeah. See, I mean, I basically don't believe in this idea of building stuff up from below.

2:27:30 Natural numbers, ordinals, or whatever, you know, it seems always that really... So, okay, so to come back, for example, to the fact that surely there should be an object in the category of categories called the category of finite sets, you know, up to equivalency. Well, what does that mean? I mean, it seems to me that there's a, see, there's a positive property that has nothing to do with building up. I mean, we dedicate finiteness. Let me see. Yeah, yeah, sure. So there's that positive property that has to do with endomorphisms and so on, nothing but below. But nonetheless, all of our experience tells us, okay, whether where we got this experience, whether from dedication. There should be one category that represents precisely those and nothing else, so there's a completely positive So analogously, you see the famous small sets of category theory. They've got to be an actual object, too, certainly. I mean, this is crazy not to let it be an object, not to allow arbitrary explanation. It's striking that the first thing Virgil said when he heard somebody named me was working on it was, OK, of course we've got to have finite types above me. By finite types, he didn't mean some kind of truncated version of it, presumably. Classes, can't be members of classes or whatever. But anyway, so we need to have something called the category of finite sets. Now, independent of Grotendieck universes, I mean, that's sort of...