Conversations FW Lawvere & D Bernardini

0:00 I think so. I think I can work that. Don't worry about that. Depends who you're dealing with. Exactly. Yeah, exactly. Behind the window. Exactly, yeah. So there's quite a bit of practice in that. And I do understand the point about David, really. I do hope I wasn't in the way. No, no. It's just that, you know, one doesn't get the chance. I'll now shut up, or at least learn what it is that I need to learn.

2:30 Yes, but the initial grounding problem is not the basic problem of mechanics. And in fact, there are many situations where you do not have even uniqueness. Describe the analogy of a stick, which is perfectly used not to. Many systems are such that the solution is not extended for all time. Heat equation, heat equation going backwards. Many times the system will explode after you've gone certain positives. So the existence and uniqueness is not a basic axiom. Maybe it's a good thing when you can get it, but you should not build it into the very fundamental theory. So I changed the formulation. I could easily, as most pure mathematicians who work on, so-called applied mathematicians who work on this kind of thing, they accept unquestioningly. And by the way, this idea of prolongation operator, one example would be, we notice that the general theory doesn't depend very much on what the so-called second order is. You could have any atom T, but this alpha could be any map from T into any other object, and you still get the fact that the category of... Prolongation algebras, isotopos, for example, the fundamental one, now that comes just from the fact that T itself has this extra right edge line, so it does not depend on the other one, so the second thing, it could even be the whole line, so you could imagine that the prolongation is just the act of solving the second order of differential equations for any initial condition and giving the whole time, past and future. This would be certainly an example.

5:00 It's just that it's not general enough, because it's only considering those things which have unique solutions, which can be continued, so that's why in the formulation that I have, you see, a motion is not itself this action, a motion is a morphism from the time to states, or to configurations, which is compatible with two, and the law itself, you see here, is dialectically, the law itself is well defined, but it only goes to second order infinitesimal, it doesn't go beyond necessarily. So you can have unique existence and uniqueness there almost tautologically by definition because that is what the differential equation is extending it further, that should stay. So that's one major change from what is commonly called a dynamical system theory. The laws of the objects and motions are maps and you do not have necessarily a group actually in certain cases. Category of group actions is related to this one, but it's not taken as the most basic one. Of course, the other major difference from almost all papers that are called, quote, dynamical systems is that they are basically not having caught up with Fibonacci and Galileo, and that they are not realizing that the state has to be analyzed as the state of becoming in some way or another. I don't mean to say this is the only way to analyze as becoming, but... Becoming could mean whole history, is Knoll's idea of mechanics, the law depends on the past history, the entire past history, so I mean this, the past history is a state of becoming, so in other words, formalize that in that way, of course losing the fact that the T is an amazing object, if it's too long it won't be.

7:30 It won't be an atom unless you, as I said, theoretically, you could always enlarge a topos, any favorite object. I mean, I have no idea about the ramifications of that idea. It's certainly true, but whether it would help, I don't know. By the way, in Newell's new foundation, the histories were now finite histories, but finite histories of arbitrary length as opposed to infinite colors. This distinction I analyzed categorically in my unpublished paper in honor of Noel's 60th birthday in 1977, I think, which was a long, long time ago. This paper has been unpublished for a long time. You mean you mentioned about Noel Fitts in some way, the paper on Noel Fitts? No, that is basically an earlier version of what we were talking about here. That was the test at the time of his retirement. He was already, I think, 73 then. He must be over 80 now. No, no, that was 93, right? His 60th birthday was already back in the 80s. The paper has been cited by Martha Boomer and some of her work. The paper by Koch and Reyes. Yes, yes, yes, yes, yes, yes, yes, oh yes, that is very basic, that's the first paper, it was developed by me, by me, by me, by me, by me, by me, by me, by me, by me, by me, by me, by me, by me, by me,

10:00 Well, it's technically important because they're using the smooth topos as a framework and then developing the distributions of contact support and so forth and so on. And it's very interesting how precisely how they formalize the idea of the fundamental solution that we need, the whole spheres that expand. Very nice vision. It doesn't need to be understood technically, but there's also a basic philosophical proposal to this that really, really, we should just work in a smooth topos because all those functions that people for hundreds of years have thought were discontinuous or not smooth, they are really extensive quantities. So they think non-smooth by experience, by practice, by hoping that they have a density with respect to volume, then it is not, the density doesn't even exist, or if the density does exist, then it could be a non-smooth function, but if you just accept it as an extensive quantity, then it's an extensive quantity in the sense of the smooth topos, on smooth functions and so on. So the question is whether this idea is really, you know, not just for pure things like the wave equation, but say in actual thermal mechanics, does it really make sense, you see, to say that actually all these qualities really are extensive and that's what makes them possible. You see what I'm saying? In other words, the tradition is strongly to try to interpret everything as a density if possible, and so many things... Many things can be smooth, but then some things are necessarily not smooth, but is it possible that actually extensive physical quantities, that really these are extensive quantities which are being misinterpreted as densities, and if we just accept them as extensive quantities, then there would be no problem. Do you think this is a reasonable program? Sure, definitely.

12:30 Because it more goes to observation, if you observe something you can discern something over some region of space, I think this makes sense to me as a representation of physical presence. I see the consideration of intensive quantity as a matter of convenience, let's say, of freeing from some dependency on the space. You consider the density of matter, let's say, mass for human volume just free on the dependence of the space. I see definitely the intensity of our drive at this point. It can be useful, maybe, for many things, but I think we should drive foundations, make it quite certain that we should stop energy or something like this. So this is a very real thing. So the techniques which they use in that particular paper, although they seem to be about one pure example. I think I should be giving a guide on how one can actually technically develop this. Yeah, actually I found it recently, so I searched it. Yes, yes. Because also they give some practical... They give an explicit expression of how the action on the art of all the extensive factors that isn't bounded and isn't signed anywhere. Yeah. How to... Actually, it's very nice because if you interpret the energy like this, you get directly to the principle. If you see any of these distributions that you evaluate as test functions, these test functions are exactly what I call a vehicle of displacement. That's what they are naturally. Because actually, what I teach in my class is a vehicle of principle. It's not just a trick. If you teach it like this, it's just a mathematical trick. You can call weak formulation of the differential equation, but it looks like a trick. Thank you for your attention.

15:00 One of the more general treatments of the principle of virtual work, in which he always interprets in terms of weak solutions and all these functional analysis tricks, so once I asked him directly, you know, I tried to explain this idea, he said, the quality is basically inexpensive, and so he doesn't want any of that, and also the fact that the space plays a role, the nature of the main space. He thought, you know, everything is really just Borel sets, countable infinity spaces and then measures, but anything could be explained with measures on Borel sets. I said, well, no, I mean, so in my paper there about categorical algebra and microphysics, there's a line which is actually... An answer to that, because I point out that the nature of the domain space for the variable quantities is important, because the domain space has got, you see, it has homotopy, it has dimensions, it has infinitesimal structure, and so forth, and that all these qualities are completely lost if you go to Borel sets, you see, all those things dissolve if you only look at algebra of Borel sets as the fundamental point. So that's connected with this idea of the extent of qualities perhaps being more basic than the intensive ones. And another question which is intimately related to these ideas is the, like I say, the body point of view. I always forget about Euler, Lagrange, you know. Truesdell says, oh, see, the other way around, Lagrange, Lagrange, I don't know which is which. But you see, there is the point of view that we sit in space and the body goes by. So really it's just some manifestation in space and the other body point of view is, well the body exists and this particular body exists and it moves through general space, but the intrinsic quantity should be intrinsic to the body and not to the space, so when people talk about the density, this is already some kind of very strange mixture of these points of view because the mass should be

17:30 The property of the body independently of its placement. It's a distribution of mass, I see. But when you talk about the density, that's relative to a certain placement. So you're taking the radon-nicodin derivative of one with respect to the other. One is variable because it's a sort of Abrams concept, actually, in that way. So, so I, you know, the question, again, I put this question to Girton at the Truesdell Memorial. So it's fundamental, the basic thing about constitutive relations is usually the contact forces. This is the essential point about continuum mechanics, particle mechanics, contact forces. So these are also extensive, you know, extensive on the boundaries. There are extensive quantities of all dimensions. I didn't discuss that in detail, but I mean, You have measures of all lower degrees, so the area measure in three-dimensional space is kind of a fundamental ingredient in all these discussions about contact forces. But that's about space. Now the body has its own internal parts which have boundaries and so forth, independently of space. It's just in the body as such, considered as a space in its own right. So the question is this, is there such a thing? Something analogous to mass, because mass is an inherent distribution of the three-dimensional, which when you place the body in space, you push that forward along the placement, and you get a mass-dimensional distribution in space from every particular placement, just by pushing forward along the placement itself as a mass, the intrinsic mass on the body. This is a very, a question. He understood the question, but he didn't know the answer.

20:00 Surface tension, you know, that kind of thing. He did not have a clear idea about this, whether there could be somehow a material phenomenon. In other words, to have something like area. I mean, not actual, I mean, in other words, the area of, you place the body, and of course all these boundaries inside it, they have area. Are these things coming as push-forwards of something in the body that does not depend on placement, in the way that the mass adjusts in this way to do every such a thing, in order to explain the contact forces? Maybe there is an interesting point of human beings that is not fixed. Are you aware of the mathematics in the works of Segarra? Segarra is a great scientist. Oh, yes. She was at Blacksburg. I think it's a new way of thinking, it can be interesting. I wonder if you heard about this. I think, yeah, I think I discussed this with him. I have a paper. No, no, it's the most recent. Yeah, I have a paper. There are a number of different types of physics, such as quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics,

22:30 Geometric, I didn't mention geometric rather than, so I cannot, the general notion. I have not seen this one. This is a resolution. Yes, very nice. You need to find the time and the length. Extensive quantity is first the density and then the equation. Without analyzing what is the measure with respect to which you are integrating it. Time is data. Of course, there are also extensive quantities on time. Yeah. And I didn't mention those things. I mentioned in my paper called Categories of Space and the Quantities, I gave this example of the definition of sojourn, in other words, a particular path through space, so if you push forward the duration, an extensive quantity of duration. It makes sense to find the timeline. And you get a distribution on the space, which physical dimension is still time, but you measure each part of space, but how long did I stay there? I mean, there was one little fragment, but actually in the formulation of thermodynamics in Rochester, it's in a book from 1982.

25:00 Feinberg, excuse me, yeah. Feinberg, right. There they actually, I mean, they speak about distributions on time crossed with some variables, so that the basic revolution of energy and entropy and so forth is seen as simply a distribution on the product space and then factored in time, you know, and during how much heat was added, you see, during an interval of time. It's distributional, both temporally and spatially, so there are several examples over time as well. It's quite interesting. In spite of that basic flaw, probably one could easily eliminate it. Yeah. Also, maybe this is going to be, if you formulate this thing technically, maybe it's... Yeah. In this book, I have somebody on for me to go into detail. So I can understand the general, but I'm not involved in the details. Yes. By the way, this Cauchy's theorem. I pointed that out already to Kurt in 1974. Actually, this Cauchy's theorem is well known to Hasler Whitney. Whitney has this book called Theometric Integration Theories. There's a dual version of what later came to be called Geometric Integration Theory. In the 1950 International Congress of Mathematics, which was held in Massachusetts, 1950, Hasler-Whitney has sort of a preliminary version of his book on Geometric Integration Theory. Among other things, he actually has what's called Cauchy's Luck Theorem, essentially. In the small, these things become linear, even though you didn't quite expect they would.

27:30 So I told Gerton that, and then a couple of years later, at the next meeting, he said, well, I looked it up, and yes, you were right. So really, it is something which is known also in pure mathematics. But again, the language is sufficiently different that maybe I might be the only person who noticed that fact. I mean, it's possible. It's conceivable. Yes, well, anyway, I think that we may define something at the end, if we can redefine the integral curve, because it's very undefined, what is the material point in terms of the integral curve of the balance. Yes, so the consequence of the balance, right? Yes, in general, very interesting, yes. I don't actually have the second edition of 2000. It's quite different from what you might have heard in the past. Geometry of Cauchy's book for 2000. This archive, unfortunately, is one of many journals that got cancelled by our library. I hope I can get a copy of this paper. Maybe I can send you the PDF if you want. Yes, please do. Please do. Please do. We have electronic. You have it electronically. Yes, so yes, please, please do. Very good.

30:00 I was very excited about some lady in Berkeley, some lady in Berkeley, I don't remember the name, but I remember that she was female, in Berkeley, California, who had written two or three papers about this, but I see he's not citing that at all here, so maybe he decided to, because I thought that was a little bit dubious. Some of them didn't seem quite right to me. Maybe he decided the same. She was this woman, by the way, who was, she was denied tenure and so she made a big case that it was because she was a woman and so she ultimately got tenure and this kind of thing. But I didn't know that until later. I looked at the actual papers and they didn't seem to be quite, you know, geometric measure theory, but not quite. I would appreciate it very much. Another point about this incident, you know probably the book of Shilali. David Owen told me that it is the new Bible. In other words, the old Bible was Hadut al-Khuzik. Yeah, I heard about it. It's not a story, it's the slogan. Yeah, the slogan, not a story. So naturally, immediately I bought this book. And then, by mistake, I left it out in the rain, so it became totally water-soaked. So I had to get another copy, anyway. And now, meanwhile, I thought I'd figure out a way to dry out the pages one at a time, so that it's actually the first copy is also legible, so I have two copies. Anyway, so there is a section there where he talks about, you know, as a general formulation of thermodynamics. That you have, you have, he wants to derive temperature and something else as rather indicative derivatives of measures.

32:30 Sorry, he doesn't do that. He assumes density, he assumes that these things are densities, but then he integrates to get extensive quantities. There are various choices of primitives, but you go from these extensive quantities as, what's the word, when Fusdell speaks about the principle of equi-presence. It means, you know, that there are certain independent and certain dependent, but whatever, I mean, it can be divided up in various ways, but anyway, this formulation, and it really, in essence, the independent variables and the dependent variables are both extensors, which is good, from the point of view that you said, but he doesn't quite formulate it. He formulates it in terms of densities, and then, you see, so there is a kind of, and also you want to say that But, you know, it's one kind of material, so that there is really just one law for each little piece. The laws are the same. All of the laws are the same. So you have this function, the energy is a function of temperature and something else, I forget exactly. No, temperature should come after. But anyway, so there's merely a function of point variables, which you feed into the... I'm not explaining it very well, but my question is this, and I don't know general formulation, if you were to postulate directly a law in terms of introducing also mass distribution, so Feinberg emphasized the idea of mass distribution, to which Coleman says mass has nothing to do with it.

35:00 I don't think that that could be right, you see, I mean, mass, we're using mass as a measure of the kind of amplitude or the importance of certain quantities in certain regions. It's a reasonable measure of such a... It's the most reasonable, since it's a measure of matter, and it's all about matter, most reasonable measure of amplitude, so I don't think Coleman was right in saying mass has nothing to say about it, but, so, in other words, so if you were to introduce mass distribution as well as energy and entropy and so on, and express this law as just a, you know, it's an input or distributions and output distribution, What are the necessary and sufficient conditions on such an extensive to extensive law that it should be possible to express it as, well, everything is absolutely contingent with respect to mass, we have a law which acts on the functions, and as when it's acting on the functions, it acts only on the function values, you know, functions of points. So the law acting only on function values is one way of expressing the idea that it's all one kind of material. Every part satisfies the same law on these function values. So, what are the necessary conditions? In terms just of the transformation, extensive to extensive, that would follow from the fact that it is, you know, Global law is defined only for values, and then you blow it back up again by integrating with respect to math. So that's some kind of a restriction on the whole process. And it's at least related to a kind of condition that you want to say, where you want to be able to express the idea that the whole body is made of the same material. And this is one way of expressing it, by saying that the law intent, the intent of the law... It really only depends on the values of the functions and not on the functionality. But it would seem, well that should make sense just in terms of the extent of quantum expression. So you would see a material as an internal dependent on the space?

37:30 Well, no, let me say on mass, you see. You can make it entirely a body, a body story. It has nothing to do with placement. Yeah. Because you think they're absolutely continuous with respect to mass. So the intensive aspect is with respect to mass, the uniformity, the idea that the law rule alone depends on the value of temperature and density and should be possible to character. You know, it's one of those abstract mathematical... Probably characterized as this thing could arise from that if known in itself as such and such property on the basis of the key ingredients in Shilohi's book. I mean I was at the very beginning but at a certain point it starts with this kind of setup and then this whole discussion of thermodynamics is in those terms. Yeah, and I knew this, I didn't see it in this direction. I saw him there at the, well, of course, at the Frisco Memorial, and he's told me he would be spending more time there. At Calvary. At Calvary. But I was hoping to see him myself, but I didn't do it today. A lot of talk about mathematics. Absolutely. I'm excited to hear some of these questions. Yes, well here you have them and I've been preventing you, you see, by this.

40:00 I've been preventing you from asking. No, but we covered this. These ones we... We've got the questions. We've got the... We've got one of the questions. One of the points, so... Okay, so it's... Italian news. Italian news. Yes, yes. No, I was trying to make a point of where I am in interpreting the thing. So, let's see this course in which, at some point, we begin to define the laws and then the categories of laws, Space and law. And then, morphism, which is motion, global motion. This would be, in classical terms, I see it as an equation. But then comes the solution of the particular problem, which comes later. Because, although I interpret the first step... When you write this alpha star, it's like writing the equations of the equation without the equations of the equations of the equations of the equations of the equations of the equations of the equations of the equations of the equations of the equations of the equations of In a slightly longer path, a shorter path, this banality is the most specific structure of the law, which is not implicit in the functor, it depends, it is a more fundamental structure.

42:30 Do they all contain the equation of the equation of the equation of the equation of the equation of the equation of the equation of the But this already contains a specific... For example, Galileo's law of the fall of a particle. This is a particular X. But the harmonic oscillator is another choice. Maybe it's the same view. But there are no constitutive equations. The space X, I say X, but U means X to the power of Y, this X could be A to the power of B. When we do another dialectic division, the state is the contradiction between the body and the space, the physical space. Morphism that puts B and I in a certain point. This morphism is a point. So the constitution is implicit in itself, in the sense that in the case where X is this type of thing, the constitutional reports... They give you, at least, the main ingredient in the definition of Xi.

45:00 You can't see it here because we don't say that X is like this. So it's in there. Yes, it's in there. But in fact it's in there, it's not the case that... The problem is that a certain level of abstraction in a single term contains an infinity of information. But we have already separated the concept. This is the dynamic equation of equations. Yes, but we have done it in such a way that we are not trying to say the greatest things. The equations of a particle, let's say, to get to a question for a body. No, instead, X has already thought in such a general way that X could also be the states of the electromagnetic field, or something like that. It's already a complete thing, but we can analyze these things more. Contradictionally, frat, corporo, et, spatio, frat, campo, et spatio, etc. In questi termini, annonezzare più specificamente o ricchiere città degli esempi di legge di Giuseppe Nino. So, after the specific problem, we come to the problem of solving the problem of the internal circumference. So, let's say, starting in a broad sense, let's say, conditions, one finds the solution. Also, to use the concept of the word, movement, for each movement, Yes, it is a morphism of the category, but with a very specific domain, that is, the pure time interval between time and nothing else. As I said, in fact, it is a complicated thing to explain time in the sense of second grade, not in the first grade.

47:30 But the same, despite this problem, is pure time in this sense. The movement is a morphism that preserves the law of time in contrast to the particular law of our body, our field, etc. But I think it is very natural to consider other general domains. In many cases, one should consider not just one movement, but the family. This would be a morphism with a more complex domain. It contains parameters not only for the same time, but also for various movements in the family. So, every morphism can be considered. On the other hand, the transformation of differential equations, many times, to this kind of, instead of directly considering the solution, a transformation of a differential equation is made into another, such that a solution of the first is carried into a solution of the second. So this is a movement, perhaps in the simple sense, We make the composition with these transformations in order to get to a new movement in the simple sense of the word, but also not to say that this is the composition of movements in general, in the same way that The elements of space, according to Voltaire, are not only points, but also curves, superficies, vectors, tangents, they are also elements. So, to use elements only for points is a mistake. In the same sense, the word movement means, in a way, respect.

50:00 We always talk about morphisms with a very strict rule, so why not also say that, at least much more generally, morphisms are also movements. The whole history of transformations of non-denominational differential equations can be transformed many times. It is important that even if all the parameters are changed, the law is preserved. In order not to lose the essence of the content. I think that this can also be interpreted from the scale. Here perhaps it is a little different from the interpretation of the scale that is here. It is a little different from the interpretation of the Mann-Caster. Here perhaps there is a difference because Mann-Caster considers the dynamics of the two scales. In my opinion, as it seems to me to emerge from here, there is a separate dynamic, and here I have actually this article that I have read, this was written on the archive some time ago, in which they make differential equations in which, over time, there is the resolution, there is a whole axiom. This is what I wanted to find out in my mind. There are some differential equations that absolutely seem to be those of time, but there is the resolution of ancient time and ancient bodies. There are images that are mapped on the Earth, on the greyscale, and then here we have the relations of various types of rest. But it seems to me, in this sense, it seems to me that This is the same concept in a different sense, so I see it a bit like a part of... No, in a different sense, I don't know. In fact, as you said, this is a process of September...

52:30 This is the explanation for everything. For example, the fact that the university has much less money. So, for example, many scientists don't come here anymore. For example, the fundamental one I haven't seen for more than two years. This one is better. We probably have it, but I just haven't seen it. It's ok, because there is a title that is easy for someone not to know. Yes, yes, yes. You probably know it too, but instead... There are other engineers. But in the end, in my opinion, this is a way of seeing what you want. Of course, the details are not necessary. No, no, but if you can give me the instructions... Yes, of course, I can do that. But the important thing is that here there is the dynamics with respect to the scale, which I didn't want to do. So I think that this is, in a way, similar. You can interpret it in a way. Yes, yes. It's completely nebulous, but the process of processing becomes more and more expensive and you think, ah, it's exactly the same parameter that parameters have to become. There is a process in time when it is realized, but the concept is that the degree of precision is variability. Very well. Almost every image on the Internet, at first it's not clear, but after this type of engineering that they want to describe, it shows a picture. Of course, this is probably a picture of the Virgin, I don't know. But you see the difference. It's a process, you see, to go from the less clear to the more clear.

55:00 Or perhaps to go from the more clear to the less clear, and then find a section of that process, and so on. But then I mean, on how to become a good... And also people becoming perceptually more clear. Yes, yes. Let me write down the... Okay. Well, I think it's probably a different one. I think the other one is the only one. Oh, there is. Yeah, then you're also introduced to time because of the movies.

57:30 Yes, exactly the interplay. Yes, yes, exactly the interplay. Yes, yes. So, I think that, given the importance of the cinema, cementing the connections between collective thinking and individual thinking, anything like this is well worth trying to understand, I think. Because, yes, it's very exciting. Very, very exciting. Well, usually the computer science point of view would be something discretized, but they seem to be something understood from a smooth point of view and then discretized. Always one important angle to consider. I don't like this interaction between the two of us. Excuse me. I'm not saying a word, I'm not saying anything true, I'll just have a look through some of them while you're talking. I could follow the gist of the Italians, but it is out of my imagination. What were the papers you weren't sure this morning? If you don't mind me asking, what were the stuff that you weren't sure this morning? We didn't know. No, we didn't. Oh, you didn't know, okay. No, no, the main one was, yeah, the paper we were discussing. Yeah, we were talking about that. Yeah, mainly we were discussing that. That was the point you were talking about, yes. Yes, yes, yes. Because I really would like to get a hold on this, but I would be very interested in hearing more about Binzlein and his analysis, because he came into so many of the ideas in the Panic Africa in particular paper. So we have clearly come to the essence of functional analysis. You were saying that you were talking a little about the nuclear phases program at Brody.

1:00:00 Yeah, but this is really very too far for me. I cannot... Yeah, I didn't know that I can... Well, if you were getting taught again, I would be very interested. I was not able to follow the technology. But this is extremely interesting. I'm going to try and fix it with the idea that... At this level, I'm going to be speaking about quantum mechanics and... It's clear to build up a concept of the question is just that's why we need a unique way at the same time. So for him, it didn't matter if it took 2,000 pages to define what is topos, nonetheless he can see all the general relationships that there should be between topos as well at the same time maintain all this baggage. So he is a very unique individual in that way.

1:07:30 The elementary topos, Maverick and Tierney, is a qualitative improvement over that for everybody in the world except Rodman. He doesn't so much need it. Everybody else needs it. He didn't need it. Which is very fortunate, because otherwise we couldn't have created the theory. So it's this... Normally we need to have some very efficient way of synthesizing so that we can continue with general reasoning without... Without always carrying along the baggage of the construction. This is what I learned actually from Eilenberg and Steenot's book on algebraic psychology. The construction of the homology functors at that time. There were many homology functors. They all required many, many, many details. And the only way to think about these things was actually to carry along the baggage and sort the baggage again every time. Making the details not necessary, nonetheless one could always unpack the thing and do the details if required, so there was this dialectical cast in my mind as I was walking back and forth here. It may not be of any interest to you.

1:10:00 I don't think they do think so, sir. I'm telling you that, just saying to them, very briefly, that when you were speaking this morning, that you reverted again to this point about how deeply Gromitik's early work in functional analysis influenced his approach to the use of categories. There was not only functional analysis in general of which he was the star. I mean he was thought by functional analysts in 1952 to be really the most advanced genius of that field. Even today some of his results are recorded and people think he was a functional analyst. But even within that context, he treated this contradiction between the smooth and analytic spaces of functions and distributions on the one hand and the approximation to them by Balan spaces or Hilbert spaces on the other hand in a way that nobody else had really done. In order to explain this one result of Schwartz, which is also in the 1950 International Congress work with me, it's a body worth looking at, Schwartz's kernel theorem was true because of the very special nature of the spaces of distributions and functions, which is not at all, nothing like it could ever be true for Bannon spaces. Even within functional analysis, he was already looking at analytic and smooth spaces in a new way. Then he attacked the problem of what is the dual space of the space of analytic functions in an open set. What Voltaire and Fantasia called analytic functionals. There was a whole school in the 30s of analytic functionals. Max Zorn, the famous Zorn, he studied these things. And Danish people and Swiss people and several, many Italians studied these so-called analytic functions, which Denae said couldn't possibly exist because there was no, there was no assumption of the default category of topological spaces, you see, but, but, but Rodenby looked at that stuff again, you see, and he saw, well, the, the, the, the linear rule of the space of all analytic functions on an open set.

1:12:30 These can be represented, again, as analytic functions, but on a complementary set, not all analytic functions, but analytic functions of exponential growth, and the reason why, you see, in this case, you start with intensive quantities, you take the dual, which are the extensive quantities on this region, but you see that the extensive quantities on this region can actually be seen as intensive quantities on the complement. And why is that? You have the Cauchy kernel, you know, 1 over z minus w, so if z ranges over this space and w over that space, these are complementary, so z minus w always is non-zero, so 1 over z minus w exists, so basically if you take a function of two variables, z and w, and divide by w minus z, then you can use that as a kernel. When you multiply that by any function of W, you get a functional and a function of Z. This was the concrete representation. So now you're getting into the structure of the shapes of the domains in this n-dimensional complex space. In other words, you're getting into analytic geometry. Several complex variables are analytic manifolds and this type of thing so in other words he went from functional analysis to analytic functional analysis to the analytic manifolds which underlie the variable extensive and intensive quantities and so now now that he's in analytic spaces he comes across the classical theorem that compact analytic spaces are algebraic spaces And that's why he gets into algebraic geometry. So with no abstract leap from abstract functional analysis to abstract algebraic geometry, he went through very definite concrete steps through which the complex analysis, complex geometry, extends upon these as a key. I don't know of any history, not even Colin. No, Colin's never come. I've told this to Colin dozens of times. This is so important. And he keeps saying that the prohibitive analytic functions only became important in the late 60s and only the formal one.

1:15:00 You see, in fact, the first TOFOS, not by name, but in fact, was in 1960, the Cartan seminar on analytic spaces. Where Grotendieck constructed the Grotopos, or analytic spaces. In other words, a topos which had the character of a category of all analytic spaces, not just the ital sheaves on one analytic space, which of course also was very important, and he studied that later, but he had both the Grot and Petit in mind, and even the Grot came first! See, and no historian, not even Colin, keeps telling and doesn't get it. I mean, it's a crucial part of the whole... Well, it's certainly neglected the distinction between the batty and the crow that is distorted... The role of the analytic case, you see, the role of the analytic case and the algebraic is just merely the compact case of analytic and real-real theorem and all those things, you see, they're just not incorporated into the whole story. You know, the idea that extensive quantities could be... Extensive quantities on X, let's say, could be intensive quantities on a different space. See, this is rather different from the idea that they're all absolutely continuous with respect to some measure on a given space or something like this. It still uses an underlying measure, but in quite a different way. In fact, you find a path around the original domain and you integrate just along that the Cauchy integral formula. The Fourier transform, the thing which I started this whole investigation in Cruthdale's course in 1959, it too has that character because the Fourier transform on the category of group objects, like a union of group objects, it goes from, given in group G, then there are two things. The distributions on G, so harm of R to the GR, let's say, or maybe R is the complex number, I don't know.

1:17:30 It would just be the specific idea of extensive quantities on G as space and so on. But there's also the character group, the homeworkisms of groups from G into the unit circle. Now that's already a contrarian functor. So now if you take the functions, the intensive quantities on G-star, G-star is the character group, that's again a covariant function of G. So the Fourier transform goes from extensive quantities on G into intensive quantities on G-star, and it's a natural transformation. If you substitute along any homomorphism of groups... The nature of the transform, too, is very simple, because you simply interpret that these characters are a particular case of functions, so you can integrate them, and that's the usual form of the transform of mu. Thank you for your attention and I look forward to seeing you again in the next lecture. Natural transformations of extensive and intensive quantities. It would all be incredibly complicated if I didn't have a Cartesian closed category. It's very simple then. See, in analysis you're supposed to constantly worry. Maybe this doesn't converge. Maybe I can't do that. Maybe I can't do that. This is the, this is what you were, this is the indoctrination. Maybe my operands are on magic. Yes, yes. This is the indoctrination that we get. So the fact that these constructions are simple and always defined is obscured. Obviously to really study this thing you have to make all kinds of special assumptions and prove deep theorems and so forth. Simple construction. And this was never made clear to these students. Forty-five years later, I was in the field.

1:20:00 I put what Pogstock gives off to you. After I first started to realize, what are they doing to me in these phenomena? And then I had the opportunity to express myself a little bit in Grisdell's course. And of course, this is all fed into the reigning, all of the framework of mainstream physics. Oh yes, Fourier transform plays a key role in particular and more generally. The whole idea of unbounded operators is totally bogus, you see. It's just because of some operatory assumption. Well, you have to work with Hilbert spaces as basic objects. And it's just crazy. The idea of unbounded, you see. Well, this is something that's really illegitimate from the point of view of the framework. It says that. Of course, they deal with it in a rigorous way. I'm not saying they don't, but the terminology already is a twisted situation. Actually, an unbounded operator does live in the category of Hilbert spaces. An unbounded operator from A to B is really a span, you see, of two bounded operators from a third space to each of A and B, C to A and C to B. Where the one from C to A is actually a bijection on points, not an isomorphism. It's only continuous in one direction. For example, the derivative operator. For the derivative operator, you define a third Hilbert space in which the typical norm is the integral of the square plus the integral of the square of the derivative. That's another Hilbert space and the projection. Where the inclusion, in a sense, back to the original space of S, is continuous in that direction, but of course not from the other direction. But then the operation of taking the derivative is just the other projections. And of course that one is not, you know, not called bijection. The span in which the first leg is an isomorphism in some more abstract category is a typical way of...

1:22:30 For example, the morphisms in O-minimal models are the same sort of thing. We speak about, they say, what they say is that the map is a relation which is in the same category, which is piecewise, polynomial, or whatever it is, and it's provably a function, see this, putting it in symptoms. Subjective terms that way. But actually it's just a third object, a span, in relation with the property that the projection under the first factor is an isomorphism of abstract sets, or an isomorphism in some lower, less structured category, so that in that lower category, it is just a function from the domain to the codomain, but unbounded operators are actually bounded spans with special property on the domain, and likewise the... The piecewise linear, piecewise polynomial maps are the same sort of thing, polynomial spans, et cetera. You can also say mono-epi, instead of saying that it's isomorphism in a lower category. In the case of Hilbert spaces, you simply say, well, this map is both a monomorphism and an epimorphism in the sense of cancellation within the category of Hilbert spaces and bounded maps. But that doesn't imply that it's invertible, a span with a mono-epi first leg and an arbitrary second leg. Now there is a paper I've never grasped completely of Wilfred Schmidt. This is really about lead group representations. But you see, of course, the representation of a certain lead group G in a cohesion linear category is itself another cohesion linear category. The same idea applies if you have the group G or you don't have it, I mean it's not really the issue, that these monoepis, you see, that you could simply invert them by the method of Gabriel Zisman, category fractions, you look at the maps, you have a category which is not very exact, it's not like a topos or an abelian category, maybe it's a linear category but not an abelian one, so you have these maps which are both mono and epi, you put in a formal inverse for them.

1:25:00 There's this calculus of fractions that you can talk about composing fractions of arbitrary maps and then reverse upon these good maps and then forward and reverse again. These words, these so-called fractions, you compose those in an obvious way of stringing them together in its relation on a category of fractions. But according to Wilford Smith, that process would seem quite implausible to many of us. It actually gives a very good category. It gives you a kind of canonical representative of these many different function spaces that functional analysts consider, you know, L3, L5 and a half, and, you know, all these various levels and degrees of smoothness, all the, all the Sobolev spaces that are approximating linear space or something like that, they sort of all tend to collapse together because those, those bonding maps. There are a number of ways you can convert these terms. You're getting sort of the, that indeed the nuclear spaces are the canonical representatives of these whole equivalence classes, or by equivalent classes, I mean isomorphic in the category of fractions. If you allow yourself, you know, monorepies to be considered as invertible, you get many much coarser isomorphism classes. Some of these coarse classes have very good and very nice representatives. Wilfrid Schmitt did this for the purpose of studying Hebrew representations. This is a very important example, but it just seemed at the first glance almost mind-blowing that this could be the case, because, you know, the monolithies, when they're not already invertible, it seems like they're sort of throwing away lots and lots of information, but actually they're not. Well, for one thing, it's clear you could never get a finite-dimensional space isomorphic to an infinite-dimensional space that way, so certainly some distinction is being maintained, but actually, in some sense, all the...

1:27:30 Yes, you know a lot of thinking in this framework, but, you know, it's very difficult to get to the last part. I'm just going to make a remark about the fundamental theorem of calculus, but you already talked about what that is. Yeah, the fundamental theorem of calculus is a crucial example of the book of Peet on nuclear space. Somewhere in there you find that actually the fundamental theorem of calculus, which is about indepth and integrals, In a certain sense, this already contains the whole idea of nuclear space. It's a well-known fact that integration is a very thin kind of operator, right? It's like a smoothing operator, differentiations. So the actual fundamental theorem of calculus itself can be interpreted to say that the C infinity functions on the line is nuclear. The sequence of Bonnock spaces with the bonding mass at your compact and all that, in a way, it's much more concrete. You have more to add to your question? No, well, no, anyway, no, I wanted to comment that this is the general picture. We are trying to think a little if this makes sense, how we can take a step forward to do something like this. For example, to try to recognize canonically a structure of the balance inside. And then, the fundamental thing is the discourse of invariance. This is a broad discourse.

1:30:00 Invariance as a guide to choose a taxi, probably. Because sometimes it comes to mind. You can define a category of the observer, I don't know if this will make sense, but there are various things that can be done, that is, to have a point to continue, to go on, if this is a concept more or less that makes sense, I say, where to attack it. Right, 6.30 exactly, so it's going to take a couple of hours, and then we'll see you in contract, and then over, and then we'll see you in contract, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, and then over, It ties up at a deep level, but one which is fairly clear after hearing Bill expose his ideas systematically a few times to Witten.

1:32:30 Other, you know, recurrent themes in his work, like the same topology program and the role of the component space, and this idea of getting sets as the category of discrete and co-discrete spaces out of homology and so on, those are underlying this deep geometrical framework, which obviously connects also with the requirement of local Cartesian closure, or, well, not necessarily local, but Cartesian closure on the categories. Did you have a chance to speak at all to Colin, Colin O'Clarty, while you were here? Yeah. Well, Colin is a very, very bright person. He's a hugely gifted expert. But he does, and he is, as it were, the, if you like, the kind of official historian of... Category theory, well not official, but he is very much, the thing is he knows far more than people in philosophy departments normally know. He's published good papers in category theory. He's published nice papers on synthetic inferences, on geometry, in logic, on topos of smooth spaces. He's pointed out a lot of things, but he does have this, I would say, slightly... His structuralist, pragmatist view of the development of the Sun doesn't really see... He thinks that there's nothing as it were below the level of cross-relative consistency proofs to hold it together. He distrusts this deep philosophical motivation of Bill. Or perhaps he thinks he's too sophisticated a philosopher to kind of take, because he seems to connect up with all these Aristotelian models, I think he's very mistaken because you can of course have the individual insights if you know a great deal about very many areas of mathematics, but I think that to have the synoptic vision is very important. I think that is what Colin slightly tends to miss. And also, because as a historian he spends a lot of time talking to the French, to the algebraic geometers, his take on the history of category theory is very much through algebraic geometry, through what Cartier and Dubonnet and Deligny and people tell him, and I think he does miss out on that whole aspect of Grothendieck's early work in functional analysis. It's very revealing to me that...

1:35:00 Both Bill and Grotendieck began in functional analysis and that in many respects I think the problems in functional analysis are deeper than perhaps than in anywhere else in the whole of mathematics except perhaps in pure algebraic number theory. They are actually even deeper than algebraic geometry which is not to say that algebraic geometry is not a very deep... And certainly all that stuff about geometric Galois theories is tremendously important and deep and beautiful and how it ties up with this sort of algebraic viewpoint on logic. But I still think the functional analysis is just as important. It tends to be the neglected part of the story. I think it is because this is the most structured. There is a lot of structured mix. This is mainly one reason, because you have the... Many kinds of structures. Sure, but there are connecting themes all the way through. Yeah, but maybe the path that he was describing in Brittany, he was trying to separate all the different structures that mix in functional mathematics. Yes, absolutely, absolutely. This was, he maybe arrived at Topos to really separate... Yes, yes, yes, I think that's absolutely right. I think that's a very... I think that's a very acute insight, I think that was a very important part, but it didn't just come out of scheme theory, it didn't just come out of the demands of the program for proving the vowel conjectures, it didn't just come out of the development of algebraic geometry, there were other important things feeding into it as well, including these deep problems in functional analysis, and this insight that the category of topological space of the gender, and what Bill calls the default category of general topology, has... But precisely because of the way that it's built on, set theory, has incorporated this, well, and this is to speak in varying language, but really this ineffective and redundant generality, which is allowed for, well, which partly, of course, also comes from the same source that the... Pathological functions of analysis come from mainly the assumption of completed discrete infinity, the assumption that no one can have completed discrete infinity, you know, the infinite trippiano algebras.

1:37:30 In other words, they're looking at the EMBO maps in the wrong category. I think that's very interesting, but I'd love to know more about it. But tell me a little more about what you were... I'm sorry, I shouldn't press this button. Tell me a little bit more about what you were talking about this morning. Things about functionalities I really have to admit that I'm not able to follow in detail, so I just had a spot or the stress of importance of not only of topology but also of bornology, so he was complaining of how people not only neglect but also use without acknowledging. This structure of boundaries I told them, trying to explain to me how the notion of boundaries in Kim's mix, that in Hilbert spaces they are put together. Yes, they are put together. Yeah, but here things are really, of course, really separated, and he was making the point that if you start with this clearly, On a topos, you separate into just the topological structure and the bottomological structure, then you can make clear this. The other thing about the Hilbert space construction is that it mixes the contravariant and co-variant functoriality. It does seem to be that the intensive and extensive aspects of the spaces are somehow... What I'd love to understand... I'm sorry, but continue, continue with what you were saying. I want you to tell me what you were talking about. I don't want you to hear me. Go on. This is just the general thing, because the details of course. And if I tell you the construction about tonic spaces that approximate the things, then I lost the point, of course, So you see that Hilbert space is a kind of inverse limit of, no, inverse space, so, yeah, but then I was lost, of course, so the main point was this, trying to separate this.

1:40:00 Five. Five. Yes, sorry, say again about the inverse limits in Banach spaces? I'm sorry, I wasn't, I was distracted by... No, really, I'm trying to reconstruct, but I'm not sure I got the point. This is what I remember. Last year, you were saying that if you consider a limit of Hilbert space, it's varying both things, the domain of definition and the number of derivatives. So if you take the limit of the direct in the category of Hilbert space, the index of parameter, the domain of definition, and the number of the number, let's say the multi-indexes, the incisor, in the sense which you're going to do, and how many derivatives you can see yourself. Yes, the measure of smoothness of the spaces, yes, in some sense, yes, then. And then if you take this limb, you get this big category of spaces that maybe, I can't think of a name in my book, so this limb... Not Sobolev? No, not Sobolev. No, maybe the point was that in this big space... But this is not Banach space, because there is no node there, seen as a limit also. I don't know how that happened to me. I lost the pen. Very high level. From my notion of fractional light, this is the moon. Not the only one. You're not aware of it. This is the main lesson I learned today. That, for me, is enough. The importance of this chronology, because it seems some... We could go through there. The sky is so noisy.

1:42:30 No, I know it's a very important recurrent theme with Bill of chronological spaces. Whenever he speaks about the continuum, he always stresses that the category of chronological spaces is one important aspect of the continuum. Yeah, it's a super functional analysis. I'm studying the basics, so it's... That's a very long time since I did even that, so you're having... From here... Yes, as you say, it is your big after jump as high as the moon, when you've just learned to... I'm studying, just... Oh, no, that's... But anyway, I think I will keep the track in my mind when I read it. So you mainly spoke this morning about functional analysis, about the... No, this was the part of the... Because he arrived to say this, because he sees this as a way to see intensity and the extensive quantity, but once again on this I didn't catch exactly the connection. It is clearly, I mean, the most profound single kind of organizing notion, this contrast between the structure of categories of space, because category of space is linear, extensive, left extensive, you know, linear categories, which tend to be abelian, and categories of intensive and extensive quantity. I mean, he just sees, he, I mean, my impression is that he sees the whole, virtually all, all structure in mathematics as fitting one within that framework, one. Well, certainly logic fits within that framework just as the study of the roots and supports of intensive quantity, so that in itself is an extraordinary idea. Numbers, this Grossmannian idea of numbers as being ratios of intensive quantity, I would love to understand more. And this all connects with what it seems to me philosophically is the most profound aspect which comes from this. This perception that once you have algebra and therefore can do calculations, you think of universal constructions rather than in terms of the structure of domains and co-domains as consisting of objects in some sense given in advance to be the values of the variables, what the set theory does. Hence the revival of this. When people hear it for the first time, they think it must be surely this very old-fashioned notion of quantity.

1:45:00 Surely nobody knows since almost 200 years that mathematics is about quantities in that sense, because he misses completely the depth of the concepts. And, I mean, what little I understand is very little. I mean, I'm a complete outsider, just struggling to understand. I mean, I'm doing what I can to be of, has really come from logic, from my early study in logic, which, and I suddenly, and on models of set theory, and suddenly seeing this extraordinary unified vision when I read his original thesis, his 1963 thesis, you know, the fun quote, semantics, multiple of theories, and suddenly the scales rolled from my eyes and I saw how. As I say, once one has this algebraic understanding of operations, then it's natural to think of things in terms of mapping spaces and of universal properties, of universal constructions and map spaces, rather than in terms of objects. And you don't need all that existential stuff, you don't need all that heavy apparatus of existential quantifiers that comes from set theory. In fact, you don't really need to think in terms of... The Fregean category of objects at all, things in some sense the same or different absolutely to the values of variables, which of course connects with the Platonic philosophy that there's a distinction between the concrete and the abstract realm, that of course these are abstract, these are just as good, these are just as much objects, because one still has the purely logical ingredient of the notion of object, but they are of course, they don't live in the world, they don't live in the world that is unified by being the content of... In a space of time, they live in some more fundamental realm that somehow confers whatever unity of structure there is on the world of the senses or the world that is subject to spatial structuration. Whereas he clearly thinks that what is subject to spatial structuration really does provide the limiting structure within which, in some sense, everything... So this is a profound philosophical notion, and I came to this from philosophy, although I did little study, a very elementary study in mathematics.

1:47:30 Did you never study the philosophy of Whitehead? Yes, though I must admit I was not impressed, partly because Whitehead is very, very... Confused thinker, I think. I was very attracted to the idea of so-called process metaphysics at one time. I mean the second Whitehead, of course. Oh yes, I assumed you meant Whitehead the philosopher, not Whitehead the topologist. Process in reality. Yes, I assumed you meant Whitehead the collaborator, Russell. I don't think Whitehead the topologist wrote anything about philosophy. But he did, I'm not aware of it. But no, no, no, I did indeed. And as a matter of fact, I was very interested in foundations of physics for a long time and did some work in that subject. And I studied under Bohm. Of course, was also very much influenced by Whitehead and by ideas of process. And I became a little bit disillusioned with the Bohm program because it was not... It really didn't lead to any rigorous mathematics. Although there is a group in London that still studies, which Bill knows about because I introduced him to some of them earlier this year when he came to stay with me in London. But you know that they still continue, they still have a group there under Haile, who was Boehm's assistant and collaborator, who study on, who work mainly on the algebraic structure of the phase space of quantum mechanics, and I think, I mean, obviously the level of Bill thinks I'm None of these speakers exist as mathematicians by comparison with that, but all the same, it seems to me that some of the work they have done is quite interesting and conceptually interesting. I'm sure Bill probably thinks it's completely confused, but there's a kind of very dim hint of a connection with some of these ideas that Bill comes out with about... There's some quite interesting recent work, I think, which connects with geometric quantization, there's a subject, so-called geometric quantization, which is trying to treat the quantization conditions for a system very structurally from the point of view of algebraic geometry.

1:50:00 For instance, looking at defamation of Poisson manifolds and looking at the quantum mechanics as the theory of Hamiltonian flow, so technical, on a symplectic manifold, a differential manifold which causes symplectomorphism. And then it turns out that there's some recent work by a Swedish mathematician called de Gaussens, who was a pupil of Serre, the French algebraic geometry school, and who is very close to his people in Birkbeck, and they spend a lot of time collaborating. He has studied the double covering of the phase space in quantum mechanics. There's a double covering of the phase space by a group called the metaplectic group. Well, it's the double covering of the symplectic group and it connects with the quantum, it connects with the crash form of the Hamiltonian and you can actually derive Schrodinger's equation in this setting, completely in the setting of rational mechanics. They think that they've even got a way of showing where the Planck constant can be derived because when you go to the double covering of this, the double covering of the phase space by the metaplexic group. It turns out that as a result of this double covering, there is, when you try to deform volumes in space, space, space, like in the Google's theorem, there is, because of the action of this double covering, there is a fundamental topological obstruction which prevents the indefinite compression. There is a constant of a volume in phase space or an indefinite distortion of the volume in phase space so that Louisville's theorem has to be modified and there is this fundamental and topological obstruction which they think is the origin of the Planck's constant. It's very pure rational mechanics. I'm not saying that it's physics, it may not be physics at all, but it is suggestive that there might be... I was interested in the general views, of course, nothing... They've done a lot of the work they're doing now on the algebraic structures in phase space and the way that the subjective Clifford algebra, which kind of allows you to get the...

1:52:30 The bosons and the vial algebra, which allows you to get the fermions, can be unified in a single algebraic structure, which connects up with the supersymmetries, which people study in their mainstream physics. And also with hopped algebras, a lot of the stuff that people do in quantum gravity. There seem to be some very interesting lines of convergence and connection, but it's all at the level of fairly abstract machinery. But de Coulson's book, I'd be very interested in your reaction to it as somebody who obviously has a very good abstract, powerful abstract understanding of mechanics, of rational mechanics. But it's of course completely unlike what's this continuum. Physics and material science, which is where Bill came from, which of course is fascinating, but I was completely unaware of it, like most people who have studied any kind of maths or physics, most people don't even know that this subject exists, I'm afraid, that's not... He got it only in the sense that he, when he was staying in London earlier this year, I introduced him to Hiley and they spent a couple of days, you know, talking to one another. I don't think Bill felt he got a great deal out of it. That doesn't make sense. He didn't, I think, get a great deal out of it because Hiley's a very nice guy. He's reasonably talented, but he's not, obviously, remotely, he's not a mathematician. It obviously doesn't begin remotely to approach Bill's level. And I don't think they got a great deal out of talking to one another. Also, of course, Hiley is very interested in this whole programme of non-commutative geometry because he hopes that his own work on these algebraic structures in fairy space will tie up with the ideas of Korn and the people of the Enlightenment. Topological groupoids, non-commutative geometry, it doesn't like, doesn't like. Well, no, because, you know, so you don't have the rita equivalents, so, you know, you don't have any good definition of a map between, you know, between one space and another. It's, you have to go to this construction through derived categories and by modules to kind of define the notion of a map, so it's not cut easy and close. He thinks it's a complete...

1:55:00 He thinks it's algebra, but it's not. He thinks the idea that this is geometry is crazy. It might connect up with the structures of the phase space, but the idea that it actually connects up with the physical space is a common thing. So it's just a confusion of levels. On the other hand, there are a lot of incredibly bright people, like Connes of Atiyah's students, and these people he was talking about, and Iker Murdag and his students. Iker Murdag, the Dutch topostherist, the big Dutch topostherist. Well, he now works on this. He's gone right after topostheria and now works on this non-committal. He wrote a book with McLean. Yes, a famous book by McLean. Yes, he's a very brilliant guy. Well, if that was the book you learned topos theory from, then you're very bright and it's a good book. Yeah, but I got the suggestion from the book. The title is scary. Yes, it's what is the title? Sheaths and Logics. Sheaths and Geometry and Logics. Sheaths and Geometry and Logics, yes. It's a good... For me, it was a... It's not a... It scared me off for a long time, actually, and then I read it. I read Colin's book, first of all, and then the introductory book. I would love to have tried to learn topos theory from Johnston's book, but they say that Iko Muradike is the only person who's ever been clever enough to have learnt the whole topos theory from Johnston's book. Absolutely. You have to have a... Yes, it's very difficult. Of course, his second book is even more difficult, The Elephant, because it's even more having to cover all the possible bases and, of course, the subject's developed for another 25 years since he wrote the first book. Well, I'm glad somebody else thinks that, because I tried to study it from John Stone's book and I almost gave up completely. I realised I was never going to be anywhere near being. I'm not able to be a certain mathematician, but I hope that I might at least understand the basics of the subject, but I got completely lost in John's.