Am session: FW Lawvere, P Cartier, A MacIntyre, JL Bell

0:00 Well, if it's okay, I'll set this recording and I'll just let that in after a bit later. Um, okay, just a very brief, uh, administrative question. Since this, obviously, profoundly Exciting and important topics of objective number theory has come up. I mean, I envisage that this was something that we would discuss in the course of what we've loosely called, labelled the Law of Irrational Days, but if you want to continue to pursue it with Pierre while he's here, put it this way, I shall quite understand. On the other hand, I had rather hoped that while Pierre was here, we might... We have one final discussion with him available to input on Grotendieck and specifically on this well-known remark of Grotendieck of points and never really points, which Pierre took as the leitmotif of his expository paper about Grotendieck and those aspects of Grotendieck's legacy that are now pursued or allegedly pursued. So, it's a question of which we, if you think that's a more valuable scientific use of the three hours or so that we still have left, especially since these things have grasped the idea. I mean, it would be nice to discuss it, but it is also on paper. Well, yes, exactly, yes, yes, yes, I take the point. Yes, sir, it would be. Okay, well, perhaps, perhaps you could run with an hour and a half with the objective numbers here, and then if there is time to come back to a group of points just for an hour or so. Would that be a reasonable compromise? Okay, we're just saying here, while you're still here, I had sort of provisionally...

2:30 I thought that we'd use this morning to discuss Greg Leek's number of points and to develop some of the issues touched on in your expository paper, Emma Mandate's work, and just to discuss Greg Leek's views on foundations of geometry, but since this issue of objective number theory has come up, perhaps you'd like to spend just the first part of the discussion of developing that further, but if you would try and keep some tonight in reserve before you leave. Okay, the recorders are running. Colin and I must go, I'm afraid, and we'll bring Leo Corrie to you as soon as we can. Okay, good luck. Thanks. Well, you're still going to find John Bell, aren't you? Oh, yeah. So it's running. They're running away. Yeah, we can just gossip for a second, eh? Yeah. Martello Fiori, I suppose you know him? I mean, he's in Cambridge, I'm in Cambridge as well. We're always trying to get together to talk about, obviously, to talk about the kind of things you were just talking about, but in the exponential situation. I mean, I know it's connected to high school algebra, but what are your thoughts? I mean, Wilkie's stuff on high school algebra, of course, the interesting thing he's got, which is that even for the positive, it's not that long for the right thing. I mean, in the end of the journey, just let me sit on my board and say a question. Well, it's not true that Bernstein did, but the guy who would find it says. Yeah, okay. And so, that would suggest that, you know, instead of pianos, there isn't any. Yeah. Elementary theory of the category of finance, yes, yes. And that's, well, this is a program also mentioned in, by the way, I don't have, in other words, you could, you could calculate the number of elements in an exponential set by using the exponential

5:00 Thank you very much for your time. It's a context, okay, I mean, in other words, you could say, well, how is the category of finite sets different from the category of sets? The sort of obvious thing to me is that they can finite this. So in other words, any two-ended work of an object which composed the identity one way also composed the identity the other way. That's a crucial thing. This, as a property of a single object, is very difficult to do anything with in a category. So, final test for each object. You formulate it, but the condition is formulated for one object. For every object. For all objects. You forget the object. You formulate it, just then, no more. And by the way, there's a little trick to it. This very algebraic way of putting it, which is confusing, implies the more usual formulation of this passenger power set. In other words, if you have a soft monomorphism, you take the power set, that will split. Splits implies it's a nice morphism, and drop back down to a nice morphism too. So the nice algebraic formulation implies the more topo-theoretic line, the more traditional way of saying either a mono-endo or a mono-epi. Now, how, see the question is, I don't know if one ever formulated a precise, there must be some analog of this statement, but I don't know.

7:30 What is the listing statement? Just the finite, everything is finite. The interesting thing about that it can find that, you see, is that it has nothing to do with induction, with building up. It's an objective property that exists from the outside, from inside. So you say epic and mono are the same thing? No, endomorphism. Endomorphism. Oh, I see. Mono implies identity for every object. Mono implies... It's equivalent to epi implies... epi equals identity for every object. What do you say about piano rhythm, Dick? I mean, this has been a very important issue at the level of, say, first-order piano rhythm practice. There is no difference between having induction. All formulas and having this pigeonhole principle for all formulas, pigeonhole principle being the following thing, that if you have, let's say, two initial segments of a model, and you have a definable map, or one initial segment and a definable map, you can't have a one-one map from the segment into itself, or one-one map is subject. Now, at the level of all formulas, there is no distinction, well... Locally, there's a distinction. I mean, to have the induction principle for one class of formulas is not equivalent to having the pigeonhole principle for exactly the same class. And it's a very, very difficult issue. I raised the problem 25 years ago. It's never been solved. If you work in some kind of restricted complexity arithmetic, the kind of thing that Nelson and people propose... Then, supposing the induction for bounded quantifier formulas is essentially the kind of thing that Nelson wants to have, we don't know whether this visual principle for maps defined by bounded quantifiers is true or not. And in fact, there's a very delicate issue with it. Of course, we're interested in the language, just the ring language, I don't mind, it's also bewitched, but what's time, sir?

10:00 And we don't know. People have tried without success. If you analyze how you prove pigeonhole from induction, you typically have to add a function quantifier out front. You're doing induction on a formula with a function quantifier, so you can't get it. Now, Itimes, a leading plastic theorist, did prove something interesting. He showed that there is no possibility of a formal proof of this principle, that if you have induction for bounded formulas, you will get pigeonhole for none of those, because he added a So, we have a generic symbol for one new function, and he was able to show that there are interpretations in which you have induction for bounded formulas and you don't have pigeon formulas. So, in fact, it's an issue that we, Wilkie and I and others, have known about for a long time. It's important for complexity theory because many arguments in elementary number theory use, rather than induction, a pigeon formula. Thank you very much for this kind of issue. It's really about the category of finite sets and the axiomatic, the first order theory of the category of finite sets. More or less equivalent, but which theory? But this again is an interesting one, which I've encountered over the years. Again, for a logician, going back to Gödel, there's not any difference between the two. But this is rather gross in some respects. From time to time, I remember once I proved something about... It was related, it was before Hilbert's 10th problem had been solved, and maybe just afterwards, but there are results, as much as we can, about models of arithmetic where you can construct extensions which solve them, are satisfied by the same axioms, but solve the equations over the original problem that have no solutions.

12:30 It's not clear that this phenomenon is equivalent to the corresponding problem for models of the theory of finite sets. Can you take the right primitives for both? So there are, probably they're not known to many mathematicians, they may be slightly out of fashion, but I've always thought they were a significant thing. So, yeah, I mean this business of exponentiation, as you say, exponentiation of the category of finite sets, well, it's very new. So I come across Ramsey's theorem, the DC dimension, these kind of things, and it seems to me they must have a direct interpretation in the category of finite stats, so the question is what is exactly this mean? So this result, this is kind of a test case at this point in time because the way that it's formulated... It seems that you have sort of two parameters, one of which could be interpreted as a finite set, but the other one still is a natural number, but not true. I think it's odd because it seems to be really a category, a property just of the morphism category of finite sets. In other words, it's another finite code book, because the effect of this natural number variable It's really to say we're quantifying over finite families of finite sets. So we want to be able to make this kind of Ramsey-like reduction at the same time. So I think it can be fun just using maps of finite sets as objects as opposed to... Which raises an interesting question, though. Why should the first-order theory of the category of maps be... You know, significantly different from that category itself. I have the impression that some of the Adler people did attempt to look at this. I mean, Kirby, who was Paris's student, in fact, Kirby and Paris really got the first results.

15:00 And then, Paris and I did both. And, actually, one of the folks who wrote the paper in which I saw, I was not doing very much with Kirby at that time at all. It must have been in the 90s. I'm not sure, but at the time I found it an interesting thing, but they didn't do much. Anyway, there's another reference. So that paper was in Theoretical Computer Science in honor of Dan Scott's 70th birthday. Thank you for your attention. I managed to bring measurable cardinals into the theory of finite automata and shit like that. Well, of course, in the end, it's... So there are numerous... Maybe there are fewer specific ideas there than anything in Scott's paper. There are similar ones. Has this appeared already? Yeah, I could see it now. I'm sure it could just appear right away. It's not too long either. It's a challenge to work with. Yeah, I mean, I haven't studied it yet. I mean, what again is your perspective on this? Well, I'm a little bit unhappy with that. Yeah, I have a vague who you might be, just, yeah. Because, you know, this time I've really had ownership feeling. Yeah. I've been working on this stuff for years. Yeah. You know, so, precisely at this David Scott meeting. Yeah.

17:30 I saw Fiori, of course I knew him already, and I told him a few of the crucial points, you know, just in conversation, and the next thing I know he's writing it out with his own papers, you know, so what can I say, you know, can I really object to this or not, it's just, I mean, of course, with Shandywell's price, it's probably not happening. Really? I've never met him yet. I mean, I should have since he's in Glasgow now, isn't he? Yeah. Yeah. But how does he been doing? Oh, he... Oh, he zoomed in. He was around at IGS, I think, wasn't he? So, I mean, it's this whole idea about him category. Yeah. I don't remember. Well, I've seen him. Was he? Was he? I didn't, I mean, see, I didn't get to meet him. At least I probably have seen him, but I didn't. Thank you for your attention. What about M-category? And he gave a response, which he was trying to make a nice little response, but he said, I prefer to keep N as low as possible. And they said, ha ha ha, then N equals zero. So there's this derisive presumption. I took a look at it the other day, but I didn't for sure. You know him, he worked with Konsevich I think. He's now in Scotland, in Glasgow, but apparently he was at IHS. I think he may have been a Hodge fellow or something like that at IHS. He has written a book on N categories. There's the whole field and then there are the individuals and neither of them are very good. It sounds like a great idea. Uncategories must be amused and aren't things really incategorical and so forth. Fine, but then when you come to the way in which the community is attacking it, they're just doing propaganda, they're not getting anywhere.

20:00 This may get sorted out at this meeting in honor of Ross Street in Sydney, which unfortunately I turned out I couldn't mention. Because there are a number of people there who think about it in categories from different perspectives. I expect it might get sorted out. Because the one who is now engaging is the one that's going to be talking about it, Thank you for your attention. But there are no other applications. Even that is likely a certain application. At least, again, what they've actually done with it. I confess I did not care for Mayer's essay. Oh, by the rush of... It was the first time I had seen him live. Before we go serious. Could you sign my guestbook? Oh, I'd be delighted. This is my book of secrets. There's nothing to show for it once you've had it. Well, I had heard there was a summer school in Portugal, a week course, which didn't have any more in it than that, about one hour, as far as I know. But anyway, so Fiore and Leinster, right? First Fiore and then Fiore and Leinster. They've written this stuff. They don't understand what it's all supposed to be doing. So what exactly do they do? Well, there's this one remarkable, very remarkable discovery of Shandorov. It sounds almost trivial, but it sounds impossible. That is, you know, so it's sort of an additive monolith in actual order. A top element is something that's greater than anything else. But of course, since you completely don't have anti-symmetry, if there's any top elements, there's a whole cloud of top elements.

22:30 But it's not a cloud at all. That's the remarkable thing. Especially if you're in a rig, if there exists a top element, then the set of all top elements is a ring. There's a new zero and one. It's the same old addition and multiplication, but a new zero and one. So there's a uniquely determined infinite zero, or top zero. There's a uniquely determined minus one. So it's like, I always think of electrons. You have essentially infinite C, and if you add one, it's really different from the other. It's not isomorphic to it or something. So it's an infinity where you can add and subtract and make an isolation. So what kind of rings can appear here? Any ring, any community of rings can appear this way. But the way that they came up originally was, as I say, what I've just said is very elementary. I told this to Fiori, he went up and wrote down it, but it's very elementary. The fact that you're in this extremely loose situation, And suddenly you have total uniqueness. It's just very surprising. I still can't quite believe it. But the way that it arises concretely, you see, if you have, you see, take one of these fixed-point equations, x equals f of x, where f is a fairly general polynomial, a degree higher than 1. You don't get it with degree 1. You get it completely different. But then you take the bird-side rig of the category that this generates, and the dimension rig, potentially with two, with just the three elements, as I said before, so the top elements are identified with those things whose expression genuinely contains the object x, rather than the finite sum of things. And so, actually, there is an object there which behaves like zero and added to many other things, and minus one. So, we have a whole theory of this kind of race. It's even, oh, in fact, in fact, what ring is it? I mean, every grid has an associated ring by the well-known high school construction of the Select Adjunct.

25:00 Or, of course, the junction map is not monic, as it is in the case of something that's cancellation. So, in fact, it's that same rig. In other words, the ring, you take the rig and you take the associated ring. This is one of those situations where you have, so to speak, negative sets, because the negative sets would sort of... Now normally minus one is totally virtual in this case here, but in Chanuil's case it wasn't quite virtual because it was an actual object that gave rise to it, even though of course it couldn't possibly satisfy the equation x plus one equals zero as an object, but it played that role. It satisfied instead the equation, what is it, x equals one plus two x. But you have to think of the higher degree as dimensionally sort of collapses, so then this homomorphism from the original rig to the ring is on to. Because the infinite minus one becomes the actual minus one. Zero and one go to zero and one, but so do zero and one. The infinite zero and one go to zero and one. But there's a section. In other words, the ring injects back into the ring by a map that preserves addition and multiplication, but of course not zero and one. So he calls this, for short, the Euler section. Because the projection, he was calling the Euler characteristic, this is a section four. It's almost a ring homomorphism. It actually doesn't preserve either zero or one, but every ring can arise that way from a suitable category. Well, every ring can arise that way by a formal construction, and many of them can arise as Burnside. You know, it's the Euler-Rammert, the Bernstein-Hiergau category. I have a feeling this may have something to do with old old paper, the clay that I never understood, which is associating to promontory of objects. It's matroids. The assertion to matroids, algebraic numbers, as in variance. How is that possible?

27:30 It's very... It might have something to do with it. In any case, some of these sort of intimate commentator categories, i.e. the ones that are generated by following the fixed point, you get the associated ranks and grades, which have this infinite zero, infinite minus one, or top zero, top minus one, and such a thing. So I have mixed feelings about the theory therefore. On the one hand, I'd like to tell him, go ahead, find out more stuff about this. But on the other hand, publishing it without being part of the group that is discussing it, he has no idea what it really means. He knows it must mean something, and it's a very amusing calculation. In the case of a communicative monoid, there's one top element, then there's a leading group embedded there. This is even just added to that. It's basically a computation of something that's just added to it. So when there's a unique element which plays an infinite zero, together with the given addition, there's actually a leading group at the top of this, at the top of this... So somehow the whole idea about cardinal numbers is not typical. The abstract cardinals, or even the isols. Well, actually isols, there's a lot in isols. Oh yeah, there almost has been. I mean, Rowling had some pretty heroic efforts at the beginning of these things. He did want to get something invisible, non-standard model, at least a good bit of arithmetic. But I think it has something to do with the ice holes. Stanley will probably explain that a little bit better. It's not exactly the same, certainly, but it's one of the same kinds. This connection with Nelson, this thing you're probably familiar with, but it was pretty slight when he discovered it in connection with this.

30:00 Philosophical ideas on quantum quantification and so on that Wilkie also discovered and so on. I made a report by a few other people in Europe on standard models. You know this system, I read Roberts, Q were finding names for that system. The thing is there's no, everything is, everything, there's nothing, it's not at all, cancelative except that there's all these funny things over there. The addition and multiplication of the community was a freakish, marvelous addition to the classical standard. Nevertheless, they all discovered that you can, in the technical sense of atomicology, you can interpret in any model of Q, you can interpret a model of boundary and knowledge. There are formulas. Of course, you need to interpret the model of bounded induction, you need formulas with unbounded quantifiers, so philosophically, I think it's less important than Nelson claims, but nevertheless, as a, simply as a, that was great pathetic observation, it was somewhat better than that. You can interpret restricted, some, in a cafe, Nelson came up with, to this original smash. You know, there's a restricted exponentiation. You take an integer and you exponentiate it to the binary length of another. It's technically important for coding and substitution. Sticking one sequence of strings instead of none is much better than using expenditures, much more economical, much more algebraic. And, in addition, it's connected with these pigeonhole principles that we... I think it's very close to the idea of John Bell. He had this paper where he was talking about a general phenomenon of bell orderings and sets, which I interpret in the following way.

32:30 If you take the power, if you take the symmetric power, if you take x to the n modulating factorial, then this typically can be comparable with x itself. This was sort of type reducing, type reducing, it's also cardinality reducing, you just divide it out by the automorphism group and you find that the, you know, so for example, even in the finite case, two to the n is pretty big, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, if you, You always typically get this plus, right? You get this additional element. You get the empty set. Yeah, that's right. You see what I'm saying? You take all the morphisms of n, natural numbers, and 2 to the n, modulo that. Just to count it out each other, right? So this is a kind of definition of cardinality. Well, that's how Frager used it. I mean, he actually gets omega plus one in Frager's construction. He gets omega plus, you know, the natural numbers plus one in his construction. No, your friend, your Greek friend, you went to work with him. My Greek friend? I don't remember his Greek name. At London, Ontario. Oh, no, Bill, yes. Bill. Sure, Bill Demopoulos. Demopoulos. Yeah, yeah, sure. Bill Demopoulos. I haven't touched many other Londoners. No, no, no. There are two Londons. No, I mean… Yes, yes. I see. He was referring back… He claimed that this was… Well, that's the term that they use, philosophers like to use, but it's a bit of a misnomer. Even Barry Smith, much like him, he immediately caught that there was something strange about this, or at least different.

35:00 The idea was if you want to study the size of a class, you look at endomorphisms of the universe, All of these terms are related to the sub-class of the universe, which is called the isomorphism on that sub-class. Now, if the universe is finite, of course, that's okay, but basically it means that you're getting the negatives or the complements at the same time. If you actually divide out by the isomorphisms of the universe, then the equivalence classes of the sub-sensors, you have the finite ones and you have the co-finite ones. So it's just sort of, it's always going to have two ends, because of the complementation, and who knows what in the middle, but it's quite different to the usual notion of cardinality and fact. Well, that's exactly what I mentioned, infinite dimensional, calculus of variation. I mean, I mentioned that I have two theorem complex, one which is finite and the other one which is co-finite, but they never join. That's a problem, that's a problem. In finite, for a character, it's infinite dimensional spaces, manifold, tempos, and so on. I mean, they reflect each other, but an infinite dimension, they don't do that. They are separated by some backroom, some backroom. Here's a representation of Descartes, you see, because Descartes had some idea to the effect that if you can see something very, very clearly, then it's true, right? Yes, basically. But Shanderwell says that he could see very, very, very clearly this ordered set, which was well-ordered from both ends, you see. Yes, I can see it, you know. He saw it so clearly. So he's just convinced that it doesn't exist, by logic. You know, this reads something like, I mean, it's something like your cardinals, you know, then you have something like co-cardinals. Some kind of dual cardinal notion that's emerging. Because, of course, that's the thing with cardinal algebra, I mean, well, Cantor's cardinals, that there really isn't any, you have to put a positive in, and there really isn't any corresponding, well, formally, I suppose you could introduce it, but it doesn't play any kind of role, this, you know, this symmetry. And, of course, that's the real difference, well, at least, between Cantor's cardinal and Z.

37:30 Well, the fact that you don't get this analogy at all with negative numbers, I mean, you get an analogy with positive numbers, which of course can't do a... And of course, negative numbers just really don't appear at all in any serious sense. It's just a consequence, in fact, the exponent itself doesn't go as fast as the equation. Yeah. Our host, as he was leaving, he asked me to try to steer the discussion. That was his parting shot as he left, so maybe we should, I don't know, we should do that. Just small things. This telephone has been ringing at least three times, but I don't know how to start it. I probably don't know. Okay, let's... You push me in, you know what I mean? Good line, but it's... Press the red button. No, no, it stops. Not the big red button. Everything disappears. Hello? Michael isn't here. He's gone to pick you, Leo Kari. Yes, that's right. Well, he's going to pick you up. He's on his way. So if you just wait. Yeah, if you wait, you'll be there shortly. Okay. All right. Bye-bye. Just think, all this is being recorded for posterity. I take it. It will be viewed as a reflection of the primitive state of... It will be of great historic interest when archaeologists, thousands of years past, come across these boxes trying to decipher what's going on.

40:00 People actually didn't have neuron operations. Right, exactly. So it will be a big blatch. One of the most primitive robots. That's right. Well, I haven't for a long time. I confess, it's been a long time since I've told you about Ramsey's theorem, but... I think this is, the way I always look at it, is that when you start with something that's relatively innocent, and then you assume you have these in the serenables which you would get for free, but you just assume you have a slightly stronger version of the serenability which you can get by this, and then purely formally, somehow with a stretching of in the serenables, which has clearly got something to do with the categorical considerations that Girard uses and so on in his books. You've just suddenly got a model of something much, much bigger. Of course, but that's the same game as the playing of large cardinals in set theory. One should still be looking much more closely at it. Of course, most of the set theorists don't admit the existence of categories. Well, there are not a great number of set theorists left, actually. By comparison, it's only the 1960s. No, but I think it's in steady state. It has become very much a priesthood. I mean, there's a small, small number, There are enough above to get enough people to maintain a very, very high level. Of course, it's gone in Britain, essentially, and quite possibly, well, it won't go in France because they've got the Serbs, the Serbs, the Pirates, etc., but it's in California. Yeah, yeah, sure.

42:30 There's no schools in there. No, Witten is a pretty representative of that school, Berkeley, I think, has his own, whether he caved, I mean, to the AMS. Yes, no, I mean, he does the subject no favors in the outside world. I mean, he's clearly somebody who can tell you, like, yeah, I don't know. It's not easy to get. Well, you know what struck me when I heard him? Well, I've met him a couple of times. I heard him at this meeting in Copenhagen. Well, I have my impression. What's interesting is that his presentation is very smooth. He looks very much like the actor Zachary Scott. He used to play Dillard in movies. It's very smooth. It is, it is. But you see, it's smooth in the way, in some sense, it may be like, it's something like a physicist presentation. You don't really get down to anything. It's sort of suggestive somewhat, you know, and you get some vague, you get the illusion, well, maybe the sort of illusion of understanding something, a bit like some of the physics presentations I've seen, and also the, of course, as I say, he does it, you know, he presents it so smoothly, it's, you kind of, well, you know, it's a lull into, it's a sort of understanding. However, I do think that there's a rather more serious, I only meant... I think there's a stronger analogy with physics in some sense, I mean the way he's describing it, it's a bit like casting a rug for acne, he's a Platonist of some kind, a realist in some sense, and you know, you're looking at something that's over there, I did tell him I thought he was looking for axioms that further, for…

45:00 For static sets, making the whole universe even more static. And he agreed. He understood what he knew. He did know that description of things. And he agreed. And then Heibert suggested, well, in that case, what you're really doing is a kind of description, rather than the way that physicists are looking for the right, for plausible evidence. Let's put it that way, right? For properties, in this case, of a really static universe. And that's something like, you know, what physicists do. I mean, they haven't got any, you know, the evidence may be indirect, because the difference in physics, of course, ultimately, one presumes you'll be able to determine by experiment, although God knows how long that will continue. It may be that you just can't do it because the energies required are too high. There's just got to be plausible evidence again. That's all you have. But I guess, I mean, he would probably argue that they have plausible evidence, too, in the sense that they get at least... There is a level of relative consistency to get, you know, that all sets, all definable sets are the big measure. For him, I guess, that's about the limits of what can constitute plausible evidence. I mean, it's just very remarkable. The issue of the game is being determined. I can never be out of my mind whether this is plausible evidence or not. No, that's the problem, of course. Whether it's plausible evidence may be somebody else's. dubious claim but but it but it's it's but i do think that's You know, that's the kind of, and I think perhaps that's why, you know, what enables him to present these, partly, you know, this sort of idea. Of course, it's complicated. Also, one has to say that the, you know, the actual detail, I don't understand. Oh, no, no, no, that's a good question. It is a fantastic idea. But I do think that's partly why it comes out. I mean, maybe the work of his own type of presentation, which is over-smooth, perhaps, smoothed out far too much. But I do think that's, there's some... The whole sort of program has this, you know, like some enormously complicated program, you know, trying to do this in quantum gravity or something like that, you know, where you really can't, the details are very, perhaps very obscure, and you kind of present it as if, well, we're not quite, we have some evidence for this, and this is plausible, and it all comes out as an extremely smooth kind of package, which conceals really, you know, the actual intricacies and different kinds of it.

47:30 Yeah, well, Conan Doyle, right? That's right. That's a good one. Well, of course, he also... That's right. We should probably try to get Pierre back. Otherwise, Michael will be very... Waste of tape. It's a con, but you weren't that right, because the problem is that Pierre, not the problem, it's for the interesting character, seems to have a very great resonance to the stuff on the very chanel rig was torn between the days and how... He left me! I mean, we should try to do something, because it is a pity. I mean, we're all here. It is a pity. I don't know what Pierre is doing with this. Yeah, I agree. I don't think that anything would do. Suppose he convinces himself, in this very static conception of set, this core set, that 2 to the L of 0 goes to L of 2. I mean, who else is going to believe it? You know, it's not really like the... It would be a very hard sell. No, I mean, I don't understand it, for example, mathematics. Oh, yeah, yeah. There's no question about that. But it does remind me so much of Zachary Scott.

50:00 You know the actor, I mean, you must have said you know the movie. Yes, yes, of course, absolutely. Mildred Pierce in Mask of Demetrius. Yes, yes, yes. He's very smooth and always, always played villains. Yeah. Well, I must say, I did enjoy it. He did guide the illusion that I understood some of it, of his approach, you know. I mean, these are, it's just that, I suppose every exposition actually takes him further away from Yeah, it probably does. It does inspire a low-level philosophical discussion by secularists much further down the... Well, I got tired of seeing articles on philosophy of sectarianism. I've been saying that stuff over and over again. I mean, you know? Yeah. I mean, it's... You don't know it. I should also know what my logic is. That's basically probably banal. It is. You know, it stems with that. I'm glad to see a philosophy and mathematics other thing. There really is a reasonable spectrum of papers now on different topics. It wasn't the same old thing. It wasn't neologism or the philosopher's set theory. And there was a danger, of course, you see, that that's what it would become. I tried to... I mean, and I'm glad to say that it's not confined to that at all now. It's a pretty good spectrum of the variety of things. Because I assumed it was set at least two different kinds of cohomology. No, thanks to the effort of time with us. And there are different inputs there, you know, in phenomenology, working with Monterey, you know, and Hilbert. It's quite a good range, not that same old stuff, you know. I mean, I don't know, give me a break!

52:30 I will dedicate myself to him. So the subjective number theory has sort of gone to three levels, right? Yeah. No, no, this is science. No, that's the combinatorial. Yeah. Yeah. Elaborations of the science. There's a tremendous range of direct interests and all kinds of combinatorial problems. So there we notice sometimes that objects satisfy algebraic equations, but then Shannon Wells showed this tremendous fact that if an object satisfies any equation, then it satisfies the following power equation, and in turn that's equivalent to being separable. Yes, yes. You see, this is a side of this property that you can detach the... Or, I guess a third way of saying it is if you define the spectrum in terms of the functions to the category of actually finite sets. Can't find it. Where's his room? Oh, he didn't know his room was downstairs. He was sleeping in the front there. Yeah, yeah. Well, he's not there. He's not here. Maybe he's having a shave or something downstairs. He's not in my bathroom. No, I was just there. I talk better than he is that sometimes. So the separability, in another way, it just means that the equations, it's equivalent to some kind of finiteness as well. Objects are very finite. Yes, yes, that's right. Those kind of equations, that, of course, the only...

55:00 The only solutions that you could get by mapping into sets, or finite sets, so that the roots of the polynomial are natural numbers, which is without repetition, of course. In this infinite case, of course, we get one of these fixed-point equations that's a suitable polynomial. Then you will find that you have nilpotent objects, in the sense that relative to their top objects and their square is the top zero, you see. Ah, yes, yes, okay. So, literally, in the associated ring, you get nilpotent elements, but only by going through these infinite, these top, higher degree equations. There's negative sets, which is a linear equation. It's tremendously interesting. And then the higher degree equations, which unfortunately we don't have really real models about. In other words, the negative sets, this is a very, this is a geometrical example. But for the higher order equations, you can get an actual category, but the category doesn't have a known... Topological or some other interpretation. An infinite dimensional space which is so twisted around that it can satisfy a polynomial equation without collapsing everything. Yes, yes, yes, I understand. It would be nice to have such a more geometrical interpretation of those categories. And then at the analytic level you see there are things like In fact, I was, I think I was saying, you know, the Euler product was nice to work with, but one of the objects, yeah, that's no doubt only the tip of an iceberg, but that's sort of on a third level, where you actually, you actually have to use the existence of the free molybdenum functor, and then also the direct limits when you get all crimes.

57:30 Yes, yes, that's right, that's right. See, the free-mortality functor is a much stronger thing than just having something that satisfies the list equation. I interpret that in a positive way in that paper, but it means that it's not clear how strong it would be to postulate a category that has not only addition and multiplication, but 1 over 1 minus x as a functor, you see. Right. Because if you just take two things that satisfy the equation of one over one minus x, there's no, or say you map x into y and you take the free one over the x. Not the free one, but the one, something that satisfies the same equation, so there's no reason to have a map between them. It's a much stronger thing than just having the existence of an object. So that's probably somewhere in between the algebraic and the analytic. I can't find Pierre. I don't know where he's gone. Too high and low. Well, I could look out there, too. I could see the garden. They could use sort of Grunbaum's method of resolving the achieves by...

1:00:00 It's a standard way of resolving a beauty in sheets, but you have to use the fact that there are enough points to faithfully...

1:02:30 Right, right, right, right. So he had this general theorem, or coherence of the problem, rather than one for which the politics... Yeah, yeah. For arbitrary Grosvenor-Deacon value models, instead of the sets, so there are enough Boolean value models, and that's good enough because, I mean, the idea is you apply the forward and backward, the geometric morphism, and you do something over here that depends on Boolean-ness and choice and so on, but that's fine, you can do that in a Boolean value model, I mean, it's construed as a Grosvenor-Deacon topo, so after we got the Boolean value models... But meanwhile, of course, I recognized that they had basically completeness there. And the fact that Gödel's completeness here follows because of this business that actually anything Boolean can be coded in positive, but more perimeters as negatives. I think there was a connection, I mean, there's a sort of, some kind of connection also, although it was a different kind of independent program between the notion of points introduced in this. I've elaborated setting in algebraic geometry and also the program that was emerging of pointless topology from, you know, in topology using lattices and then trying to replace, what was the name, Dauker. Dauker, right. The Dauker, which was sort of a different, somewhat different level technically, but there of course all of this is now. You can see the kind of idea of trying to get rid of points. You have, in some sense, gotten rid of points when you work with topos, or at least when you go from a topological space, right, to sheaves and so on. You've essentially got spaces which may or may not have points, and the problem is then somehow to reconstruct the idea of a point.

1:05:00 But when it's a Petito pose instead of a... Yeah, okay, but right, right. So in some sense, yeah, you expect that the terminal object no longer separates. That's right. It's true in that sense, but that's an internal sense of points, which is rather different from the external one. But yes, but even so, before these distinctions were... It was possible really to make these sorts of distinctions. It was already this proto, you know, sort of program of trying to do topology without points, which was a, well, it goes back, that idea actually goes back in some sense quite a long way. I mean, you know, the idea of replacing the topological space by its lattice of open sets, and that goes back all the way, but back to stone at least, right? And working with that and those theorems that he proved on representing distributed lattices back in the 30s. And there is a kind of... Well, Peter Johnston makes these sort of points in his book on stone spaces. Well, anyway, I mean... This is all Eilenberg could say in his last day. So you could talk to him and he would say, if you liked me, you could say, yeah, yeah, yeah, you could say that. But the other answer was... Distributive lattices are great, right? There was also the work of Hoekstra, who characterized the so-called spaces underlying the spectrum as being really just equivalent to distributive lattices, the coherence basis of just distributive lattices. I had run into that quite early on. Well, what was the purpose? I mean, by the way, there's a, you know, there's a sort of, okay, how do you construct a Groton-Dick topo? So, you have a site, small category, motion of cover, fine. And then, of course, there's the dual math thing that I think Tierney and I were the first to point it out, but essentially a quotient is described by a left-exact homonet.

1:07:30 Yeah. Okay. So that's, you know, left-exact homonet. But there's a third method as well. Namely, you can take any category which has filtered preliminaries with some set theoretic. I mean, maybe there's a small set of objects such that everything is a... And now you simply look at the filtered co-limit-preserving functors into sets. That's going to be a great big topos. I mean, at least it obviously satisfies Giroux's axioms because the filtered co-limits commute with both arbitrary co-limits and finite limits. That's what it's all about, you see. So this is kind of a Fourier transform, topos. You take the category of all points. Take the category of all points of a topos, that's a category of filthy equivalents. And by Lernerheim-Scholem, there's a set of... So you can then go back and take the punctures to sets from it, from the category of points. And that's sort of another topos that's connected with the original one by a... Geometric morphism. If you had enough points, then that morphism is conservative on one direction or the other. In a surprising number of cases, it's actually an isomorphism. In other words, every, for example, the Shenro-Mahill topo, the one that we thought about quite a bit, in other words, it's usually thought of as, it represents the notion of infinite decidable objects. Which is probably the most restrictive notion of, I mean the broadest notion of finite available, namely that given any map from a concrete finite thing, it can be extended to a bigger finite thing. So infinite decidable objects are classified by the Chanuel-Meyer topos. The site being the opposite of the category of finite sets and monomorphisms to take care of the interseperability of all finite sets.

1:10:00 Well, that topos is also equal to the category of... We take the category of all infinite, decidable sets, so to speak, in other words, the models of that thing, points of that topos. And take the hunkers back into the category of sets as equivalent. So there's a kind of balance between, if you think of it, within the category of all sets. On the one hand there's the filtered co-limits, the infinite things, and the finite things with monomorphisms. And these are in some kind of pluriated duality with each other. So I have this number I'm going to explore as you see it systematically restructuring sort of like this, well it's like completeness and definability and it's all in one. If you recapture it, you can sort of identify the topos with the classifying topos of a certain theory. Because you're recovering that theory from... The models, provided you think of the models as a category, but only as co-limbs, not as open powers, then you can sometimes recover it. I have no idea in what degree of reality. So that's points out of topos. A special case of which, of course, is points of locales. Yeah, exactly. So that's sort of, even though it's a much smaller thing, it's part of the same general thing. Points in algebraic geometry are something kind of different. First of all, it's going to wind up inside a big topos rather than in a category of topos. With all the notions must have, well, some deeper connection then. Well, they're both glorifications of the idea of a map from a terminology.

1:12:30 Yes, but also, I mean, perhaps the purposes are different. No, what I mean is if you look at the source of use of points or areas in this earlier setting, the notion of point and locale, point of a distributive line, that was really some way of introducing, reconstructing or characterizing those locales or... Distributive frames with complete hiding algebra, which do arise as the open sets in a topological space. In other words, it was a way, of course it was an early, very early thing, quite an actual question, you know, in the usual pattern. I mean, when you have an abstract structure, when does it actually arise from one that you regarded as concrete? That was the program there. I don't know how to see, I mean, it's difficult to see how that, uh, that relation holds in the algebraic geometry. That's right. Well, maybe if it does, then you do have a parallel. The distributive lattice case was very misleading. In fact, it misled Grotendieck via Gelfand. You know, among the distributive lattices, there's only two. It's the only one where you would reasonably get points. So Gellifan could identify maximal ideals with points, but this is basically wrong, I mean atypical, because in this slightly more general situation it's not a matter of maximal ideals, or even prime ideals really, because again it was Gabriel who explained this to me and I haven't seen it in print anywhere else. But basically, the points, let's say the affine scheme, the sort of simplicity, they're actually a functor from the category of fields, namely, given the ring A, you map it into every field K, those are the K-valued points, that's a functor of K, as a functor of K, it's a sheaf, you know, and so forth, it's an object, it's a single object, really, moving in atomic topos, actually, so it's the underlying set, you see, even though it's not abstract.

1:15:00 Now, if you want to get prime ideals, you just take the direct limit of that functor, because given the map to a field, the kernel is a prime ideal, but it isn't onto the kernel's prime, and you know, if you have two different equivalence relations, there's lots of points, and there's automorphisms of points, but those get washed out when you take the... And that way you get just a set of prime ideals. But you see, notice what an unnatural thing this is, because this particular direct limit is not filtered, doesn't even preserve products, let alone left exact, you see. So it kind of destroys the elementary set theoretical operations that you might like to apply to. So it's again like the slogan of Verdi and Grotendie, don't take cohomology, don't take this direct limit. So you have a ring and you consider the k value and you consider this as a factor from the category of field to the category of set. Right. Okay. And then what do you do after that? Well, I mean, let's say it's for finite field extensions. It's a small category. Okay. So along any small category you have a notion of co-limit. Any diagram of that shape averages out into a single set. So if you take the direct element of that particular kind of function, it turns out to be isomorphic to the set of prime ideals of the original ring. Because taking the kernel of a given map is a prime ideal, right? And then the equivalence relation identifies these. You can mod out, get an integral domain, it takes field diffraction. It's the kernel of that. So as long as your category with fields was large enough to include the field diffraction, so they obtained it that way, then it's going to be isomorphic instead of primal ideals.

1:17:30 So, as I said, Gabriel explained certain things to me, so I already had pretty much developed it. You know the ideology is different from the standard one. But then Grotendieck himself came to Buffalo in 1973 and gave a lecture which was exactly, guys the old definition of scheme was wrong, throw it away and use this new simple one. He was very emphatic, you see. Use this new definition. That a scheme is a set-valued functor on a group, which is actually due to you, in a way, at least in the group case, where you emphasize that an algebraic group is not only not a topological group, it's not even a group. Precisely, if you think of it as living on this direct limit set, taking this direct limit destroys products, and therefore destroys any algebraic structure. This is the approach that they take from the beginning in Gabrielle and Gabriel. Right, right. Which they take from the beginning but then unfortunately abandoned to a certain extent. I always thought it wasn't really a completely consistent execution at this point. But they certainly start with it. Grokny was very emphatic. This was all a mistake. We shouldn't have done it that way. One of which was just this fact, that one of the prime ideals is this non-exact direct limit of the perfect limit of people. Well, the trouble is, we were late. I didn't realize we had to go. I thought it was coming this evening. I didn't realize that. I thought 5.30 was 5.30 again. Yeah. No, he arrived at 5.30. Well, I didn't know. Yes, they've been going at least an hour here. Well, you have to save our honor by telling us all your fault. Yeah, no, it wasn't my fault. I mean, inadvertently. Nothing intended.

1:20:00 In particular, after his lecture in Buffalo, I wasn't at Buffalo at the time. It just happened by tremendous misfortune to be visiting. And after the lecture, he went on to say, well, points have a lot of morphisms. I think he didn't say this, but it's just a matter of terminology. The idea of speaking about the b-value points for any ring b, I think it's not quite right, you see, to call it points, because points are, those are b-shaped figures. Among the figures are special figures that deserve the name of points. Now they may be quite a bit more general than just maps from one, you see, but still. Well, I like it, and it maps to simple objects. If you have an algebra, you can match it to simple objects. Subdirectly irreducible objects. Yeah, yeah. You get better. To atoms in some kind of sense. Some kind of sense. Maybe atoms with little clouds around them. It's still something much more special. Sort of irreducible minima of some sort. Yeah. Yeah. Yeah. Yeah, yeah. So, but, I mean, of course the... Speaking about the b-valued points, in a way it makes sense if you pass to the category of spaces over b, because then you have this old Greek idea of a locus of a moving point. So the figure as defined over a base is the locus of a moving point. But if you say the same topos, the meaning of points should be more restrictive. That if you consider a general scheme and you consider, again, the b-value put for all of our four commutative things, before you consider the k-values, the bonds with the conduit field, and I think they deserve to be called, and the b-value part, but that is something that is a local thing, and now the remark, which is, I think, more or less the same as the one you said about the direct limit, is that if you consider for a given scheme x,

1:22:30 The B-value point, where B runs over the local range, is a disjoint union of representable factors, so it's a direct sum of representable factors, and each sub-factor, which is representable, corresponds exactly to 1.90. That's another way to do it, but more or less the same. But it says that, yes, I should consider these, I'm moving over. But king in this set, I don't understand. Meaning a local ring. Yes. But the point is that if you take this point of view, I mean, so if you take a ring A, you associate the factor from the local ring to the set, which is, so take the factor from the category of all, instead of taking the category of all commutating rings, let's take to a subcategory of local rings. The third category is not representable, but it has died from a representable factor, and each third factor which is representable is exactly one primarily, and the local ring which represents that factor is exactly A over B, I mean, the localization. That's one way.

1:25:00 When I came out to make a long detour, you could not enter the town. There was public work on the streets. Yes, so it gives, not only the board, but the road. There are several, but there is one, I think, I'm pursuing it now, actually, you know, this intermediate. For example, as I was saying, all the objects that are orthogonal to one infinitesimal object. Well, the maps into it are all constant. In which sense? X to the infinity is the model or the other way? X to the power of T is isomorphic to X. X to the power of T is isomorphic to X. What's the definition of discreteness? Yes, a kind of discreteness, except it turns out to mean, in this concrete case, it turns out to mean precisely these functions on the fields, rather than abstract sense. This was very appropriate. Extanties, right? So you take this object, which satisfies this equation, and then? Yeah, yeah. So then, so you have a large topos. In fact, I'll...

1:27:30 You get a sub-category which is connected by four adjoint functors to the big one. You start with the big one, you carve out this sub-category of so-called discrete. Four functors. Four functors, yes. Yes, there is... Embedding in the adjoint. There is the inclusion. Which has a left-eye joint, which is to take components, to take the space of components, or in the algebraic geometry case, it's the space of components equipped with the Galois action, may or may not be there. And that function preserves products but not equalizers. That's the only exactness property you have to mention because all the others fall from the joint. Which is the Cantorian idea of extracting from a cohesive space just the discrete space of the points, the points and the outstructure, the points without structure, except this very, very small structure, roughly speaking, just the Galois group, I suppose, so that if you were working over, if you worked over an algebraically closed field, then this is abstract stats, it's just a set of components and a set of points, rational points. But in the general case, it's not just the rational points. So you have these two, actually. So the inclusion has a left and a right. Left and a right. That makes only these three factors. But then there's one more, which is the co-discrete space. The inclusion is like... the inclusion is you start from a discrete thing and just view it as a very special extreme case of a cohesive space. You can also start with a discrete thing. It's like the co-skeleton in the Simplicial case. The co-skeleton, which is sort of wildly filling out all the holes, all the things that look like they might have been holes, getting an object which really intrinsically has exactly the same information as the discrete space. But it's plugged into the cohesive world in a very different way, so that this, you know, it's the maps to it that are trivial rather than the maps from it, so that there's no...

1:30:00 And so by contempt or by don't function... It's the furthest right, you see, so that if you want to look at it in a row, then there's the components, discrete inclusion, points, co-discrete inclusion. So, now... Okay, and this is a full subcategory, so various composites are actually the identity here, as we retract, and I said, well, I want the leftmost one to preserve products, that I have to put, it's automatically true when I define the one of flogonality, but I didn't mean that as an axiom, and now the, another axiom which makes the following construction easier is the Nostrandsatz. But now, the null set of the sats does not mean that every object has a rational point, of course, if we are not over the algebraically closed base. I take it to mean that the natural morphism induced by the adjointness and the full inclusion from the Cantorian points to the components is an epimorphism. Points to components should be an epimorphism. That natural transformation is induced because of the adjoint system, because of the fact that the equilibrium is full, basically goes through the original space, right? Okay, so now we can... So each component has a point. Yes, that's right. If we were actually surjective, that's what it would mean. But you say epic in the given category. Epic in the given category, right. And I don't really need that, but it makes the construction easier. So now comes the construction of the intermediate category. So that's a new statement, that's your formulation of the new statement. Yeah, the one at least which is suitable for this new context. So the intermediate category is simply those objects for which that map is an isomorphism, for which every component has exactly one point, which doesn't mean that it looks like a point, because it... It has maybe a little infinitesimal, the point was a little infinitesimal cloud around it. And you got a lot of action also.

1:32:30 And you got a lot of action, yes, that's right. Box, box, box. It's quite different from an actual point, but for any given space, the number of these is the same. The map is a bisection. So now it turns out that this is now a bigger subcategory. It includes the original mysteries, of course. It's another subcategory which plays the role of something like local or infinitesimal or... It also enjoys three adjoins. Which I have invented now the colorful name of hot and cold adjoints. Left and right, you get tired of left and right. Hot and cold. I'll tell you why. Because you, and this is quite precise in the case of reflexive graphs, I mean everything can be exemplified there. To this category where points and components are the same. And on the right, and there's a map between those even, and it has sort of this flavor to it that you, when you take the portion of an interesting space with that property, it's sort of like catastrophes. You've got something that's defined by a curve and you just look at the singularities by themselves and you throw away the global cohesion or the physical. You've heated up this material to such an extent, you've superheated it to the extent that there are no more interactions between atoms. But the atoms themselves retain their, you know, their electrons and, you know, they have their own... So high temperature. Yeah, this is the high temperature adjoint. Ranging, I think. Where you throw away, see, connections. On the other hand, the other one is the supercooled version. Because you collect together all the atoms that have any interaction at all into one atom. It's like one of these Einstein. I suppose Einstein. You collect all the atoms that have any reasonable connection at all into one. And then there's a huge atomic number because all the original bonds have become particles.

1:35:00 Again, this idea that, as you can admit, that forces materialize into particles. People talk about this all the time. So that's what is just literally coming out of this idea of the hot and cold adjoins of this inclusion. And there's a map between, which is kind of the prediction, you see, that if you have the superheated experiment and you have the supercooled experiment, then everything in the gas, you see, will coalesce in a certain way into the fluid. And you can't predict that from looking only at the gas because you've thrown away the connections. No, the interactions are negligible. But from the original object, yes. That's why it's a natural transformation, a parameterized by the original. I like this. And it applies in every situation. In every situation there is exactly this kind of thing. It's just astonishing. It's a good allegory. It's a good allegory. But it exists in every such situation. Whenever you have these four functors, that's satisfied in Milstone's results, which means, for example, whenever you have this infinitesimal theory, so it has many, many, in principle, in every field of mathematics, there are things like that. So what is superheated and supercooled? What does the cooling map mean? So, the second principle of thermodynamics is that the map goes from the superheater to the supercooler, not the opposite. Right, right, right. So now we need Van der Waals equations. Now we actually make the transition. At present we have only the qualitative extremes, but a map between them. Van der Waals equations would have to be compatible with that, whatever that means. But then there's a strong, I haven't proved this, I don't know what generality is true, but there are sort of two classical ideas about the nullstrahlung sass. One is that you have enough field value points, but the other one is really essentially the Birchhoff subdirectly irreducible thing, which in algebraic geometry is really just about these Gorenstein rings or...

1:37:30 It has three different names. Norenstein, Virchow, Provenius. Different groups of ring theorists study the same set of rings under these different names from a functional and analytic point of view or, you know, a stochastic point of view that means that there's a distribution of global support. That's a local ring, both maximal ideal and... Anyway, so the stronger Goldstone sets, first of all for the rings, it says that the finitely presented algebra has enough maps to those things, not just that there exist maps, but that they're jointly monic. You can separate quantities by evaluating them at points and all the derivatives. Two different elements of the community of ring A say finitely presented algebra over page K. We'll be distinguished by some map into a local, to an algebra over k which is finite dimensional as an algebra over k, and hence has one of these stochastic sections. So this is sometimes called the null-shown size, but it's stronger because you're getting not just the existing, you're getting it mono. Of course, it will never be an epi because that's where the Zawiski topology comes in, to glue these sections back together. It's really quite exploded if you just look at it. We had less exploded than if you just looked at the yellow action. So there's an intermediate picture there. Which is a definition of one portion of a nearby point. Yeah, these nearby points.

1:40:00 So there are enough nearby points to actually distinguish. You can see in the middle here, we find that gradient dimension is 2, which is stronger than usual. But then, very often, this is exactly what you need. Exactly what you need, yeah. So, it's basically equivalent to... Now, I mean, the Luce-Telzat just says that the intersection of the maximal ideal is a set of nil potent elements. Yes, that's one way of formulating the standard Luce-Telzat for feet, but this form is a little more precise. You can get control of the nil potent element. Yeah, yeah. So, you know, so I conjecture that the analysis thing can be applied, that this intermediate level should play, that the things were, because those finite dimensional local rings, you see, they have an interesting property, they have an inoppotent endo map. If you divide out by the maximal idea, you get a field extension, but that has a section. Yes, I'm always assuming separability. That's what came out from the other definition. But then you have this idempotent, you see, which means exactly that the left and right adjoints agree. Because by splitting the idempotent, you get simultaneously the left adjoint and right adjoint. That's why, you know, the saying that the... Components and points are isomorphic. It's the same thing. In fact, in any situation where you have left and right adjoints isomorphic, really it's all coming just from a central input from the category. How does this look in the case of... I call this quality. This is intensive quality.

1:42:30 This feature, having a central hidden poet, or equivalently having left and right adjoints that are equal, is true also for homotopy. Completely different. Extensive homotopy. You know what homotopy is? The homotopy category is over sets. There's an adjoint. You include the discrete spaces, of course, have their trivialities, the same homotopy as themselves. But, with that structure coming back, there's both left and right adjoints. At the first level, if you look at maps from one, rational points of the homotopy type, those are the same as components. Those are the same as components. But that's just the beginning. In fact, that carries on to every level. The functions are represented by one, which... Secretly, it's really components. It's also morally the points, but not only morally, in the precise sense of the switch which actually it is, it's the sort of points and discrete inclusions. Also, so you take the protection from a multiplicity category, you said. The left and the right are John Cranston's. You have to, yeah, I mean, it's of course, but it's almost like a trivial statement, but then you realize, well... Other adjoints are not like that at all. No, no, no. I understand geometrically why it is so, but of course it's a new feature to how to left and right have to put it into a general adjoint. Another example, by the way, is graded commutative co-algebras. And then ungraded ones below, and the underlying is just to pick off the zero component. So then, you can include, you have a single thing, you just put zeros, and that's the inclusion, but then you can, given the general one, you can extract the zero part, and you can also mod out all the higher degree stuff, so this modding out...

1:45:00 It gives the same result as plucking out, so again left and right joints agree. Well, it is more or less what you had before about the section which is a sub-V, which is one variant of Ensen-Lemaitre, by the way. You put some Ensen-Lemaitre in the finite epoch, and Ensen-Lemaitre goes to the limit, but the point is the limit is off, the main point is to have it at finite levels. Yes, all finite dimensional looping and yes, and we separate with some, which isn't a dilemma. Yes, but what I'm waiting for, it is there at the point. Yeah, yeah. So according to my weird philosophical terminologies, homology is also a quality. Homotomy is a quality. Homology is a great co-op. Quality is just A functor that has left and right agilites to green, but then, I mean, that's sort of floating quality, but concretely, if you have some sort of interesting category, you may have a functor two quality, and that measures, you see, given in the object, its value is the quality of that object, which can again be that kind of quality of that object, two levels, that can then be measured by steady numbers. So there's both intensive and extensive qualities. You see this business about the hot and the cold or the local rains and all that, that's one completely and seemingly different sort of from the homotopy and the cohomology.

1:47:30 But they both share this feature of being quality when you attach them. One's coming in on the left, one's coming in on the right. Well, there's no objection, because it's true in the case of complex numbers. It's just misleading that people then think, oh, the way to define points is to take some kind of ideals. What is immediately wrong is that you go away from complex numbers, or from logic where you have two. But in any other case, you have really a whole category of things where you shouldn't take them. And then moreover, since there is group action on the points, the maps won't be determined by the ideals, only up to isomorphism, so by taking the ideals you've formed a... No, so, you know, Gelfand was perfect. Nothing wrong with what he said. It's just that... No, it's a special case, isn't it? Because, you know, analysts, you know, they go around saying maximal ideal, maximal ideal, really a maximal ideal there, when in fact that may be a bad... It's assumed, for example, that's right, people try to imagine, well, what could be non-communicative algebraic geometry? Well, this is really just a kind of speculation for the most part. So they say, well, we should take some kind of ideals, but you see, they've already been possibly misled because one should have taken homomorphisms into, it's just kind of a ritual which may or may not be justified, I don't know. Well, first of all, it was historically a little misleading because it was, well, it's... Too much emphasis on the maximal at the extent of pi and the next step was to consider pi and I remember when Searle first considered something in his first series of lectures and he insisted that we should call the spectrum of the intercept of maximality and then he had to I remember And then he had to make some assumption, what he called Jacobson ring, which properties at every prime ideal.

1:50:00 I mean, you can detect everything through the point corresponding to prime maximal ideal. And then it was both Martino and myself who convinced him to. But in non-competitive geometry, when I heard the discussion about non-competitive geometry, at the moment what they considered in codes is a derived category to start, let's say, from a projective value. And then you extract some features of that category and you say that's what they buy by exploring its exploration. But some good reason that we don't know. But in some sense it's not really the space. No, no, no. It's precisely some elaborate version of the extensive quality of it. The substance of it has been forgotten. Yeah. It's not the point. It's the idea to have a non-commutative topos. So the hot and the cold is a nice picture, but they together constitute what I call the substance. In other words, what the space is made of, independently of its form, you throw away that and you still retain what kind of stuff it is, and so therefore its behavior in going from gaseous to supercooled and all this. But that's part of the... pretend you could build a house out of it.

1:52:30 It has a form, and even the plan for the house is still the same form, but neglecting the substance. But that's still a quality, namely the homotopy sort of thing. The homotopy is just a new category where the new homes are the components of the old homes. It's a definite construction also. That's the form neglecting the substance. So the way, I think, these things about the right categories, which are very, very interesting, but really it's just a form aspect and neglects what the substance of the space might be. No, no, no, no. I'm trying to interpret your image more than it's going to want to be. What happens to the notion of reconstructing an object, a given object, from its points? I mean... Well, that's the main challenge, of course, that's the challenge. Well, you know its form and its substance, i.e., say, its homotopy type and its hot and cold manifestation, and we still reconstruct it. Well, I don't know, maybe in the case of reflexive graphs, it was just thinking. Maybe you could reconstruct in that case, I don't know. It seems like it's... Well, there'd have to be some condition. There's something, it depends on the topos you're working on. So the topos you're working on, there might be some... Additional feature, which is actually telling you how to assemble a substance into the form that you're throwing away if you just extract the measurement of those two qualities. I don't know, it's a sort of a... the next thing I was going to do was try to calculate it in the case of reflexive graphs at least. If there's a clear example where you... I mean, is it something like... well, I mean, a sort of simple analogous... well, cases are... You know, as with sober spaces, I mean, where you reconstruct, essentially, sobriety means that you can reconstruct the space from its points. I mean, that's a very simple, you know, in that case, you have a very... Well, only up to a monomorphism. Yeah. In other words, that's a stronger no-show and such type of thing, but still a long way from reconstructing the actual object.

1:55:00 Just having enough points doesn't even really allow you to construct a resolution of the object. Having enough projectives in a different situation, that means you can cover an object by a projective and then you get an induced equivalence relation that you can again cover it by a projective and the idea is that that resolution really does determine the object. Points behave differently. They're partial information which might turn out to be, if you have enough of them, to be faithful in what sort of only very, very extreme cases to you. There's that business about Dania is not always sober. Yeah, that's right. Scott is not always sober. Scott is not always sober. You know what I mean. Yeah, yeah. That's very much because, of course, the points have some kind of structure. As points. They're not just bare. And so if you take that into account, then you can say, well, can I reach out? And you would think, right, the Scott topology on a... What about the case of synthetic differential geometry? You have these very adjoints, you know, from the, say, smooth tapas to, that's an interesting case to look at, to discuss.

1:57:30 To tapas too? Yeah, to the category of sets. Those are good. Yeah, I mean, you have, right, and you can get sets in two different ways, you know, by taking double negation sheaves or... Well, I don't know, I don't know, I was thinking about that last night. Actually, I was always hung up on this idea that sometimes... Rudd-Bick has this construction of real and petite toposes, and it always turns out to be a subtopose, and often turns out to be a quotient topose as well, with a retraction. On the other hand, I had a construction, a very simple case, with points, and it turned out to be a quotient, not a sub. So if you have a subcategory, a full subcategory of the topos, it's closed under finite limits even, so all the simple algebra sort of looks the same. It could be that it's a subtopos, it could be that it's a quotient topos, it could be that it's even both with their retractions, but it's sort of confusing from an IE point of view, you can say, well, these objects ought to be playing that role, well, you suddenly find they're on the other side. So this idea of trying to extract for a general topos, an external way, some approximation to discreteness. I mean, there's taking the double denation scheme, which is part of the sub-tokos, then there's Johnstone's netoyage factor that gives you quotient tokos, inverse image being inclusion, but the so-called direct image being the extraction of the locally decidable portion of any object. But maybe it's like hot and cold, I was thinking. Maybe I have to accept both of these things and of course there's an induced map between them which goes to the original more interesting topos, but what you have then is this Boolean one and this QD one. Boolean is a special case of QD and a map between them. So maybe that is what, in a very general sense, would be playing the role of, quote, discrete, end quote.

2:00:00 There's really just enough non-classicality, so to speak, to allow for the presence of these, well, there's only just enough, you're not... You really haven't produced a kind of non-standard. It's not like non-standard. I mean, you could do non-standard analysis and such things, but that's not fundamentally what it is. It's been standard for 300 years. Yes, that's exactly right. Recognizing it. No, sorry, that's just a little bit. No, no, I think that's true. So you're saying that the logic is... But it's very close. Well, my point when I wrote that little book was... In fact, I didn't want to say anything about, well, I would have said something about the logic, but, you know, just parenthetically, that you don't really notice any of this. Of course, one of the reasons why smooth infinitesimal analysis, or analysis, right, I mean, infinitesimal calculus, right? It doesn't use non-constructive reason. I mean, even though it's necessary, you know, to introduce... Oh, well, maybe that's your confusion. Yeah. Because hiding logic is one thing. ...constructive philosophy and all that kind of thing, sure, but you do need to introduce, you know, if you, when you look at the, there were puzzles, of course, in the use of infinitesimals, there were actually also logical puzzles, you can see the history of it, that there were difficulties, but actually it only required a very minimal, you know, it sounds like, of course, difficult to construct the models and so on, but it only required in some way a minimal... There's a sort of recognition that, well, actually in this case of the algebraic, you know, the fact you could use, that you used, you know, potent quantities, that really doesn't change the practice. What I mean is it doesn't change the actual practice of these computations with, you know, derivatives and so forth. It just really simplifies them. You know, after hundreds of years of trying to get this straight, it's interesting that with full classical logic, you had to do quite drastic, you know, very drastic, perform really drastic operations, right?

2:02:30 You had to throw out infinitesimals, you had to turn everything into discrete sets, right? Fundamentals, you would call it cohesion, that was already there and so on, we read, right. And yet all that, it looked, and then later on, right, in synthetic differential geometry, which is moving to the decimal now, you realize that in some sense only rather minimal... There were a lot of arguments that went on at the beginning of the 19th century that people didn't really want to use limits and all the rest of it. Lagrange, for example, didn't, although this is the 18th century, but a lot of people were quite reluctant to jump on that. And so on and so forth. Yes, and it only meant in a certain sense rather minimal, rather minimal adjustments, I mean, which doesn't actually affect the practice. I mean, you're still working with more or less the things you had before, which were, it's true, not very, perhaps there was some vagueness in the notion of the infinite test, but there was some definite intuition, if you like, a feeling of cohesion or smoothness or continuity, which does not form out with the discretization of it. If you have a better word, I'd be happy to enter. I think cohesion is a very good word. I couldn't think of any other. No, I think it's a good word. And it doesn't seem to... You don't want to say connected. It doesn't mean it. Oh, no, no, it's not connected. It means hanging together. Continuous, but it's... It means hanging together. ...tuzamen haut. It's translated, you know, cohesion. No, cohesion means hanging together. Adhesion is putting two things together and sticking them. So cohesion is definitely... I've always thought so. And it hasn't been... Co-opted by, that's right, that's right, exactly, exactly, that's right, all the other terms have. I'm in good shape. Yeah, yeah, nobody, it's interesting, no, that's a good point, I mean, nobody, it's true, that word, that term hasn't been appropriated, sort of co-opted by, I've never seen it used in mathematics, you know, there's connectedness and connectedness of the small, you know, there's all these terms they try to use, but funny enough, cohesion doesn't seem to have been one of them.

2:05:00 These two qualities could also be called large and small. The substance is in the small, the quality is in the form of the large. That distinction, that dialectic, was noticed long ago. Anyway, can I make this more precise, your point about how this has become so mild? A mild extension of classical logic? Well, it's just that you don't notice. The point is that computation, I mean, just looking at it from a level, just say basic calculus. The principle that using infinitesimal, the infinitesimal never appears at the end of a calculation. It's a basic idea. I mean, it's a sort of intermediate step, which is discharge. And this idea, of course, of the use of infinitesimal, is very old. You never do the, of course, it was mysterious just how they entered, right? But the point of smooth infinite, and this was, of course, there the difficulty was precisely that, in fact, it violated the laws of long, long contradiction, and this had been pointed out long before, you know, that you assume at some point that it's zero, but not zero, and then you divide, right, and then you divide by, and so on. I always meant to follow up on you, because it was actually Nick Goodman. I think Aristotle already, I always meant to study that. Actually Aristotle considered hiding logic and co-hiding logic and said, well let's start with Boolean assembly, rather than the law of excluded middle was really a conscious oversimplification as a starting point in the whole enterprise of formalized logic. Rather than, you know, an eternal wall laid down by Plato, that's a different thing, right? Yeah. But I think in this case, the reason why... In fact, the fact that we have both hiding and co-hiding was sort of needed as an intersection. Yeah. So I guess maybe we can take the push out of it.

2:07:30 Yeah, perhaps, perhaps in the... I got the idea from what Nick was saying that Aristotle already thought about that. He did talk about it. In the middle of a practical program. Yeah, but he does talk about it with these future contingents. You know, the idea that you may not have the law of excluded middle. You know, he does recognize this. For tense statements, Aristotle. For tense statements. For tense statements, it's quite natural that you don't have the law of excluded middle. Of course. Right. But I think one interesting fact is that I think it was felt that, well, you know, of course there was analysis also of variation by Pagel and the idea that somehow you would... That this would, that you would need some of the logic of contradiction, a dialectical logic in the sense that you would have a law of contradiction, the law of non-contradiction would be violated or would be violated in some sense. The interesting thing was that the, it was argument about the law of non-contradiction rather than the law of excluded middle that dominated, you know, all the discussion. I mean, certainly the basis of dialectical philosophy, you know, the whole idea of... The law of non-contradiction is the thing that's really questioned. Not the law of excluded middle. I looked into it. I always want to see where can we find examples. You find it actually curiously in Plato's Parmenides, which is a very difficult dialogue. It's very hard to understand. And where there does seem to be some kind of recognition that there's a connection between the law of excluded middle and the law of non-contradiction. To violate one is something like violating the other, and then you find it in Aristotle with the future contingents, and then, well, you hardly find it at all, it's the law of non-contradiction is the one that's, you know, of course it talks quite a lot about the law of non-contradiction, but not very much about the law of the middle, and then of course later on the basis of dialectical philosophy is somehow the unity of opposites, you know, the idea in some way of violating the law of non-contradiction. Whereas in the case of smooth infinitesimal analysis, it's very interesting that if you actually look at some of the earlier arguments, well, you know, these quantities, these infinitesimals are really... It's easier understood as being things which are neither zero nor not zero, rather than being both zero and not zero.

2:10:00 Yes, hiding and not co-hiding. That's right. It's definitely hiding and not co-hiding, at least in many of the arguments you can see. If they recognize that, in a way, the objections that Barclay and others have made... Descartes, too, I mean, sometimes. He was unhappy with infinitesimals. Probably for this reason. But they did not recognize, Fermat recognized in his determination of minima, you know, his principle, that he really, he's sort of fudging it. I mean, he's assuming the quantity is sort of zero, not zero at one point, and then setting it to zero. Well, of course, you know, he realizes that there's something somewhat fishy about this, but it's interesting that all that can be resolved simply by assuming this is a quantity that doesn't satisfy the law of excluded middle. And yet that point was not recognized until, well, as far as I know, who's the first to recognize it? Peirce sees it. Peirce actually had the idea of rejecting the law of excluded middle for positions. And a continuum before Brauer. He says he said this at the end of the 19th century, but it's quite interesting that it was the law of... I think probably because the dialectical philosophers hardly mentioned the law of excludable. It just doesn't play, of course, a central role, as far as I know. It certainly doesn't in Hegel or in his successes. Marx doesn't mention, you know, in his work in the Calculus. It's very interesting. I'm sure you've seen it. His notebooks on the calculus. Well, I mean, I gave a specific model for you in my paper. Right, right. I mean, he doesn't also really consider this... I don't know. It's an interesting point. ...an identity of opposites. Yeah. Oh, I see the way out. But you're saying that it's a... somehow the infinitesimal calculus. All of these are mild extensions. Yes, yes, I think it is. I mean, they made this drastic, of course, it was recent. I mean, one way in which it's modern is, you know, you have this first order infinitesimal object in a topos. Yeah. But the topos is the smallest topos containing that object. Yes, that's right. That's a very important... Because if you, you know, if you take, if you take T to the power of T... That's a beautiful... That already contains the line as a retract.

2:12:30 Yeah. I make multiplication by scalars is faithfully represented just by multiplying. I mean, I call this Euler real. So just having exponentiation, which is any sheet subcategory is closed under exponentiation. Then products, equalizers, so you have all the algebras, varieties, and so on. And that's normally taken as a site for the gross topos. So there is no topos in between. There are lots of subtopos, but none that contains this kind of object. You see, for me... It doesn't regenerate it. Yes, yes. Well, I make this point also. But I don't think that really makes the logic. No, no, no, really, I don't think that has much to do... I don't think that makes it... No, no, no, this really has nothing much to do directly with the question of the log... I mean, most of these extensions turn out to already have the maximum... Yeah, yeah. ...complication. Like, as Peter Fry points out, the sheaves on the unit interval, in the usual sense, the unit interval of ordinary sheaves, faithfully interprets the full intuitionistic development of higher order calculus. But I think this is an important point. So in other words, it seems like a small extension, but it's already maximally complicated from the logical point of view. Yes, but I think this is an important point for the methodology, well, philosophically for the methodology of science, because I remember... Hermann Weyl was a great advocate of the idea of infinitesimal physics. I mean he says it in Face Time Matter, he says it all over the place, that somehow you gain an understanding, right, of the physical world by analyzing its infinitesimal parts, and he, of course, he says this goes back to Riemann, and of course it may go back further than that. The other Hermann Weyl does say also that you have, he admits, At some point, that you have to have limits. He says this, and I can't remember the term because I've got it all in my chapter on my own book. You mean in Space Time Mathematics? It's not in the Philosophy of Mathematics and Natural Science. He says the purpose of limits... That's not part of the continuum, is it? Yeah, but in the Philosophy of Mathematics and Natural Science, he says really the purpose of limits is to link. You know, for him, in physics, the purpose is to link this micro-world with the macro-world, because in some way you don't have any, this very simple infinitesimal world where laws are essentially linear, you know, and you have great simplicity.

2:15:00 Just isn't linked. It's just not connected with the nonlinear world, so to speak, the macro world, except through limits. Well, I think the idea that it's generated, that everything's actually generated by the infinitesimal object, establishes a complete link, I mean a complete linkage, without the use of limits. Exactly. I've often wished Weil had lived long enough to, you know, to see these developments. He died in 1955, so... And he was very concerned, you know, that there should be some very natural way of doing, of, well, really of founding the continuum, of course, that was what he was looking at intuitively, and I think it's a pity, you see, that he didn't see this, because, well, I mean, there's, this realizes, you know, the idea of the connection, that the macro world, you know, the world, the physical world of observation, somehow emerges. All of this is generated by this simple infinitesimal world, and of course that's what was something that would have pleased him greatly, because he thought limits had to be the linkage. He couldn't see, of course, the way that you could get it by this means. I think it's a very interesting, a very important observation. This link, what I said, you know, is presently generated. This could not be... We assume this being is here, this category is here, and then we investigate it, but it gives its own notion of natural transformation, you know, the morphism of natural transformation, and that's going to affect everything, right? On the other hand, I mean, well, you always speak of the first order infinitesimal, but the whole hierarchy of infinitesimal is more complicated, I mean, but it is a second order infinitesimal in your opinion, because these are just exterior powers, not symmetric powers. So you get those, but it's very pretty that it really is generated, you know, that they're generated essentially by the first order infinitesimal. It's the simplest case. And then, and then, well, real numbers, real numbers are, you know, are quotients.

2:17:30 They're quotients of infinite decimals. I mean, like the way there's zero over zero these days. It's not a joke if we interpret the word ratio correctly. It's not a calculation. No, no, no, no, no, it's an attempt to invert multiplication, so it's a whole story about that. Yeah, that's right. Well, actually, the theory of second order is not... As it is, you can say that the test space can be generated by iterating the tangent pattern, t to t to t to dx, and in that you have room for the tangent pattern. Yeah, I was thinking that actually this business of dividing by n factorial is just a special case of dividing by a finite group. Of course, of course. And I think there's been... For a long time, the realization that taking the orbit space of a group is a much more controlled kind of quotient than a general co-equalizer. At least for the discrete group, probably you can tell me, for more general algebraic groups. The reason it's so concrete is that you can average over the orbits. So again, it has one of these distributional sections that I was talking about before. There is a sort of relatively concrete type of epimorphism, which has a distributional section, and I suppose that for some nondiscrete groups that are small enough to integrate over in that. Well, I mean, it is known as the notion of integral in the whole function. Right. Well, it's not. For Feynman, they are basically just...

2:20:00 There are two that seem to be close to the class of Oppenheimer and Haydn in general. And basically, if you're Oppenheimer and Haydn, someone will count you with maybe some extra assumption. Now that's exactly it. One more thing. You said, don't go to the cohomology, keep the complexes. But in the quotient, what I've learned in recent years is that Let's say if you want to, you have a finite group or a group of a geometry on a space X, you want to take the orbit space, but it's only in, but what encapsulates really is the equalizer itself, I mean, or the groupoid, I mean, there's... Thank you for your attention. One of the basic ideas in non-commutative geometry, as I see it, is that non-commutative space is a quotient of a reasonable space, commutatively one, by some equivalence relations, some completism. If you have x over z, as I said, you can make this group of it, so the x is a set of points of this group of it, g times x is the set of arrows. And, of course, there are the two projections, g times x, two projections over x. First of all, g, x goes to x, and g, x goes into g, x. And the composition is, of course, the groupoid. But then, what we have is that, for groupoids, we have a reasonable notion of Mori-type limits, and that is the question of bimodules.

2:22:30 In a sense, when you have two group weights, or Morita Morpheus, I would say, Morita Morpheus is a bimodule over the two group weights, it can give a different sense. And so, once you have this category, so you have, the category you have, I mean, one way to construct non-commutative geometry is to start with a category of group weight, So, when you say bimodules, you mean linear ones? Well, but there is a non-linear version. These are actions? Yes. So, what you have is that, I mean, if you have, what is a generalized morphism between two groups, or even two groups, or two groups if you want, I mean, it's a space with two actions which commute. And then, if you have, how do you comport, up to equilibrium. And so, these are the morphisms, these are the morphisms. The morphism from G to H is a space X which has a left action of G and a right action of H up to isomorphism of space over G times H and a two-action commute. So it's really a space zone where G times H operates. But then the composition is just that you have G, H, and K. And then you take X, E, Y, and then you make a contraction, X times Y, all over the intermediate, you eliminate the tensor product, it's a non-linear version of the tensor product. And then, now you have this new category. And of course, now in this new category, that may become isomorphic, as is exactly more spectacular. Two rings are multi-element, if and only if there is a bi... In this category, there is an item of information between a bimodule over AB and another one over PA, which is a proper correlation. And so, now, but to have a quotient of a space, it's given for us, it's given tautologically. As I said, if you have a simple situation of SpaceX and a goopji which hang, out of that you build a certain gooproid. But you consider this gooproid in this new category. As I repeat, the strategy is that you take a category whose objects are familiar, but you take an unfamiliar notion of morphic, which is exactly how you build the right category,

2:25:00 So, most of the examples of non-committal space arise in this way, and the advantage of that, from the point of view of Kahn, is that if you have a group G, you can take out of that, you can make a Banach-Hajima, the integrable function, the L1 function with convolution. This is not a C-star algebra, but there is, of course, the fact from C-star algebra, the embedding of the category of C-star algebra into the category of Banach algebra as a reflexive subcategory. So you have the C-star-enveloping algebra of any Banach algebra. If you apply that to L1, then you get a C-star algebra associated group. But the remark is that this construction, which is well known for Goop, extends to Goopoids, if you have a Lie-Goopoid or a topological Goopoid. And so, but now what you have is that, so you have a pure, I would say, a non-linear theory of Goopoids with Morita morphism or Morita equivalent and so on. On the other hand, this Seastar functor brings you back into the world of Seastar algebra. And it's almost ontological that if two group weights are more equivalent, the corresponding system algebra will be more equivalent in the algebraic center. And that covers most of the examples, almost every example so far has been obtained. And now the point is that as Gabriele has shown, as Gabriele has shown already in his thesis, If you have two... I mean, you cannot reconstruct a non-commutative ring from its category of modules, but the center of the ring can be reconstructed. So, the interest in ordinary spaces correspond, or commutative spaces correspond, not to commutative algebra, but to algebra which are more reticulant.

2:27:30 I mean, that's really what we learned from Kahn. For instance, if you have a space, let's say a compact manifold, a smooth manifold, then you have the algebra of continuous function or smooth function, it's a commutative algebra. But now if you replace this algebra by the n by n matrices over that algebra, it's a non-commutative algebra. But more or less by definition, the more it happens. So when you go from the ordinary functions, scalar valued functions, to the matrix valued functions, you don't change the space. That's the idea. And that, but it was, I mean, can't use that to, because now if you have, if you have matrices, you can do a great deal, I mean, by characteristic classes, churn classes, etc. That was the original remark was, came from, from Karubi, that you can calculate, you can define the churn classes and all these things by manipulating not ordinary differential forms, but matrix value differential forms. And, of course, the practice of differential equations and a completely integrable system in the sense of Frobenius and what physicists call Gagefield is exactly that. So the practice of Gagefield, and more and more people have recognized that Gagefields are matrix-valued differential forms or differentials with value in an iagema. An iagema usually is a sub-agema of the matrices. So, of course, when you consider differential form, I mean, well, going from the function to the differential form, well, the differential form from a grade, an algebra with a grading mode, integer mode, it doesn't make much difference, but then, but of course, the differential form are commutative in a graded sense. That means A, B is up to a sign B, A. And so, I mean, in a proper, with a proper coherence, McLean coherence condition, it's still commutative, well, you know. Yes, yes, certainly. It's common. Relative to the natural symmetry of the tensor. Yes, natural symmetry of the tensor, but you change the symmetry of the tensor upon it and you get from ordinary commutative algebra to graded commutative algebra.

2:30:00 But then, when you really go to... If you really go to the matrices over differential form, you are really in a non-commutative world. In a non-commutative at least for the computations. You do non-commutative computations. But you know, because of monetary equivalence, that the geometrical content has not been changed. So, it's one way to do non-commutative geometry on and out in high space. But then, suppose that you have a quotient by some group or something like that, then you are in a genuinely non-commutative space, which means that, let's say, if you take the simple example by Kohn, the starting example was foliations. So, you have a foliation, you have a foliation, and in the wide foliation, in the sense that, suppose, in godicity, there is one leaf which is dense, every one dense, and more than dense, and godly, in the sense of that. Of course, if you take the quotient, I mean, it's nothing, it's nothing. The set of leaves, the set of leaves has no structure, no topology, not even a measure of fear, yes, because, why? If there is a dead slave, I mean, the topology on the space of life is trivial, I mean, on a different level, I wouldn't call it a, I mean, the topology is only the empty set and the whole set, absolutely, chaotic space, chaotic space, chaotic space, but more than that, due to the algorithm, you cannot even do measure of pure and that, I mean, all the sets, I mean, the measurables, the measurables, the Bohrian measurables are either trivial in this set up to nulls, up to measure zero sets, they are. And nothing or all, nothing or all. But then you build a group void corresponding group void, which was introduced by Hefligel, and many people in the 60s, and then you have an interesting group void, and this group void, and this group void, and this group void, and now, of course, you have your different foliations. If you have different foliations, each foliation carries, they define its own group void. But now the more it occurs between group voids. It gives you an equivalence among foliated space, or rather among the, under quotation marks, space of leaves of the foliations. And so, really the basic insight. And you are not very, while the points are not, don't play a very important role.

2:32:30 No, no, no. The points are not an interesting thing. So, Broden proposed the idea of état du. Yes, exactly. So, what's wrong with that? In other words, I tried to convince Kahn that... No, I mean, to me, one of the big challenges. But I see the crucial point, I think. It is the linearization, really. Because you're looking at by-actions on modules, not just on sheaves. So, in fact, there is this kind of coarse equivalence. But you could say for topos, namely, well, for example, to put it in a concrete context, in every Grotendieck topos, there's a well-defined complex number object. I mean, you could talk about range topos in general, but let's say in each one, there is the sort of the Dedekind version of the complex numbers, which was probably the appropriate one. Now, if you look at the category of all modules internal to the topos, over that, It's of course a Grotendieck AB5 category, but to have given two toposes, look at these internal complex vector spaces, as it were, and ask for equivalences between those categories, this is precisely, in other words, this coarse equivalence makes linearized coarse equivalence by bimodules, because of course these functors are going to be implemented by bimodules. To me, one of the big challenges for the coming years is to reconciliate the point of view of Goethe and de Tantin with his point of view of Kant. And very little has been made. In that direction, very little has been made up. Tarpiaou is from Toulouse and was, I think, a mathematical grandson of Erasmus. He was a student of Pravin who was a son, a mathematical son of Erasmus. I mean, had worked rather hard on that, but he went into very complicated... The technical question about, I mean, fresh algebras and so on, and I'm not so happy about this solution. There is something there, but I'm not so happy because...

2:35:00 Again, I mean, we discussed it already, I mean, this point of view of infinite dimensional manifolds and, I'll say, using this approach by a banal algebra of generalization, for example, doesn't seem to be the right approach, and so we should have to take it into account, let's say, maybe taking into account the idea about calculus of variation, that means that... You approximate an infinite dimensional space by finite order, they say. Instead of working with the space of all functions, you work with the space of the jet of finite order and the tower of jet spaces. Let's make it this way. Instead of working with the functional space, you work with the tower of jet spaces. And as I said, the work of Chen, K.T. Chen, K.T. Chen provides a model of the loop space of a space, by such means, and this is, I mean, the Chen work looked rather isolated in the beginning until people like, well, first of all, his main disciple was Dick Heine, and then... Many people still live on Darlene and so on understood what could be known and that's a great work and I think you could say the idea is replaced in the function space by the tower objects of finite order and that's the inverse system of the general space of finite order. But comparing, so it's a linear, it's somewhat linear theory, linear theory, but Conne works, of course, with, I mean, he came from, his field of expertise was operator algebra in Hilbert spaces, so of course he assessed algebra and algebra and so on, but more and more he takes a distance from that, more and more he's taking a distance from that.

2:37:30 And it's clear that you don't work with a full algebra of continuous factor or smooth factor. You work with, let's say, some sub-algebra which is finitely presented or something like that. And most of the calculations you do there, well, I mean, sometimes you have to go to the limit, but not often. Of course, I know in physical terms what is a program, and it's connected with what I described about having these integrals with a good period of time, that means, how can you integrate functions... Not only without knowing an integration, clearly, but even without knowing the real numbers. That's a challenge, that's a challenge, that's a challenge, and that's a problem. Are you involved with motivic integration and so on? That's slightly different. I know, yes, yes. No, not myself. I observe that from a distance. I think it's more or less the same basic idea but with different technical things. And of course, as I said... The goal is really to prove new transcendence results. That is how far we see rational or something like that, transcendence results or ambitious conjecture in that direction. But, as I said, I mean, in my opinion, you cannot do a series and Grottenick fully agrees with that. I mean, if you want to make a deep technical progress, you have to think about the conceptual aspects. It's not only a question of techniques. It's not only a question of techniques. And you cannot, I think, you cannot... Really prove new results in numbers, I mean, on the nature of numbers, whether they are algebraic, transformational, transformational, without a deep reflection on what is a number. But then, ok, I finish. Yes, it is, it's a bit more structured. But then, now I come to the punchline, I come to the punchline. As I said, so you have... A purely algebraic model of what is various classes of integral or various classes of solutions of differential equations. Everything is a purely algebraic theory of differential equations, integration and so on. But then you want to carry it out in the world of real numbers and complex numbers.

2:40:00 And the crucial thing is that you have some positivity conditions. But in physics we have exactly the same problem. All the math, I mean, is fine-minded, functional, intuitive. I mean, at the formal level, that force pays fine. And I've made a fool to put it on a, I say, half axiomatic basis, because it should be futile to try to put it in a completely rigorous axiomatic basis. But half axiomatic. To me, half axiomatic means that you are... You are careful enough about deciding the computation rules and being as explicit as possible on what is permitted to do in a calculation. It's not a complete axiomatization. I would say it's calculus prior to the 19th century. In the 18th century, people knew perfectly well how to deal with integration by part and so on. Assume that there is a class of functions that you can manipulate according to certain rules, but they did not know the boundary, they did not know the boundary, they did not define the extent of the world of functions you manipulate, and I think we are in a situation where axiomatization to me means that, I mean axiomatization, so you have a first step which is, I mean, you have, I would say, algebraic calculus, and people speak of algebraic analysis and de-module and so on, you have an algebraic analysis. And which is a certain rule and you manipulate. Well, there is some infinity. I mean, when you say sum of one over n squared, that's infinite. Bad, bad, bad, bad, bad. Not really, not really. And then what you have in the next step is Dedekind and Cauchy and so on, which is to delineate exactly the domain of function you are. And at the moment, I consider the first phase half axiomatic and the second one half axiomatic.