Discussion — 0-Minimality program, tame topology, ramifications & other topics

0:00 The alternating sun is more invariant than the indivisible frame. Yes, yes, there is that. This is true. But nothing is standing with this either. Of course, this then precipitated a great movement in logic of passing to all the characteristics of general theories without giving... I've never mentioned the word side of it. Certainly not. I'm sure in many cases really arriving at... ...results which are literally correct, but must be wrong. I mean, they're just not the right... the right question has not been asked. I mean, it's... You get a general idea of different grades sometimes. Yeah, yeah, yeah. They have a natural value. The pianic question, they... Yeah, they're a correct example. Yeah. They claim everything collapses, but I think that something is being ignored here. The dependence on P is being ignored. You really need to look at it a bit more delicately. It collapses, it does collapse, but the process that makes it collapse depends critically on P. If you pass to infinity, it doesn't collapse. I mean, to me, that is the deeper reason. That's somehow behind some parts of material integration and so on. You know, I think let's look more closely at a lot of these things. All sorts of quantities. I think that the lesson was clearly made by K-theory. If you ask, you know, what is the dimension of a vector state? Okay, that turns out to be an actual number. But if you ask, what is the rank of a projective module? Well, it turns out you first have to calculate k0. That gives you the entity in which the values... The particular values are going to lie. It might turn out to be the natural numbers, it might turn out to be something else, these two levels in the memory process. This is a strange thing in logic. I mean, the only dimensions we have are those coming from the set kinetic environment. They're ordinal. They don't seem normally the right kind of thing. Whereas they ought to be some kind of formal structure. It's not even an iterative notion. You might calculate it as something in a different way. Continuing with Douglas Vanagry's, I mean...

2:30 Tchaikovsky, you see, now this was, I kept telling Steve, it was kind of a delicate situation, he came to Steve as an advanced graduate student, he had these brilliant ideas in a certain direction, all about basically group objects, and the amazing groups of cardinality zero, which actually has a content, it wasn't very, but to see the relationship was, That this guy was, you know, just totally locked into the logical terminology. And Steve would tell him these things, would even go to the trouble of translating what he actually thought. The idea was he wasn't going to disturb this guy, so this guy was, you know, really doing good stuff, so to call him aside and teach him a little elementary category theory might have deviated, and therefore Steve didn't do it. But that was bad, you see, because in the end... In the end, the student didn't learn any category theory whatsoever, and he developed an actual aversion to doing so, and so his results may be known to you guys, but they're not known to the category theory. Exactly. I mean, I personally have remained very detached from that particular movement. There's a lot still going on, but they're doing it in a tremendously mechanical way so far. They don't even seem to understand basic things like concreting duality and so on, you know, which would simplify a great deal of what they're about. I was shocked at a meeting in Banff, which is quite far, but they do take significant notions and try to translate them into words. It's never done, I mean, I've almost given up. It's hard to see any character. We interpreted the play around a bit with all the characteristics, non-standard models, extended parts.

5:00 You mentioned in that feedback comparison of Chevrolet's theorem of direct images of constructible sets to direct images of closed projectile subsets. Are these theorems equivalent or just the motivation? Well, it's not difficult to get the one for constructible sets once you know the one for projectile sets. I don't think it's straightforward to do it the other way. ...was known earlier, much earlier than the Chevrolet, many people would have known this in the 19th century, but the point is that the machine now, that their notions of dimension, what Zilber did, when he basically defined this form of machine, when she knew the theorem, then it was a game for getting a situation, getting some nice dimension machine. The same, exactly the same group of work analogies. It's an argument which is spelled out in Matsumura's book on communism, and I noticed a number of years ago that this was not in the command of this gentleman. He needs something on lithianity, but you've got it in both cases as well. You've got lithianity in these complex analytic cases because of the Barstras preparation theorem, that the conversion power series is the point for gerunds, which is the theory of ring, etc. So basically, that's the point. There's a lithianity thing hidden in it, and then there's images of those sets of quotes, and there's a general theorem that will give you something on it. But the question is a good one. I mean, I now remember when I first went to Yale, somebody sent me a paper for a three about the logic of projectors space, and I must have missed the whole point of that, I think I rejected it, but I often, I don't have the paper anymore, I didn't keep a copy, and I keep thinking, oh, I made a big mistake there, this guy really was doing the right thing, we were doing the wrong thing.

7:30 I mean, you don't require to introduce negations to deal with it. It's a theorem. By the way, if I just might mention it, it's really a whole notary result, but, you know, in Bourbaugh Key, any book on general cosmology, it talks about proper math. Yeah. There is the, you know, the obvious condition that the inverse image of a compact is a compact, and also the condition that it's closed. Yeah, yeah. Now, the point is that the second condition follows from the first, actually, provided you work in one of these directed categories, like Horowitz's case spaces. Horowitz defined lectures in Princeton in the late 1940s. This notion of k-space, which is later discussed in Pelley's book, various are used by Seymour, Brown, Spanier, and all sorts of people simply because of the basic need for function spaces, for arbitrary pair of objects, function spaces that have a basic lambda transformation. Lieberman pointed out, I think, a well-known apologist, but rather late in this whole discussion, he had a small short case, if you get where, just the theorem that in the category of case spaces, if the inverse image of compact is compact, then the map is closed, which is almost a tautology, if you remember that the case spaces are characterized by the fact that a set is closed if and only if it's closed in any compact. At the intersection of compact and stopover, or if you take a map from a compact space and the inverse image will be fixed. That's the condition for a closed set. The very definition of what a closed set is, in other words, is determined by the fact that you have this site for the big chatting room consisting of complex networks. It's almost a tautology. That's a condition. Closed is automatically true. And that may be a similar argument. No, I mean, I think your question is a very good one because, I mean, another way of saying it is there's no easy way back.

10:00 Perhaps no way back at all from the construction category to the Zariski, those things. I don't see how you could deduce the projective version readily from the very avant-garde. But I certainly don't see any formal pattern whereby you can do it. But I'm still on, maybe I just need to think about this myself, but you do see this as the projective and applied version of the same thing. I guess, I don't know, it's difficult to say. I really think one should see, one should be in a situation as the algebraic jobs are, where proper maps are somehow central stage a great deal of the time. I mean, the fact that the cohomology of projective spaces and so on is projective algebraic varieties, not single ones, is nice. I mean, you can do, there are theorems, of course, you then have, for affine, for non-projective things, non-compact things, of course you can get cohomology theories, but usually you have to pass via compactification or something like that, and you change the sheaf, et cetera, et cetera. It becomes infinitely more complicated, and you then only get theorems under strong assumptions of a proper math. They have in the cohomology theory similar things. Direct image under probable maps of various nice sheaf things are nice. There are analogues that can show me things higher. These are typically very important theorems. Like the Grauert-Rundt theorem. Yeah, exactly. I mean, this sort of... I don't know. It seems to me unquestionably true that they... Well, it may not necessarily be a projective theorem, but the theorem about probable images of something nice being nice, these are always deeper corresponding. I don't know, you see, I don't know how much of the dimension theory is on the Morley.

12:30 It can be done in that situation. I just don't know. No, you don't use navigation. It's just that you recognize and see that they live in the topos even if they don't live in the site. In fact, I was going to put up a theme about the evils themselves. There is no such object in the site that we have no animal on the site, but we can always create a broad and deep co-post on that which will inevitably have such an object, which is even individually related to the first, but it's kept a bit outside, it's seen to be, I like to think it's seen to be a higher, you know… The higher level of idealization, and in the same way, the negation. It doesn't mean you shouldn't do it, you should have done other stuff. I think that it might be something that relates to one of the more difficult periods of the subject. Wilkie, I think it's in 1996, managed to do something for a while. I think John and I had noticed that somehow the dominant thing that made it possible to get a minimality for this was its differential equation, the so-called Fafian equation. I mean, some of the Fafian chain of functions, you essentially start with some basic stock of functions, the polynomials, and then you start closing under just a form of operation. You take, well, it's better to say it in terms of differential forms, but essentially it comes down to you take a system of partial differential equations. In the end are knowns and you express what basically what you do of the simple form that you take your new function f and you express all its partials as polynomials in itself and every of created functions and this is a this is a factory operation for creating more functions and Nowansky noticed that function statements may have a have good finiteness for that in in real situations that we have only finally many

15:00 In fact, he proved that the zero sets have only finitely many connected components. Now, so one expected that if you took all the Fafian functions, all the Fafian functions of the reals, that this would be an O-minimal for you. Not responding to the usual methods from Sovereign Minic Theodore Wilkie's earlier work, Wilkie eventually proved it to be an O-minimal by stating an outline, but it's interesting because it gives negation a very special note. You start with the zero sets of your terms. You can also take positivity sets, but start with the zero sets of your terms. And now you allow yourself the following operation. You allow yourself direct image. You also allow yourself closure. The logical closure of things. And you start building up a high function, certainly C-infinite functions, but less even. At any rate, you just start generating sets in this way. So you're not using the conventional apparatus of first-order logic. You don't have negation at all, and the quantifiers, one is the direct image, and the other is somehow the bunch of quantifiers that gives you closure. So he built up sets. Actually, the method had been attempted before by a Frenchman called Charbonnel. Look, he built up the sets this way, and he showed the following. If the original zero sets you started with had uniformity of the number of connected components, actually varying families, then the same is true all the way up. And moreover, that you will get finiteness of connected components for all the sets in the Chagall hierarchy, moreover, you will get cell decompositions, and in the end you will then be able to prove that the classes closed under negation, because you take a set, make it cell decomposition, well you can show you can get cell decompositions compatible with other sets and so on, you just get the negation out of complements of the cells.

17:30 My 64 paper on category sets was just re-published with those columns. It's in the reprint. And I realized, reading it, I noticed that I had done something that I had totally forgotten. ...constructed the negative sub-objects, and taken union of all the things that don't intersect with each other, but you see it was, you know, to do that was definable even then. Yeah, that's right, but this is exactly what Wilkie does. This paper is a very hard paper. I think it's in the Journal of the American Math Society. I rewrote some parts of the lecture notes in PISA. I think I improved on this totally. But I like this very much, you know, that you just deviate a bit from first otology at the beginning. You get negation at the end. But this remains isolated in our culture for the moment. No, I mean that somehow, I don't know, loosely, philosophically, morally. In real life, you have to really work to get to those negations, whereas this way of formalizing logic just makes it easier to say, you know, that's it, so somehow, concretely, you can sometimes annihilate the enemy, but that's not, that's writing down the symbol in a piece of paper, so somehow this idea that you can, well, it's the same thing with the alternation of quantifiers. In real life, you never can arrive at an arbitrary, long alternation of five players. How do you see this? I mean, this low-key thing is very beautiful. I mean, it's quite intricate. It's somehow different from the things that Jensen did in his alternative construction of the constructible sense. He took much more delicate operations than Gödel did, regenerated much more slowly, and had much more information at the end. And that's how you get the source of the tree and all the rest of it. You can never have seen the source of the tree by Goethe's way. This is similar. You just do it very slowly and you use the operations which are geometrically, topologically natural and work.

20:00 But, you know, you have to use Sarve's theorem, genericity arguments, things that the topologists use on a regular basis. But that remains isolated. No one has pushed that. It's also been quite difficult to really extract complete, constructive content. Yes, well, this is quite refreshing to me because I certainly have wanted to see the environment, our community, and I don't know why, I don't know what the block is. Well, you're not even talking about decidability anymore, that was the first... Well, yeah, I mean the thing is... To what extent is decidability equivalent to no infinite components? Good question. Yes, this is the point. I mean, of course, typically what has happened, there have been some developments, I don't know what the philosophical significance is, mentioned, because they also connect the group. At the schedule, which is cute, I mean, the, so for example, the case of the real exponent, once Wilkie had proved that it's minimal, one at least knew that there was no question of a kind of. Gross undecidability results of classical kind by showing that arithmetic was interpretable, because arithmetic couldn't be definable, and it couldn't even be interpretable, because you have definable choice functions, because someone gives you a definable function for all of these things. So, one at least knew that the theory may still be undecidable, but it's not going to be Godelian undecidable. There are other examples that make it undecidable that don't interpret directly. For example, they might interpret more common for groups. There are some theories of modules which are known not to interpret. It's known that no theory of a module will interpret.

22:30 But there are many theories of modules which are undecidable because they code the more common for a general group. So that remains a possibility, had remained a possibility before. And then, that's how I understood it, it was doing, and we showed decidability assuming channels. And the proof is quite constructive, Donald E. Shandling's computer science ability that would be useful to modern day computer science or something like that. The bonds are certainly many explanations. I think they are definitely elementary because just recently I came upon something which I think is potentially more interesting. This relates to, suppose you take, okay, so again the model theory, so this is another defect of the way that... As I said, they have this idea that they work with complex manifolds, but they typically don't, they work only usually with their analytic subsets, they don't work with functions, and they don't work with shapes, I don't know why, and this is really a glaring thing to do, because for example, you take a torus, it's got elliptic functions, through a real interpretation of that, it's curved situation, and various... Blah, blah, blah, blah, blah, blah, blah, blah, blah, blah...

32:30 I would say there's sort of a naive reputation of the idea, because you would think, well, let's take the idea of polynomials. I mean, we all, in a purely affine situation, we look at all maps from the line to itself. Well, that's a certain space, but after all, polynomials have degree, so there's a map to the natural numbers. But is that, in fact, amorphous? We have to check that. Spaces of functions and states of distributions in a way that's closely tied to the basic low-minimal geometry of quantum dimensions. You can clearly do it. You can clearly construct composites like, go and decrypt composites like that. But what sort of specific properties do they have? What sort of good properties dare we expect? Just, you're mentioning the comparisons here, and they're reminding me... Right! Yeah, yeah, I know, I mean, this is... There's a lot of many, many vocals. What can things mean in some species of functions, are you saying this? Functionals, also. Functional, yeah, yeah, true.

35:00 Especially functional, because you... This idea of universal naturality gives you an idea of a structure on a function space, hence the idea of a natural outcome. Say that again. Sure. I want to make a special case of this. A special case of this, which of course would definitely come up in any attempt to do any sort of learning. Yeah, I mean you integrate functions depending on parameters against... And really, we don't know the answer to this. I mean, the question has been formulated. There's only one result known, but we're near enough. I start in a, in a tail universe, like I say, it's a no-minimal universe. I've got a lot of nice, I mean, everything in science is still very measurable, I mean, that's just part of the self-evaluational functions and all that stuff. So now I have a new process, I can do this. In this parameterized period thing, if you like, I don't perform closed curves, I just integrate functions or a function depending on parameters over sets. Now the question is, to what extent can I create new functions, to what extent can I add these functions and preserve them in an algorithm? For sure the functions are not going to be in the original theory. If I start in the semi-algebraic category, I can get a logarithm. On the other hand, we now know that the law, combined with physics, is perfectly timed. The actual domain of the law. Well, that's another question. And this kind of question is beginning to come up. I mean, the Havansky result sort of said, you know, like, special case of Havansky is integrating a function. But it's really between definite and linear answers. Yeah. So, the Khovansky method would give you a function of x by integrating, say, between 0 and x, a function f of t you already have. That's especially, it's not, that's a fundamental theorem. That is known to preserve, it's the only case, I think, where we know for sure that it does, except for the following important case.

37:30 So, the semi-automatic rate category is defective. You suddenly create a logarithm. There is the issue of which logarithm. Havansky's results, or Wilkie's, et cetera, typically say that you could really add all branches of the logarithm. But they've recently come up with other examples for other differential equations where you can add one branch or another, but you didn't add them both. For the Euler equation, I mean, it's, there are, this is an interesting thing. What does the equation mean? There's a big singularity. And they know, you know a lot of them, there's a small family of them, there are, the solutions are different, seen from the point of view of a woman and a man, and there are two which are incomparable. I mean, they don't understand. They have known the phenomena of these generic cases, but for classically studied equations this has only become, they don't have to be addressed, because until now, in the Fabian cases, they, you could just add all the solutions, you could add all the, but now... The subanalytic, so that's the one right here, now I'll take the projective subanalytic, and now you're going to say you may have a family of functions, a subanalytic family of subanalytic functions, and you want to integrate those functions over a family of subanalytic sets, and you'll see you get functions of the two sets of parameters, the parameters of the function and the parameters of the sets. What can you say about these functions? Well... They needn't exist in the subanalytic category, but the two guys in there exist in this thing that Van Der Wies and I and Marker studied, namely the Tame Universe, not by adding the real global exponential.

40:00 Out of the subanalytic category, you don't have to go beyond the subanalytic plus exponential category, but there it rests at the moment. And of course one will have to understand these things to have a decent theorem theory, current theory, anything. There may be something else, but one just doesn't know. People are looking at it, but this is very hard. There may be some natural class inside here, some kinds of variables. The only way you can be sure of getting R, just getting a function of one variable, by integrating from zero to x, another function of one variable. That is okay, by Havansky and then by things that Speisiger and Karpinski did. You can't even take a function of two variables and create another function of g of x and you write from zero to x. We don't know what they're all potentially going to be. That's a big gap. But generally, about decidability, well, I mean, I suppose everybody's perception of these things is now a bit changed by computational complexity, because even if one proves these things, it's rare that it's a useful bound. But it's still, I still think it typically, it very often throws out interesting things if you try to go all the way to show that there isn't any kind of whatsoever. They have probably been dealt with some very deep conjectures, not yet formative, of Shanyo time about the valentines. I think I suddenly remarked in here about not really ten spaces. It might almost have been correctly, it was ten topoi, but in fact, I'm not sure what he had in mind.

42:30 It's probably going to get it later. It's more of the end of it, it's more of getting notions of tubular neighborhood, et cetera, et cetera. Also this idea of somehow working with what he calls isotopic categories. As far as I know, we haven't really done that. Do you still look for Tame Topos in you? Yeah, yeah, yeah, yeah, sure. Especially Tame Topos, but he makes a distinction at some point, I think, between space and Topos. I'd rather make a huge difference when he's talking about the so-called growth, the so-called fatigue. Yeah, yeah, yeah. Because all spaces are... Yeah, I just think... ...some sort of hibernation. ...on one space. Imagine either one in this context. Yeah, sure, absolutely. Of course, unfortunately, in the old minimal thing... The sheaves haven't really made any appearance. And this is a big defect also, right? There's also this passage from, in some sense, local to global, which come up there. One of the problems these problems can have with exponential derivative functions, but also even in terms of understanding what sense the theory of the unrestricted exponential is a limit of the theories of restricted exponentials. These are theories of different kinds. I mean the unrestricted is an example of it. All of these are known as polynomially bounded because the functions can be very very close. Polybounded, functions behave... So when you say restricted it exponentially, you mean just the domain?

45:00 Yeah, on a compact. So I decided I'm going to just cut off my... First of all I'll do it with minus one, then minus two, two, etc. Adjust on a compact. Right, and so each of these theories have been known to be a bit hairy, and also very guilty. But it's not the case that natural science has to follow theories. So the important things it is. Yeah, exactly, exactly. And what I think is stupid is that's the major defect in the culture of our students. They know an awful lot now. It's enormously important, the topology of planetary gatherings. Yes, exactly. This is coming up. There are a few of us who are looking at these issues. There are all sorts of things about art. I have categories how do they behave with all the products. You have to work in a systematically softened way and you certainly have to work with it. You have to work with the softening of the kinds of colorings you use. I mean, formalism at the moment is pretty ugly, I don't want to. But it's just in a similar way. Not finding tang-toe. Well, I don't think that's a discussion of this. But there's some distinction, if I remember towards the end of it. Towards the end of this discussion of this is the... I mean, topos. Can you locate topos anywhere? Is it in French? No, it's in the English version. Translator, I see. Yeah, it was on the, I got it off the, it was in the circle. Oh, and that's not a word that's used a lot.

47:30 Stratified topos, for example. He stresses that he needs, he wants to get stresses stratified. I agree, a nice stratification. So, what is the notion of stratified topos? I wonder. Yeah, I mean, he's talking about, of course, I guess in these stratifications you have to be able to move to a tubular neighborhood to do various constructions. So he says, the question of the daily science of a stratified torpus is even considerably simpler in that framework than the attained topological one. True, there is foundational work to be done here too, especially around the very notion of the tubular neighborhood of the torpus. It's actually surprising that this work, as far as I know, has still never been done. I guess it refers back to the work of Art and Maser, the early creators that they had, and very hard-going, since the context of Italian topology had been more than 20 years, so they were dealing with homotopies. No one, apparently, ever felt the need for it. Surely a sign that the understanding of the topological structure of schemes has not made much progress since the work of Akinmeza. I guess that would be superseded now with the work of Bogdansky and so on, but I guess maybe that's all there was. It was a notion of stratified topic. Once I had accomplished this, more or less heuristic double work of refining the notion of daily science of a stratified space or topless, which was a crucial step in my understanding of the module of multiplicities, It actually appears that as far as these are concerned, one can actually take a shortcut for at least a large part of the theory via direct geometric arguments. Well, there's more, but I just wondered if it meant anything. It's sort of a connection with moduli spaces and multiplicities, but I haven't understood it. The motivation there is that moduli spaces, often all you need to know is how far the moduli set has collapsed. Yes, yes, yes. To know about the root structures to know the laws of position. Yeah.

50:00 And that's the kind of thing that was happening very much at the time when we settled the composition and we had linear situations and the number of roots will suddenly change and stuff. And that is usually a manifestation of non-problematism, a coefficient drop. Yeah, that's right. Yeah. Which in the same way that Tarski quantified elimination in a smaller number of... That's essentially how we get in the middle of the world. Oh, about the number of variables in the program a few years ago, and there were a few non-trivial papers written about it, but it hasn't been since the lady who wrote one of the most interesting ones became president of the Canadian Math Society. Ah, was this Christian Rousseau? Yeah, I've never read it, but I've looked up these other papers. The basic you mentioned the Weierstrass I thought of this this program is still not carried out at all. I thought of it in terms of the Grauer direct image theory you see that this would this would depend on the real development of the logic, internal logic of it. One could give definitions of compact space and complex numbers and all this in a good way by the tokos and so that, for example, the Grauer theorem, which is about a proper map of analytic spaces, in fact, there's a more general, there are more general versions where neither of the spaces is an analytic space, it's just the fibers are a compact analytic space. It's a perusal thing.

52:30 Which, in some vague sense, lends support to this idea, namely, see there was the original theorem of Cartan-Sitter, which is just about a compact space, and it said the cohomology of a coherent sheet was finite dimensional, so I guess the idea is to actually use Grotendieck's There's a general idea of passing to a slice category in an even more serious way by using some of the higher order logic that's going along. So the slogan would be to prove the Cartan-Sayre theorem. If you use, quote, arbitrary topos, then you can deduce the Gauss-Rather and the trivial corollary by applying it to the topos of sheaths on this basis. Right, right, right. Some of the ingredients of this can be verified. You can get a definition of compact, so that if you externalize it, it doesn't mean proper math, for example. So things like that. I mean that's very attractive. I would really like to see somebody. I discussed this thing with Ruben. It was interesting. And also with Roussel. Oh yes, yes, yes. Some people said he wrote one of the best papers. He wrote one of the big papers. Not a big paper, but he wrote a paper on the Traceformer, was it? I mean one of the SGA's. I used the N-Weave. I thought it was very informative. I think it was around the S-G formula. I can't remember. It's around the trace formula. I regret to be professed not to know who Huzel is. I made just a momentary laugh or simply my ineptness at pronouncing the name. I don't know. Anyway, I don't think it's an S-G. No, I would be surprised. I've seen Huzel in... There's something called the Seminar Bala, you know, the functional analyst. I don't know far as well what's possible. Well, finally, I mean, I still have to admit, I cared a huge amount of time with this paper. Anyway, I went to this because, well, there's John Gray's history issue. Yeah, yeah, yeah. Where he mentions various expositions, more and more general and more and more clear.

55:00 Now, Huzel's treatment was preceded by some papers by Duody and other people, where they extracted the, the thing is that there's a functional analytic component to all proofs of this theorem, because you, you, you... You construct, you have some sheaves and you construct a resolution, and then you look at the sections of these sheaves, and the sections are given a functional analytic structure, like a topological vector space, and then you show that the... The homology of this complex would have to be simultaneously compact and open, you know, and therefore not finite-dimensional. This is the thing. So the theory is not compact and open, but bounded and open. Well, Doody observed that this is all much better if you think of these as more logical. He used the boundedness structure. Which in many instances is more or less equivalent, but it sort of works better in terms of the contorial variance system. And Kuzel incorporated that, among other things, into his expositions. So there's this big exposition, and he even uses toposes, but in another way, just in terms of the sheaves on a given space. But, I mean, that gives us the topos theory. The topos and derived categories, phonologicals, everything is in there. Some of it I don't think is really necessary. But anyway, so I discussed it with... The problem was that... Grotendieck says, well, I don't quite understand logic. You know, he didn't know those words. This was the point, you see. So somehow bridging this gap is really needed here. Nobody's in a position to do it. I'm in a position to preach about it. It should be done. I don't have another position I can do it. Nobody else does that. But I think it would be marvelous because it would really be a vindication of Grotendieck's general program.

57:30 On a higher level, because you can use the internal functional analysis, whereas the internal algebra is one thing, but the internal functional analysis is another thing, because it genuinely uses. It's a striking thing that we have. We have like, what is it, 35 years of elementary topos theory now, and we've always known that exclamation is one of the two axioms. Nobody uses this. Nobody actually looks at the examples of the functional side, and the quiet is just to skate around it. Well, people look at it a little bit to see whether you could put the hard inverse function theorems into SDG. And he did verify that these exponentials naturally had the Frechet topology. Yes, I agree. There have been a few series, there have been a few series favored on this, but it's something that most topos care about. I'm not sure that produced any first function, you know, so... Right. ...the seminar at Guantanamo, and again, he said, oh, this looks very, very interesting, but he couldn't quite judge, you see, whether it was crazy or not. That's right. I think this has been neglected by some of the magicians, that in the end, once you have these 10 universes, these 10 categories, well, there should be somehow only one isotopic kind. I suppose he means that the stratifications of the triangulation should somehow rather be the same in all these categories.

1:00:00 Classifying things up to be finable, but not a projection, as any must have been to be present in the semi-algebraic category. I don't believe that this is what Hooke is after. I'm just trying to figure out what it could be after. I mean, he's clearly much more interested in the specificity of the neighborhood. I don't know if logicians have really looked at this. I mean, what is, what in the end is the difference of entities? Singularity is true in the cases where you have the global exponential. If you're not predominantly bounded and you're in a no-minimal theory, you have the exponential. And almost every attempt at this can't be true. The first time I learned Robert McPherson, I told him once. It's a tip of some iceberg. The thing is, I feel like I've seen it, but it was almost staring me in the face. Chris and I had... Discussions, very often, we spent six months together in Southern California, and I was keeping encouraging him to study a paper of Rosenleaf, a formal paper on Harvey Fields. It's completely formal, and I love the algebra. It's sort of the algebra of L'Hôpital's rule of infinity and stuff.

1:02:30 I didn't actually know this material, and I figured that's about all he used to get this theory in the end. But he told me, I thought. I didn't exactly kick myself. I was like, how stupid can you be, really? You have this dichotomy. And you find that this genuine old man has got super polynomial codes. On the other hand, there's another limitation. We don't know any example on the reals where there's a function of trans exponents that grows more rapidly than any iterated exponent. And it's known that that is not so. In terms of the Hardy fields, I've rolled out this kind of question. Nixon found examples of Hardy fields, but we don't know if they... That's actually the free rig. Yeah? E to the x is, of course, the summation of x to the n over n factorial. You interpret over n factorial as modulo the symmetric group, and so that's the free monoid. Now, the concept of rig is simply what you get by applying the back-distributor law to a composite, but in this case a composite is a thing of itself, so that's e to the e to the x, and that's the free rig. That's the only time that E to the x ever, E to the e ever occurred in my life.

1:05:00 That's interesting. Yeah, yeah, that certainly is interesting. Because normally it occurs, I mean, it occurs, you know, it occurs in some sort of derogatory statement. I mean, the, the, unfortunately, in the Segal's book, from one morning of his series on, that is a nice thing. I haven't been aware of it until I started working on elliptic. He has this, um, he clearly thinks elliptic function is financial. The point is that you can get the exponential function by, from the elliptic functions, by, you see the Varsha's elliptic functions depend on a, on a bit, but essentially a Varsha's function is given by, take the number one is one of the, you know, scaling, and then the other one can be a, so a Varsha's function depends really on a tau and then on the variable z, so it's really a function of two variables. And Siegel shows that, and it's recently been rediscovered by someone of his, that if you let tau go to infinity, you can pick up the real exponential form. He points out, so he somewhat thinks this is his first award. The elliptic functions have this nice algebraic generation, you know, you can get everything from the vastness function and it's derivative, you know, for a given language. No, I don't think so. I mean, I don't think I could produce one offhand, I mean, I could do it artificially, but I forget how much Anitzen did it. The papers and the transactions are quite a long time ago. I don't necessarily think that Lee Rubel would have seen a natural example of them, but I can think of them that don't occur.

1:07:30 Yeah, I mean, that's of course connected with the story of Burton Lee calculating the... The dual space of the space of analytic functions on a bounded set. And that turns out to be analytic functions on the complement of exponential growth. So there's a natural way that that limitation to one exponential only comes up just in the idea of taking the dual space. It's a funny way to get the dual space in some sense. You take the complement in some sense. It's like the alexander duology and homology, cohomology, homology on the rounded open set inside the sphere now it's it's it's in terms of the cohomology on the couplet. Uh-huh, uh-huh, that's what you know. It takes the latter from the former, that's sort of the Iran method. Yeah, I don't know, there was a lecture at the Newton last week, last week I was there, by But this lecture was more charming. It was about Maxwell and the problem of Maxwell's, which is still not solved, and a problem with critical points and certain charges, and about the number of connected components. And he claims, Gabriel Hodgins as an individual, that in this paper basically anticipates Morse theory a lot.

1:10:00 But it's somewhat clear that these long-term intuitions that we're eventually going to get right. He gave, he began the lecture by saying this is joint work with Sir Isaac Newton. It was! Did you see that it's mentioned in his book of history? Yeah, yeah. Did you see again about the, I mean just citing this theorem of Grudenbeek. Yeah. It's also, also Kurtz approved the same, more or less the same thing about the same time. Mm-hmm. So I think it's the early fifties after, after Grudenbeek's initial. And he was considered the greatest functional mathematician. Yes, yes, no, I remember this because I was in Aberdeen by the late 60s, because they didn't know there that he was working on algebraic geometry, but they knew very well how normal it was. It wasn't just a question of analysis. So you see, so I think there's another day reserved, another hour or two, that this is really how Wittenbeek actually moved from functional analysis to algebraic geometry. That's a natural question when you study dual space. What about these concrete spaces and then what do you do with them? So if you see, if you could only freely use the topos, which envelopes the category you're interested in, you could immediately write down stuff like this and see if it's true. The space of functions, the space of distributions on the given pieces and so forth is all there. In the same way that the external natural numbers, you can use them to do induction on formulas and all this stuff.

1:12:30 You can't do that inside the geometrical network. This, of course, is more geometrical than that. That was the first question. This is more... This idea that dimension one governs the higher dimensions, this recurs in other ways. I call it the Hadamard Principle. Hadamard, in some lectures, gave the definition of a differentiable or smooth function of n variables just by the fact that However, it's differentiable along every path. So, in other words, the only real information is the monoid of endomaps of the one-dimensional space, and the rest is just general objective logic. They have a bunch of spaces, and they're quantifying, and so on. So the real information is just in the one dimension. And that turns out, of course, this is the correct definition. You could always make this definition, take any old moment and immediately do it. So I call that the Audemars Functor. The thing is that the Audemars Functor actually agrees with the known correct one in so many cases. But it's also for continuous functions. Continuous functions on a cube, if and only if. And even polynomials. They have to have a ground field, I think, which is not countable, can be either finite or incountable. In any case, if you consider a function of n variables, an abstract function of n variables, which is compatible with every polynomial in one variable, then in fact it's a polynomial of n variables. This is a little more surprising, but now we're talking about actual concrete coefficients and stuff.

1:15:00 Again, this is a remark of Shen. It might even be useful to you, because you don't have to really check the definition of invariable, which is sort of a reverse of the usual point of view, because you also want to feel, well, the international space is important in itself, but this is a criterion for it. Let's see, what are some other examples? This is kind of opposite to what I was first thinking. You're talking about maths again. I was thought you were talking about vibrations. The one-dimensional... No, that would be another... But that's another... That would be another approach, yeah. You build up things that are repeated vibrations, but the fibers are one-dimensional. Yeah. Yeah, and so it's not, yeah, it's a different, I think, yeah, observation that real numbers are just ratios of infinitesimals. It's pushing that down to an even more basic level. There are many examples, like, let's take the category of partial order of success, for example. Again, a map is in the category of an abstract function, it's order-preserving, and you have to know that for every map you get two elements of order of success. So that's a very obvious example. But still, if you look at those terms, maybe those... Higher dimensional things are entirely so complicated after all that I could get at them in this fixed level. It happened there often enough to be considered as a non-random phenomenon. I have no idea of any common proof of this. It's trivial. It's a common statement. But since they're in different categories that have totally different properties, it's not clear what the common reason might be.

1:17:30 Yeah, there's an interesting example in model theory, Judith Schallach, There are families of sets in, say, n dimensions, but they may depend on a lot of parameters, and the parameters may depend on n dimensions. It's about the independence of sets in a given set, using s and o. There's a dual property. So this first notion was discovered. Shalech was looking at the dual notion, where somehow the role of the variables and the parameters is split. And Shalech proved with considerable, passing into non-standard model of set-fugue, where the particular property of derivating completions was true during a calculation and dropping back out again. He showed that if this dual property held in dimension 1, for instance, if held in dimension n, it would conclude from that that if you verify Chalice's thing in dimension 1, then you get this Latinic-Germanicus thing. I don't think that could have proved that by any kind of straightforward combinatorial. I don't think it could have done it, but Chalice did it by this. And that's probably the most difficult example or something that you would expect in this geometrical example, if you can still understand it, isn't it?

1:20:00 Yeah, I wanted to quiz you in a slightly more technical way. Yeah, sure, yeah, sure, certainly. Probably we should take a break. Yeah, let's take a small break. We should not. This one also? This is the foundations. I mean, everything... I mean, there is an industry of actually going down to a group structure, possibly even less. I'm working with a man, and he's probably... It's, like, worthwhile, I think, but somehow or other, the one who's doing geometry most of the time, the one who's willing to be... Well there's quite an industry and sometimes it's delicate to do things without the article on it. Well without one of them, they usually keep the plus, it's the times that they... What do you compare it with the Duke Journal article? Yeah, the Duke Journal article is very much to my taste. I mean they... because they really do an awful lot of important things. Uniforms, basically things that... things which are...

1:22:30 For example, business is like getting closed sets, bounded closed sets, and zero sets of, say, C1 or C2 or C3 functions, etc. All of these things can be done in boomy math. They're not totally trivial by any stretch of the imagination. In fact, they got proof wrong the first time round. They just basically run through all the principles which have proved useful. There were savage inequalities, stratification, self-decomposition. I don't know if they do this trick. I don't know, I can't remember, but it's very much, it's very much metrical. It's not very efficient. It might be a better thing to go to. It's not, it's not an obvious answer. No, I don't think so. I wouldn't say it's out of the equation at all. I mean, there's certainly room for improvement. I think we discussed at the very beginning this insistence on torturing things into the... The single source. Yeah, yeah. But beyond that, it's... It's done extremely quickly. Yes, yes. They pretend they're dealing with these important mathematics. They're dealing with the projected spaces of the different dimensions. And this is not a simple matter of critique. That has to be sorted, but I don't know. Yeah, I will try to. I mean, this is really a very fascinating development. This had not been it. But there have been a pre... A pre-history, not a VC dimension, but of something called the independence property. This had been a, like this had been well studied in the model theory of the 1960s in Kiesler's.

1:25:00 Certain theories are, from this standpoint, it's Boolean algebra. Or you can get some kind of independence. That's to say, you can, for example, the Chinese remainder theorem is a very bad thing in the Now these are independent things essentially for the Chinese remainder theorem. That is to say, if you want to, if you start with a set of elements, you can get a set of elements of size n such that private subsets can be described in terms of, can be defined in terms of divisibility. But I want an efficient way to define each of its subsets. Okay, well, I can do this from every set of size n, but there are sets of size n such that I can use visibility to pick them over the change remainder. In other words, I can, if I want to get, it's just obvious basically, I can take n primes. So essentially what it means is I can take a set of n primes and I can define every single subset of it in terms of them being the primes in the set, which divides some specific element x.

1:27:30 Okay, I can get, I can pick up, I can get a set of size n, as big as I like, and a set of n elements of some kind, such that I can pick up all the, I can give a uniform definition for each of the subsets of that set of n elements, using some fixed family, but it's really pretty, in this case, p divides x. So it's really the x which is going to be the defining. I take a subset of... And then I can take X to be the product of the primes in that set. X will then pick out exactly that set. So the parameter X itself is out of the set? No, no, that's the point. See, so I've got two things. I've got the prime bit, I've got the double predicate, P divides X. It turns out that I can use different, I can get a set of them. The thing is, I can get a set of size n, you tell me any n you like, I can then say there is some set of size n, if I have a set of primes in this case, such that every single subset of that set can be characterized naturally by an x, with the change remaining. That's the classic example of invariability in some sense. You know, I'm not sure if this is important, but I'm interested in the tangency matrix here. And what you're saying here, it uses much less than that. Oh, yes, yes, yes, yes, yes. No, yeah, it's just that, it's just basically... But that's not... Yeah, it's not, the tangency matrix is too strong, in fact. I was going to use it because it's a classical mathematical example and it comes up in... Now, this would, for example... You would say that this particular formula, the particular formula here is a formula of two variables, y divides x, that you really get one. Now, I would say that this thing had the independence probability with respect to y, in the sense that I could find for each n, I can in a model of the earth, find a set of n elements, such that Any subset of those elements can be got as the y's in relation to some specific element x.

1:30:00 So, and there is a way you can reverse the order of the variables, you see. Now, so you would say that a theory had the, a formula of the independence property. You then have the specimen. The specimen, it carries variables in two with respect to some partition of the variable. Each n or arbitrarily large n. Let's say for each n there is a set in which all its subsets get produced in this way uniformly. This is called by the problem of shattering, that there's a set of size m which is shattered by the formula. In other words, all the possible discriminations you could make set theoretically can be made using this formula and putting a second variable equal to some specific constant. There's a simple way of using this formula of defining all the sets of the subsystems. So this is known to be in some ways an undesirable property. It's from the classical standard of this type segment. The other example of this... You're representing the power step. Yeah, that's exactly right. It's kind of a weak version of yet another second order logic. See, this is the point really. You can pick up all the subsystems. All of these by a first-order thing, but only for primary things, but still it's a very bad thing. It was known to be a bad thing. Now, another example, more abstract, plus monumental, is Boolean algebra. You see, again, there's an infinite Boolean algebra. You would like to get something which comes out to less than or equal to that, yes? You can pick up all the subsets of that set of atoms by taking a subset, you see, and that will be above all the prescribed atoms and it won't be above the others.

1:32:30 Again, I can't believe you said in terms of Boolean rings. So Boolean algebra is a more abstract, I mean, every infinite Boolean algebra has the independence problem for a very simple formula. So this kind of Boolean-ness is, for technical reasons, connected with the model theory of the 60s and 70s, a sort of undesirable property. It is a property held by arithmetic and by set theory and by Boolean-album. It's so obvious that what the... Well, in ring theory, typically, there's a lot of primes involved. So it's not so much Chinese remainder theory, but divisibility is a... Most of the rings you meet are going to have the independence property, at least if they've got primes. Okay, that's just a heuristic, you know. I've stayed for a long time. But Sherlock, Sherlock investigated then this issue of the independence property, because for him it was a, somehow if you had the independence property, you know, it's a very unstable thing. You would get a lot of models, typically if you take a theory of boolean algebras, So the matter, but then show off that this combinatorial thing, which at the time he needed to do, he was vague with the way I did it, I didn't devise it, he said the following, suppose I've got a formula with n plus m values. The examples I've given you to know n was equal, I could be shattering a set in n space using parameters from m space, parameters from m space. That typically can happen, you can devise examples for it. So you have a notion of a formula that is n and m.

1:35:00 Having the independence problem. In the same way, if you do the shattering. Now, shall I put the remarkable thing that if a theory has some formula with the independence problem, then it has a formula with the independence problem for n equals 1. In other words, if the shattering occurs, it actually can occur on the underlying. Oh, yeah, yeah, yeah. This is a nice article. So this was done by Schellach. The combinatorics for this were done round about, you know, and he proved other things, for example, you can ask, you can look at it more, you could ask yourself, well, that's the independence property, and then there's the dual thing. Problems came out of this too. The Russian problem was Vatnik and Chevenet. Vatnik is a player nowadays in mathematical learning. You detect the measure. I've got my probability measures. I can make tests to check if things are in or out of the cell, throwing darts on a board. Suppose I take the following crude approach. I start producing at random entities to test if they're in the cell or not.

1:37:30 So I take the relative proportion of those that are in the cell or not. I get a certain number which I'm going to use as a method. An approximation to the, hopefully an approximation to the measure of K. Now, of course, this number again depends on what I point in the n-space. I'm choosing in the n-space as n is getting bigger and bigger. So I start, I've got a probability measure on my original space, and now I start sampling points, and then sampling two points, and then sampling three points. So I've got the product measure. The Q test, I've got the square measure. The 3 test, the Q measure, etc. It's a bit tricky to say, I mean... Suppose I've done, S has given me a certain portion of points in the set I'm after. Okay, so to my run of n, I've got a number of that. That number may or may not be near to the actual measure of the set. Suppose I fix a tolerance. So I'm interested in the number of points in my n space such that this relative frequency I've detected is within epsilon of the actual measure of my set. That itself is, under mild assumptions, a measurable set in the n-space, and it's got a measure. And so what you're really after is some theorem that says that this measure is tending to 1 for a fixed epsilon in some predictable way. But as you do more and more tests, you're getting closer and closer. The chances are you're getting closer and closer to the measure, because this goes back to Bernoulli as well.

1:40:00 As for a given event or a given set... There are a lot of idioms that say under reasonable assumptions this thing is, you have this convergence to the measure, in a sense, with estimates for what the, if you start with the epsilon, I mean, basically you want something like this, the measure, I mean, the limit as n goes to infinity, the set of n points such that the difference between the genuine measure is the zero, as n goes to infinity, you then want to figure out what big N is, what the big version is. So one, but now these guys became innocent. In doing this uniformly across a class of events, suppose I want to follow the same strategy for a class of events, a class of hypotheses, maybe depending on parameters, I don't know the parameters, I don't know what they are, under what circumstances can this be done uniformly? Is this conversion, can it be done uniformly? And they detected a combinatorial condition called shattering, which turned out to be the exact dual shell-axe. All of Schaller's videos were placed by their parameters, and they showed, basically, that you would get this condition if, after a certain point, you could not shatter a set of size n, but after a while, you couldn't pick up the full power set, by instances, if they had a problem with this. And it turns out, there's then a combinatorial element that they knew, and that Schaller had proved. That says that if you suddenly drop, if you're no longer picking up the power set, after a bit you're only picking up polynomially many subsets. After some point you're below polynomially many. Various numbers come up. Now, the bizarre thing is that Shellac's, the combinatorial thing that Shellac had detected in the first order of theories is exactly the dual of this finance in that picture.

1:42:30 Then you will have this uniform Bernoulli phenomenon. And then, as we know, it was proved that this was, well, the phenomenon that had been looked at by Stanley at MIT, who had written a book on it, but they knew hardly any examples for the, for your finance of that nature of an exhibition, except for things like linear arrangements in space. And this is a very important example. I mean, you take, yeah, yeah, you take, for example, lines in space, you take one, so lines are parameters. It takes lines, say, in the plane, and you take, say, two lines, you see how many cells that break the plane, then three lines, etc., etc., and you start counting, the more lines you take, you would expect to get more, at the beginning you get exponentially many, but after a certain point it cuts out, you no longer, you don't get, if you take n lines, you don't get two to the n cells, you get below it. This is the most, the best known example of the general phenomena. Yeah, it's true for linear was known to the probabilists and so on, and that's the basis of some of these crude machines like this Perseptron, I think. Is that so? Yeah, well, if you look at the book on mathematical learning here, it gets parpered and so on, but it's just a game with sides of algorithms. Two semi-algebraic geometers succeeded in proving the finiteness of the next dimension of the semi-algebraic theory, and then it's actually true in all minimal theory. And you see, in all minimal theory, it's not the regular theorem. But this theorem is true for any measure. It's nothing to do with the regular theorem. It's true for any measure.

1:45:00 There's an extraordinary feature of it. What's most interesting is the fact that this has implications in the case of Wilkie's theorem, the exponential. We first of all proved an elementary result. We just used the finiteness of that nature of the next dimension to prove that certain concepts could be learned in what they call sigmoidal neural network, in mathematical learning theory, that could be learned at all. Prior to that time, we didn't know that the dimension was finite, we didn't know that. And then later on, we looked at it more closely with Karpinski and used it back more to the roots of what I've made work is proof work. There are other bizarre things which have appeared, but there's uniformities involving the number of persons dead, the number of hours needed to find families, it's quite a well-developed There's not a big literature. Van der Weest touches on it here, but as far as I remember, he doesn't do a great deal. He does the combinatorial part, really, and I suppose he states, maybe, I'm not sure, he even does state. He does the shellac thing, following Laskowski. I don't remember, I don't think he actually, no, he doesn't even do the measure for that. And you said before something about Ramsey experiments. Yeah. Well, shall I use Ramsey? In addition to the fact that Ramsey numbers actually exist rather than... Yeah, you see, the problem is if you use the Ramsey thing, you'll get gasped or bounced. It's an inductive argument, using Ramsey, using basically unknown Ramsey numbers. So that was the first, I mean, I knew that these numbers, that the VC dimensions could be written down in terms of, could be written down in terms of Ramsey, but this is hopeless as far as the...

1:47:30 Then I chanced upon a paper Mark Jarum told me about. He was a computer scientist in Edinburgh. It was written by a functional analyst in the 1960s. I forget his name at this moment. He was trying to mimic... Milner had given very good estimates for the number of connected components of a semi-algebraic set. In this Vatlich-Irvinakis thing, you're given finite-mini functions, and you're basically trying to figure out, depending on parameters, how many possible configurations of sine do you get? These are functions of all the same set of variables. You want to figure out why, if I instantiate the variables, each of the functions require plus, minus, or zero. How many configurations can I get? Now, we, Karpinski and I, in our paper, we managed to somehow play a duality. I've stressed this duality already, because there's two kinds of variables in a formula, in a VC dimension formula. There's the ones that, going through the space you're working in, and then there's the parameter space. There's a K space and there's an L space. What we were able to show was that the VC dimension for K space could be translated into a question of the number of connected components. For varieties in L-space, in Khovansky, and we eventually got this kind of ideology, which I think has not been properly explained. This functional analyst was essentially using Milliner's theorem and related things by Petrovsky to, from a number of kinetic components to something about sine configurations of sets. Once we had understood their theorem, his theorem was, but the bounds are really very sharp.

1:50:00 So the particle, you don't recall the name of it? Come back, I can sound refined now, let me know. This gave insight into the duality. Yeah, yeah, I mean he was using elementary, he was basically using elementary algebraic pathology without knowing it. You know, it was very, very interesting. I mean, I thought it was a good, I discussed it with Woodley at the end of my lecture. It was quite, quite remarkable. It's interesting because you have to use general position arguments a great deal. You have to stop, to get the estimates to work, you really have to stop this. Again, the SARS theorem, the regular values of a function, the special nature, most of the values are regular and stuff like that. It's really that game, very, very systematic. Yeah, there's not much being published on this, but I think it's really quite a startling phenomenon. It also occurs beyond the equation. There are other important mathematics. There are a number of mathematical theories which are known not to have the independence property. For example, all the theories of periodic analytic functions don't have the independence property. Sometimes for very deep reasons. The, that the periodics functions don't have requires you to pass to the, which is a delicate thing, it was done by Tate. I mean, he uses logarithmic topologies absolutely explicitly. You cannot get the theory of continuation you want if you weren't just googling it. And to understand properly the ordinary periodic case, you have to pass the limit. Out of that, you get the finance that you see that I mentioned. It enables you, in principle, to cook. It's quite a strange thing. Is this independence somehow strictly related to the usual statistical independence? Well, this is what I don't really know.

1:52:30 I don't really know. I mean, certainly not in Chalock's case. I'm probably not at the time he was studying Jerry Keesler's case either, but of course Jerry later would have certainly recognized this, but I guess, of course, the fact that he's Shamanenka certainly is, but what was remarkable was how he was involved with that. Yeah, I mean, they, that's how they reduced it to this finite combinatorial thing, which is so amazing, you know, that you really get down from the measure theory all the way down to finite combinatorials, and I guess that has not been properly used. You can get it for the periodic measures, and that's relevant for periodic integration, and I guess there are other cases too, but I'm not quite sure, but I find it remarkable that the real and periodic analytic theories have this uniform, I mean that's an extreme form of payments, of payments, I mean that you can really somehow sample, I mean that you can't define the measures. In terms, as far as we know, you can, well, we know you can't define them eternally in a given case, you can preserve tainless, but at least one knows that the actual measures are, can be obtained by, with up to as high a degree of accuracy as you like, by very simple sampling processes, and that's a bit strange. It's, in principle, it's relevant for randomized analysis, it's not. In principle, it is. ...as avoiding the ideality of avoiding, you know, you know, you want to conceptualize... ...I, I, I, really, very remarkable, but, you know, the moment you step into a strictly geometric situation, at least in real life, I suppose you can also call it, you do somehow just avoid the good or the wrong, the natural cases, but even, there's a tendency, like at some point, not now, it's...

1:55:00 The case of the complex exponential is an interesting one, because it is credulity. You can interpret the integers by i, z. Nevertheless, it's probably... That's another form of chain that's there. Yeah, it has. I mean, all of a sudden, conjecturally at least, and possibly more than that, and it does relate to Shannon's conjecture in a much deeper way than it has detected. I mean, conjecturally, the complex exponential, when you halve it from having the integers... And then if you choose that to stay with the integers and concoct all the ghastly things you can define there, it's wild. But as far as the broad scale of Georges is concerned, it's not wild. I mean, we believe, from this standpoint, what would be wild would be to actually get a real topology via the confidence potential. That would then introduce second-order number theory and God knows what else. You see, the point is sine and cosine on the realness. These are inherently wild. It's not merely that you get arithmetic. You get much, much worse than arithmetic. In those cases, because once you've somehow got the rules, you can start to code up sets of rationals. You've got the reals and the integers. You can start to code up, by simple methods, sets of reals. You can get the projective hierarchy. You've got those wilds. Well, you don't expect to get that in the complex exponential. We don't know for sure yet, but we don't expect yet. So it's an intermediate one. Yeah, so the idea that the kernel of the complex exponential is the integers, you see, I mean, it seems to me that that might not always be true.

1:57:30 Yes, yes, yes. So that has this, that has this, so there's a certain topos that has a spin line and all the C-infinity and manifolds. Yeah, yeah, sure, sure. But then there are all these other things. Yeah. Including function spaces. Yeah, yeah, sure. But now in particular, there are lots of natural number algorithms. I mean, there's the natural number object, which satisfies the Dedekind-Levier, you know, you could integrate an arbitrary enum map of an arbitrary object, you know, that's, you know, I pointed out that this is a left-adjoint, but it's entirely Dedekind's idea now, so, okay, that's bad, that's pretty bad, but there are a lot of other natural number objects, and, and, and, and... And some of them partake of the smoothness, so in some sense the whole topos is partaking of smoothness. More specifically, well, I mean, yeah, okay, they describe it as the zeros of the sine function, but I prefer to think of it like this, in a more general sense. Again, actually using the exponentiation of objects, as Lewis is available to show, and this you can do with any ring, you see, but we define the circle. In the square of the ring, that's the beginning of the loop object. You can take the space of endomorphisms. It's a sub-object carted out in a simple way from the exponential. So there's the space of endomorphisms, which is automatically a ring. The endomorphisms are anything that's true in its internal sense. So you have this ring. It erodes with a smooth structure. Now, from the classical point of view, it is the integers. This is just a more structural way of describing it. And yet, and this is definitely not. It's definitely not. Now, it's only got maybe a smidgen of smoothness, but still it's...

2:00:00 Yeah, well, this is an interesting point, because... I mean, you're investigating how much you could ask, how recursive is it, how... How dedicated is it, in other words, what kind of endo-maps of spaces can be iterated, can the iteration be parameterized by it, so there's some kind of smoothness condition, even if our endomorphisms of the line, whatever you like, the idea that their iteration should be parameterized by this natural matter object, so actually it's a disease, an energy object, a grating object, not just a rig object. But it plays a definite role in the whole theory, because in showing that the Dirac theory is the dual of the singular homology, which is the basic theorem, they have to use, the singular homology has its coefficients in this range, not in the discrete, you know, that you can move here. That's necessary. So, another way you could think of it is the fundamental group of the circle, right? The fundamental group of the circle is obviously going to play a role in algebraic topology and all of these things over time. It wouldn't work in the usual national numbers, but it works nicely. Well, this is a fascinating idea. I have been, in fact, I've quite often been digging into it. ...to that book, I put it for a while, and I've got it with me in Cambridge. Again, the task of pursuing a settling world, in fact. Well, you can just look at it synthetically. Yeah, sure. Assume you have an exponential puncture, usual rule of inference, and assume you have a ring object, a community ring object, then you can construct this other ring object. So the original one was smooth, pretend this one was smooth, and then you get... This other one is presumably discrete, but no, it's not. It's still particular. And you see there, just from the general point of view, everything we did to construct this one was an inverse of it.

2:02:30 Took a function space and equalized it, whereas the usual integers are entirely, you know, from the left, in a way. So there's sort of no a priori reason why... Of course the discrete integers have a unique homomorphism to this one, because they, you know, they iterate anything you like, so they really are the initial objects in the category of all ring objects.