Am session of discussions inlc. FW Lawvere, L Corry, A MacIntyre, JL Bell, C McLarty

0:00 Now, okay, and it is the 16th of June, and this is our discussions, and what will be the last full day, and going into the final term, I think it would be helpful, since the time is now short, if we tried to stay within a, as tightly as possible, within a prescribed type of discussion on each of these subjects, so that we can try and cover as many of them as possible. I'd certainly like us to try and get on to the broader philosophical picture that emerges from these discussions and to some of the very general themes as to what are the fundamental positions that drive the conceptual organisation and development of mathematics which John suggested for discussion but I think before we do that which might be the subject for this afternoon or this evening's discussions. We should just look a little bit more closely at one or two loose ends left over from the discussion. Mr. Goethe-Lasbeth, if you could just take us through a brief account of this later development in Grothendieck's ideas about how one ought to define schemes and what he thinks is particularly important from the point of view of the... Increased ontorheality of that discussion, the way that it fits into the book, brought about through Gromitik's ideas, and that might naturally lead us on to a discussion of this extraordinary memoir, Shark, that Gromitik left with Jack Duskin in 1973, which I know Angus would like to hear a bit about, and that in turn would lead into the broader philosophical topic, kind of, in general, as to how logical structures, I know it's a subject we've touched on before, but it's a very rich one, and one that I think we could... As to how logical structures do in fact fit within this overarching view of the right framework in which to achieve their ethics as a whole, particularly the way the principles of choice gets expressed in terms of projective understanding of those,

2:30 and indeed the way that choice and extensionality principles reflect the way that, in varying different versions, From the specific and the technical through the, well, more issues which are direct organization of trained into all the multiple themes. So I think they might go ahead and say a little bit about Groten, Deacon, Skeet, Latter-day. A little bit about all that. Yes, I know. Well, the impossible always takes a little while. We've been through some of this stuff. So first of all, it's not based on logic. It's a form of algebra, a form of logic flows out of it. And then you have the Cartesian space or map space construction adjoint to Cartesian product, and then you have the power set function, which represents arbitrary monomorphic sub-objects that just ask for truth-value objects, aren't formulating it, but in any case, it flows out just of those two axioms that sub-objects must form a lattice and indeed a hiding lattice as an implication operation. And in a natural way, that is, that operations are preserved by arbitrary substitution, otherwise known as pullback or composition, and also the quantifier.

5:00 So there's all this, what has been described as higher-order intuitionistic logic, which is flowing out of the algebra just by defining things and deducing what the universal properties should lead one to. ...aspect of the logic of topos, namely that the, for many reasons, not completely evident in the definition itself, the morphisms between two toposes, which arise in practice in geometry, are only rarely strictly presumed in these operations, whether they preserve them up to uniquely in comparison maps. Formalized by Leon Henkin and his proof of the so-called completeness theorem for higher-order logic. Now there are so-called logical morphisms of toposes as well and Peter Fry is perhaps the leading expert on those. For example, you know, it's said that Gödel and Wooden believe that the continuum is out of two. Well, that sort of condition is easily expressed, I'm not saying... But there is such a thing as the free topos having an object, a freely adjoined object, intermediate between the natural number object and its power set, you see. Probably in that free topos there are all kinds of other stuff that's happened that the set theorists would rather squeeze, squeeze way down to constancy in order to get a sort of non-trivial result, but at least that. But anyway, that has to do with logical morphisms. These only arise in geometry, I guess, in sort of two ways. For one thing, if you're just looking at representations of groupoids and sets, then all the geometric morphemes automatically have this property that the higher order structure, function space, and the power set are preserved. But that's just for groupoids.

7:30 A local homeomorphism, of course, is a very special kind of continuous map, and it is essentially described, I think, by the fact that the inverse image functor strictly preserves this logic once again, but of course local homeomorphism is a special case. Between local homeomorphism and general continuous map, of course, there is the open map, and again, this has a special relation with logic because it means that... Logical negation, that is to say, universal quantification and implication are preserved even though the higher types are not. But the portion of the logic which is preserved by the general geometric morphisms is in some sense already visible in the very notion of a topological space, namely arbitrary unions, finite intersections. Well, if you move outside the lattice of sub-objects and look at Cartesian products and so forth... What that means is essentially a very special kind of formalized logic, which, as far as I know, no book really does correctly, according to my likes of what's correct. They write a universal quantifier in front, and then they say what comes after is very special. It contains an implication. Precisely, we are not, you know, in universal quantifier and implication, how precise are the things that are not part of this positive logic. So rather to formalize it in an accurate way, one needs to go back to the idea that it's deduction rather than truth, which is a fundamental idea. If you can deduce truth, you know, then you can prove something, but that's a special case of arrows. So instead of arrows, we use various kinds of arrows, but these are, I call them entailments. So you're saying that that would be the best way of presenting geometric... This kind of logic is called geometric because it's preserved by geometric morphisms. It's called by me, positive, because it has a positive character.

10:00 It's called coherent, completely, erroneously, in some sense, because... Because coherent had to do with finiteness originally, and then that was generalized to alpha-coherent, and then that's the quantum. But the coherent came from finiteness, which came from cartons, a Boolean situation. And the fourth name is dynamic logic. I don't know if you've run across this. Well, I don't see that one. I only came across it recently, but it's our friends down the road here in Rennes who like this. It's good. I mean, it's nice. But what you were saying is that it was meant to present really as a deductive system. It really should be presented as a, you know, that there's a binary relation. Yeah, yeah, a deductive system. Yeah, yeah, no, it's a deductive system, essentially. A deductive system, and moreover the... That makes sense. So in that sense, the formulas that occur on either left or right... Right, right. All of these are quite positive in that they only involve existential quantification, which only makes a difference if it's on the right, because on the left it can be eliminated. I'm surprised in a way, I mean, thinking about it, that it hasn't, you know, I mean, all these people are working in deductive, well, you know, Sanbin and so on, they're working in precise systems of this type. That doesn't seem to have been done. So, there is false, you see. In other words, there is no implication as an operation, although, you know, you can assert an entailment. Yes, exactly. When you studied logic, this may have looked like some strange sort of quibbling, but in fact it comes out in an objective way in this context between entailment and implication as a binary operation on formulas as opposed to... There are lots of logical systems without implications, I mean you have to add additional conditions in order to introduce it, so it's very natural to set it up in this binary system. Okay, so notice that the and and or are true and false, because false is a consonant, you see.

12:30 So a lot of, if you actually want to assert a negative statement, you can do it. It's just that you can't have not a free-floating thing. You can't consider a negative statement, but you can assert one. You can't put it in an Amazon, but you can assert it. Right. So, for example, the idea of a field, there are at least three different ways to formulate the idea of a field. One of them, which is paradoxically called the geometric one, because it fits into this logic, but it's actually not geometric in that it's not the one that comes after, as a general, fibers or something like that. So because it just says that something entails false, so basically what you can't do is to have a formula for all that implies on the right-hand side, but you do have the false on a lot of the things that you would normally say by considering the negative statement. Although this logic looks like only less than half of a fragment of ordinary logic, intuitionistically, from a classical point of view it includes the full classical logic because if you present a particular theory with generators and relations that are called primitive terms or atomic formulas and axioms in that context, you can just introduce new primitives. Whenever the axioms that you want to state call for negation, but then these negations are stated in the Boolean way, as being the union is one and the intersection is zero, as additional axioms. So any classical theory can be faithfully encoded, but not every intuitionistic theory. It turns out that every topos over a given place u can be interpreted as the classifying topos for some theory, some kind of structure described in that language, so that the classifying topos, for example, if we consider the presheaves on the category of finite partially ordered sets, then that's the classifying topos for distributive lattices.

15:00 In the sense that if you take any topos at all over the same base and any distributed lattice object in it, then there's a unique geometric morphism whose inverse image, such that the inverse image of the generic versus generic element of that pre-sheathed topos is two to the power of blank, or two to the power of blank is order-preserving math from a variable finite post-ad into... And then in that way you see that the various stronger theories in this positive logic correspond to sub-toposes, so the simplicial sets, obviously a sub-topos because linearly ordered sets are special posets, classifies, it turns out, the totally ordered, or in general, totally ordered objects with distinct endpoints in an arbitrary topos. If you consider just the discrete post-sets, then there's the sub-category, which is the totopos, which is the self-appreciation on finite, just on finite sets, and that classifies the Boolean algebras in any totopos, according to the special distribution. So in that way, the classes of models turn into subtoposes. But now, so I've described these structures in a semi-traditional way in logic, namely accepting various changes, the binary relation of deduction, instead of merely the class of true statements, only the positive operators, and so forth, but Grotendieck didn't think of it that way at all. He said, well... Geometric morphisms have the property that inverse images preserve arbitrary co-limits and finite limits. Therefore, any type of structure which can be described in those terms will be preserved and will be in principle classified.

17:30 But essentially, any type of structure which is describable in those terms will be classifiable. And he shows real delight in this discovery and real delight in seeing how many structures are. That's right. That's right. That's what I was leading up to because Mike mentioned this amazing sheet of paper in Jack Duskin's file that some day, I'm presuming in 1973, Grotendieck was in Buffalo, and he started off on the theme of the classifying topos, and it's well known, standardly used, that the notion of local ring is classifiable by a subtopos of the ring classifier, and this is used very extensively in algebraic geometry, as well as the classifying topos for, what, Pencelian local rings with separable... There's a classifying topos for that, complicated as though it may sound, and that again played a central role as they grow topos. So he got into this mood, well let's see what kinds of rings are classified. And so he wrote very fine handwriting, you know, a list of maybe a hundred, you see, and then he turned it around like this, he thought of some more, like this, and then he wrote the terms again, like this. So there's this piece of paper which, you know, is in Jack Duskin's file at the University of Pueblo. So we have to send it to Reville Nets. Who has the technology for reading these things? The only problem I have is that they are so often that it's mostly illegible. But I think I've seen the original one. By the way, you know that... It could be detected. It's like Civil War letters or something. You know, it's an interesting thing because this is a historian of mathematics who is reading all the Archimedes palimpsests. And the people who do the technical things are at Buffalo. They have a department of digital something. That's true, you're right, they do. And I was in contact with this guy because we needed a picture. He's a very, very nice person.

20:00 And possibly with very, very sophisticated techniques of reading all kinds of palimpsests. So you can send me this. It's a terrible... Jack must know that. No, I mean, the lack of communication between different parts of the universe is amazing. Another instance is that I discovered a few years ago that there was a group of graduate students in computer science. Well, there were some common acquaintances, but it was only after quite a while that they realized it. It must have been before the internet was like... But one thing about Google, it has to occur to you to try. They could easily be reading about Topo's theory and even reading about LaVere's rule without it ever occurring to them. If you looked up the individual now, of course you'd immediately steal it. Anyway, basically the level of communication is appalling. I think I've seen a physicist once in the last five years, because I made an effort, unlike any of my other colleagues, to go to an attractive-sounding lecture on the new version of David Bowman's. Anyway, getting off to the side there. So yeah, apparently that's one thing that might want to be mentioned in this sheet of paper. We were discussing with Pierre Cartier, Pierre Cartier wrote this article, wrote indeed, and he mentions somewhere along the way the contribution to logic that we made, that we're getting from someone that wrote, he had never dreamed of it, which is true enough, but he told me himself that he was really floored by the fact that there was this sub-object classifier, that all the other stuff, you know, he could understand, he'd use that, but that particular way of looking at it, he just did not. Of course, he immediately saw the virtues of it when it was made explicit. But anyway, the point is that I say, in response to Pierre Cartier, I wrote, you know, the slogan, well, Grotendieck didn't need logic, at least not this technical formalism with there exists and or and and and so on.

22:30 I mean, the role of this technology is essentially just to organize the definition of sub-objects and talk about inclusion of sub-objects. Well, when you talk about these structures defined by direct limits and inverse limits, there's really no special role for sub-objects. I mean, why should a monomorphism be any more special than some other one? Now, of course, there are technical reasons for this. Well, any map can be factored, so we have to get mono, and any map has a kernel pair which is in turn, you know, a monomorphism into the square and so forth, so anything can be analyzed in terms of sub-objects. On the other hand, if you have some simple diagram and make some simple statement about co-limits or limits in it, you see, you don't necessarily reduce everything to the discussion of sub-objects. So, I mean, what Grotendieck was clearly doing, you know, I can understand some of these examples in the list, many of them were so technical that you'd have to figure out what he's talking about. I never knew what a Japanese... I was just about to ask you if Japanese are excellent. Excellent, yeah, excellent, certainly. I've seen the likely. Practically every adjective you've ever heard of comes up for... It was clear that he was not thinking, you see, in terms of relations in the narrow sense of sub-objects. We just see directly from the definition of the thing that the only things involved are directly inverse numbers and therefore must be classified. So in that sense, the Japanese has to do. In fact, when you started mentioning that, that was the first thing that came into my head. That seems to be quite reliable. These do play a very good part. The question about Henselian and separable conditions and all is if you can put this in terms of solutions of some equations, then...

25:00 I mean, there's existence of solutions. Yeah, if you can show that certain kinds of... this is equivalent to certain kinds of equations having solutions, then it's... This is the whole enterprise of Michael Harshton. Algebra is algebraizable. And it's exactly that, that you don't lose... I mean, you can better understand the structure you're dealing with by doing the completion in the sense that you can retain the solubility. Well, I mean, it should be pointed out, of course, that it's precisely this class of structures which have been shown already in logic and model theory. In spite of the claim, you know, that in practice it's the positive morphisms and so forth that can help Kiesler's characterization of the requirements of Morley's work on that. Ultimately, it's sort of worked out that these positive... Whoa, but you're Robinson. What does Robinson call it? Well, there's the Robinson test, and it's the limit. I mean, there's a whole lot of stuff around this issue. If you've got the universal existential prefix, that this is typically preserved under indirect limits. It's just for the map, for embeds of structures. And then, of course, if you have... if the maps are elementary. Well, I mean, it comes down to this Robinson test, which is... But if you want to detect that in a class of modern science, quantum theory, all the maps are elementary, but the mandatory for average albinism is that it will suffice to show that they respect a certain batch of existential problems, and that's enormously powerful, and it's always seemed to me that that should be done in a more, that one should look at, and without going through all this ghastly stuff about the albinism of quantum physics, which is usually done, much of modern theory can be based on modern, much of the useful modern theory. In other words, there are reasons coming both from geometry and your practice of model theory that we used to see that, in a certain sense, this is really the central concept of first-order logic, and then the other things are, you know, strengthenings in different directions, but this is what's starting to vary. In fact, even quantifier eliminations, you see.

27:30 I've often thought that, well, I mean, for example, the real closed field, this type of thing, that one views it as an ordered field, so order is an additional predicate, but in fact it's definable by, there exists a sum of squares, so in some sense, if you only quantify it around as there exists, It's just one hair's step beyond the equational and its own... That's right. I mean, that in fact is the phenomenon in the vast majority of the theories where it has been useful to make, in many structures, where it has been useful to make logical analysis. I mean, you're just a hair's breadth away from being in the equation. You'll typically have something that you've got... The important basic sets are just projections of the equation that are triggered from one dimension. That's how it goes. I mean, I propose it's a slight thing. Generally, it's a sort of curve thing and then doing it and doing it. I mean, this is much deeper than, but it's the same kind of thing as standard technique that Robinson used to use in the 50s for establishing that Robinson's test was true in certain structures. It was going to be true if you were solving equations in one variable and so on because of the basic facts of the real force field as well. It's not quite a universal phenomenon, but in practically all the cases, the pediatricians and so on, you understand something, you have a thing. I mean, it's much crueler than what we expect. Somehow it's the same thing. Yeah, in other words, it seems to me, I don't know whether this would be a correct description of the general case of quantifier, but what it certainly can do in certain cases is... What it really means is that you can define universal quantification in terms of an existential project by the, you know, in other words, the well-known double negation, which of course is not true generally, but it is true in certain cases. So, for example, with the theory of fields, which as I just said before is a purely positive theory, in fact, even a relatively simple one, you have an effective definition of logical negation right there.

30:00 And so you can define higher alternation quantifiers just in terms of this positive theory. So I think that sort of thing is true. More examples of general characterization of what's meant. Not by quantifier, not total elimination, but reduction to the case where you have only the existentials. It's more of a convenience. Again, when you say AE formula, this is... This is sort of a mystification, because the for all a is in practice just a free variable entailment, and not a real use of for all x a as a variable. I mean, the whole thing is, it is more than complete. It's all maps in the theorem. Right. It's just a positive logic. It's sort of the division of labor between the general and the particular logic. Right. The general logic should be this more general one. And it's particular theories on the basis of the mathematical content of the theory rather than pure logic that you get out the internal expressibility for all. All maps are on that trick, yeah. Yeah. That's a beautiful category. Yeah, yeah. And this is, I mean, of course, Mackay and Reyes already say this is why this is a nicer logic than in the first order, because in first order the maps won't all be elementary, because it's like, okay, yeah, yeah, yeah. Right. Morally, morally, one frequently does this moralization process to make, to make the academic element, which I think you were supposed to say. Like making it positive. Yeah, yeah, that's exactly it.

32:30 You can be presented with different categories, but you took your weight into that one. Well, you get nice limits. Unless the judge thinks, does it? Yeah. And when I read the Congress, I didn't get the stakes. I didn't understand the issue here as well. I mean, I saw, yeah, we don't have negation. Ah, but we can actually entice a compliment to a given predicate. Yeah. Yeah, that's pointed out in some paper of mine. Oh, yeah, yeah, yeah, but I didn't take it into a little better read than that, because that's something all mathematics are all about. I didn't work out the details when they did. It's still a nice book, I think. Oh, I didn't remember that. Right, okay, so does that take care of the logic part? Maybe. Trying to follow Mike's outline. About the points and about the, yeah, okay, Grotendieck's, why did Grotendieck insist in 1973 that... Except me and I was already converted, because combining what Gabriel had taught me with my own speculations, I had arrived at more or less the same point of view that he was promoting, of course. I didn't have the faith that it would all really work out, but he knew that it did work out, because you're making such a major change in what seems to be the received doctrine. You have to check all the crucial points before you can really go out and say this is the way it is, so I was much emboldened to promote what I had already partly thought of because of this lecture. And, well, I think I had already said this before, and I don't remember if I explicitly mentioned this in the lecture, but one of the crucial points is if you consider the idea of scheme.

35:00 The topological space plus something, chief of rings or whatever, then you're limited to the underlying topological space function. But this underlying topological space function does not preserve Cartesian products. So, therefore, I remember, you know, you've already, in graduate school, one professor of... Algebraic geometry telling a student, you know, I wasn't in that, I knew what they were talking about, but he was saying, you've got to realize that an algebraic group is not a topological group. You see, despite the fact that we'll say a lead group, it's not, somehow it's not a topological group, and the reason is just not some technicality in a way, it's just this bold fact that the Cartesian product is not preserved, and so... It's the Cartesian product qua schemes, it's on which multiplication must be defined, and yet that doesn't correspond to the Cartesian product of spaces, and so there's no... So, you know, so the underlying set function is not, I mean the underlying space function, much less the underlying set function, does preserve products. And on the other hand, neither does the space set of components function. Components, it's a crucial property for category and spaces that the connected components functions should preserve. These are two different functions usually to the same category of less cohesive spaces. But if they don't preserve products, then really everything becomes incredibly complicated. So essentially, and there are many other things, I mean like one slogan that Draven Deke told me on the side after his lecture there was, don't forget that the points have automorphisms. Points in a topological space don't have automorphisms. What can that possibly mean? Well, the points of a scheme should have, namely the Galois groups, basically.

37:30 Algebraic origin of the mark, but then one has to see that in a geometrical way. What does it mean geometrically? So the essential solution of it is to consider that the underlying, the drastically less but not totally cohesive, is the Galois topos, rather than abstract sense, rather than topological space. It's a topos inside which you could consider topological space. And then it would make sense, you see, because essentially the prime ideal, this is to get it back to a community of algebra, if you have a community of ring A, then you can consider for each field K, you can consider the homomorphisms from A to K, you see, so those are the K-valued points. Of course, those homomorphisms may not be subjective, so the kernel is not a maximal, it's a prime ideal. And of course the point of spec A in one will be turned into the point of spec A in the other one. And so you have a, for a given ring, you have this diagram of sets parameterized by the category of field extensions. All field extensions, all homomorphisms of fields are actual monomorphisms of course. So for each of these you have prime ideals. Two different points that have the same prime ideal as kernel will become, you can amalgamate and see that they become equivalent in the direct limit. So you take the direct limit over the category of fields and that's the set of prime ideals. So this direct limit is not a filtered direct limit, therefore it doesn't preserve products or equalize them. And that's where the unnecessary discontinuity is introduced.

40:00 So the more geometrically conceptual, the reasonable thing is to recognize, yes, indeed, we have these slightly more general points. Both the set of components and the points of space are really objects in this category of all functions on field extensions to sets, rather than just sets. Actually, they're always sheathed with respect to a very strong topology in which every map is a covering. Logically speaking, the meaning of existential quantifiers and disjunctions is modulo-arbitrary field extensions. So they're sheathed with respect to that. Now, if you want, you can consider topological spaces that you can associate to a scheme. A topological space object in this topos. It's a Boolean atomic topos. We call it the bar topology because bar pointed out this fact that if you have a category that consists entirely of epimorphisms and has a certain amalgamation property, then you can just take every map in sight as a covering and the topos of sheaves that you get will always be atomic Boolean, that is, lattice of sub-objects in atomic Boolean algebra. So, in other words, it's much, much, much, much closer to classical logic, so you're almost doing set theory when you talk about topological spaces and whatever in that toposet as a base, a Boolean-valued model. Well, it's not a Boolean-valued model exactly because it's not based on a poset. It's more like a permutation model because it's got actions. Finite Galois groups. Again, as I mentioned before, the practice has been for a long time to pass through a limit and so you have pro-finite, a single pro-finite group rather than just this natural diagram of finite groups, which in my opinion is in principle just introducing a complication only in order to struggle to cancel it out again because it's...

42:30 The actual choice doesn't hold. So it's really something like a symmetric model. Yeah, yeah. Because it's a Galois group, in some guises it maintains. I'm saying the most natural guise is it's not a group, it's this little category of all possible finite field extensions. I wonder if Lisa and I and Chandler have an unbublished way of combining humanity with the notion of work, where we're telling it's been always going to be an observed principle in axiomatizing interesting classes. If you really, I mean if you don't know the axiom in advance, you will detect them by reflecting on the nature of the finite extensions, on what the Galois groups look like. But one thing you cannot do, first of all, is you can't do it. Nothing can be understood. ...would be to attempt to go all the way to the topological group and try to quantify over its elements. I mean, it's far too complicated. You can't code it back. But each element, each of these finite extensions you're talking about, is basically coded back in the field. This is the whole point also of much of the class theory. You can detect any little bit you want back in the field, but you certainly can't detect the whole thing. I mean, the whole thing is a convenient reason for studying these pro-finite groups. In other words, I think, just speaking sort of psychologically, there's the idea that, well, this category of all finite field extensions is impossibly complicated, so we can't sort of contemplate that. Well, it is complicated. But then to go and, you know, think, well, if we have just one group, then we've simplified it. But in fact, we've complicated it before. I mean, we stress this. You better keep this system in front of you and just do a little bit of softened quantification. That way you'll get decent information. The moment you actually do the completion and try to do logic in there...

45:00 In this one topos, in this one small site, whichever way you want, the unity is already there, you see, rather than just, well, the same thing happened with the Shenandoah topos, which is the most simpler thing, classifying topos for infinite decidable objects, which is basically the category of... Covariant functors to sets from the category of finite sets and monomorphisms, equipped with the topology suggested by Myhill's observation by the intersections. I call it actually the Shanwell-Myhill topology, because what happened was that when Shanwell heard from Myhill about certain constructions, I think involving isols and certain kinds of constructions, He noted that this condition, Maillot already had this condition that a functor from finite sets and monomorphisms to sets could preserve intersections, but this actually had all the properties of being a sheaf with respect to a certain notion of cover. But then Maillot had this unique expansion. The objects are uniquely expressed as co-products in certain ways, in certain ways, a very concrete sort of expansion of these. So, you know, just a few days before that, I'd been telling Shannon about Barr's construction of the atomic topos. And so he said, oh, this is an atomic topos. And it turned out, you know, then he immediately proved that it was. And so there was this Daniel Topos, which has this, which has this very, if you think about this category of finite sets and monomorphism and the role that it's playing and why it's, and why intersection preserving is equivalent to the sheet condition and so forth. Just a little bit. Then it becomes very intuitive. But what happens then is that the machos who love these pro-finite things, they describe this classifier for a decidable class as the continuous actions of an intimate group that is immediately obscuring elementary character.

47:30 It's the same topos, but they're incredibly mysterious. It's very elementary and intuitive character. It plays a key role in its independence results and stuff like that. There's no various glorifications of it. So yeah, that's... It's only, you know, it actually has taken you years to convince people. Again, that's what it was. You've got to be civil with things at first, but they've always described it in terms of the actions of pro-finite. It has this elementary meaning, intersection preserving, the operation of intersection on finite sets, but there's a subtlety, you see, that it may not be the pullback in the category of pro-finite sets, because if you have... All of these terms may be related to a particular category, whether it's sums or products, but it happens when you restrict the maps in a category that already has, say, coproducts or products. You'll still have the functor corresponding to addition to multiplication, but it may no longer have the universal property because the universal property requires that you get a unique map every time you have to test the situation. That unique map may not preserve the property. Of course, this Galois topos, like the Chandelier topos, it has this further property in the site. Not only is every map an epimorphism, not only is every map an epimorphism, but the offsets are finite, so that implies, of course, that the endomorphisms, all the epis of an endo-epis of something finite are necessarily invertible. So the fact that there are all these Galois groups just follows from that extra axiom on the... Still, I mean, coming from this general vantage point of atomic toposes generated by finite objects, you still don't have some of the key features of Galois.

50:00 If you have these groups, the endomorphisms in the object are incredible. It's a property of the topos. I'm not one of them, but many of the topos. This idea of having that kind of an atomic topos rather than abstract sets as a base for algebraic geometry, in principle, Rodenbeek already in 1960 had proposed a similar thing. For analytic geometry, that is from the study of holomorphic functions, analytic spaces, and so on. In the 1960 Cartan seminar, under the title of Techniques of Construction for Analytic Spaces, he, in effect, introduces a similar roto code. He doesn't call it that at that time. The appropriate, because the appropriate base is a long struggle, but it is the appropriate point and Bowen and Funke's do preserve on that. Something which is not true for most topos, it's received on a topological space will essentially never satisfy. So the fact that you have qualitatively different type of a model of say all spaces of the circle.

52:30 Max Kelly once tried to teach me that you don't save the same milk. The ilk is already the same for both of them, is that right? Yes, it is. You could pass the equivalence question as the ilk, but it's not normally done. What do you think is the translation of the ilk? Well, to be of that ilk is to share the relevant properties. So to be of the same ilk would be to share the same relevant properties, which isn't what I'm trying to say. The two comparisons would be of the same oak. This would be in respect of the same property, but this one would be of that one. Yes, this thing is of the oak of that one. And it shares a property. Not that it's of the same oak. Yeah, yeah. So this led to some of my papers in computer science and so on about qualitative distinctions. Always having a qualitative distinction in mind between the two talks. I knew about a growth peak in 1960 in the analytic case and then in 1973 in Buffalo. Although, of course, the 1973 talk is, as I say, quite consistent with Gabriel in 1965, or Gabriel's book on algebra. At least they start with that. Basically, that the general spaces are just covariant factors from the category of algebras into sets. Figures of the type, you know, circular figures or cubical algorithms. The figures of those various shapes in a general space, as well as the incidence relations between them in the general space, is equivalent to the definition of the space.

55:00 So you may as well just say so right up front. The figures and incidence relations can be calculated to enhance the principle of all properties. And this pulls us close to sort of general philosophical issues, right? I mean, we're used to the multiple reductions of the real number. They could be this, as Ariel O'Frankel said, or that one. It could be Dedekind cuts or columns. But here we're saying about schemes, it's not that they have multiple realizations. Are they topological spaces with sheaves of rings, or are they set-value functors? Completely different kinds of things. And Brode wants to say, at least, that it doesn't matter which they are, and is favoring the set-valued factor approach. Because it's from that that certain properties, like those I mentioned and the others, are more readily evident. Yeah. André Joël put it to me one time, he's explaining the two, but he doesn't want me to worry about the distinction. He's saying, there's just no ontology here. There is no ontology. It's not just a multiple reduction. Because he doesn't do that kind of thing. He doesn't do ontology. So, I mean, here's this growth in deep move. Can we take this move that's really seriously fundamental and say that we've got this notion of a space where the space isn't really anything at all. It's represented by a functor. We don't ask what it is. We know what it's represented by. And that's all there is to ask about it. And that's fundamental to a lot of relativity stuff is to, on a working level, say, but this goes way back in mathematics too, we know there are as many of these as there are, say, this kind of ring. We don't say these are this kind of ring. We just know there are as many of them as there are of, say, this kind of ring. I think that's the basic definition of geometry, you know, and it's insufficient in general sense to include its figures and its observations. So you even get a picture, and sometimes a very low-dimensional picture, curves and surfaces, sometimes a very geometrical picture, of what doesn't have an ontology at all.

57:30 This isn't a picture of something. This is a picture of the relations that hold. There's a third picture, of course, which is the discrete vibration. The set-value functor is equivalent to the discrete vibration, but what does that mean? It means that the generic figure shapes, figure types, and the generic incidence-relation, they form a small category. Category, again a category, but fibered over that. So in other words, you have the concrete figures in this space and their incidence relation. That's a category. But it has this labeling function. You can think of it as a labeling function, because each of the concrete incidence relations is labeled by the type of abstract incidence relation that it is. So that's what the fibrations is, that layout process. You start by thinking of set-valued functor. You've got this domain category, and it's wandering through the whole universe of sets, but you can actually bundle the parts it wandered through as just another category, no bigger than this one was, however big this one was, and you bundle it into a... So instead of looking at one thing down here, you look at a sound fibered over this. So it's fibered over this. There's a lot of situations where people talk about labeling, labeling graphs and so forth. So the labeling is actually, you can think of it as one category that's slightly more complicated in a particular way, in many different particular ways, over the fixed categories of labels. For objects and maps, and then you have something slightly more complicated than that, slightly in that it has a certain set of figures of each type, of each label, and then the fact you have this forgiveness labeling functor that has a property of Negan's Greek vibration, it's completely equivalent to having a set value functor, and sometimes that's more appropriate. There are a lot of situations which aren't normally thought of as geometry when you have this labeling process. There are situations where the arrows are thought of as processes, but then the processes can be labeled by, for example, how long did it take, the duration of the process, as again...

1:00:00 And this gets to the principle that you talked about in Florence, and you mentioned there also that, and it's in Zermelo, but this idea that whatever totality we're tempted to talk about, if we're tempted for reasonably coherent reasons, I'm putting this more in my words than yours, we can. We don't want to say, oh, that's too big, that's not a set, that's a proper class. Because we know perfectly well how to talk about proper classes, so they're there. Then we know how to talk about classes of them. Well, there's a simple-minded model theoretic. If you have a consistent first-order theory of these things, then there'll be models. The models will have power sets, literally interpreted. And then the defining point is, well, there are several different ways that you could interpret what is definable in the power set, so there's not a unique model of the higher order theory, but certainly the consistency of the higher order, where you've collected these things, itself follows from just a low-level subject. Yeah, yeah. I guess that's more or less Pfeffermann's. Well, it's a kind of a reflection principle. I mean, it doesn't follow from ZF that there's models of ZF, but people who use ZF take it that there are models, and so since there are those models, there are those things, you know. Yeah, I noticed if you sort of take the fact that consistent theories have models as a... Thank you for your time, and I look forward to seeing you again next time. Subtitles by the Amara.org community

1:02:30 But the International Logic Review gave an interesting proof that the net, that parallel arithmetic is inconsistent and that you would look at an article and it would refer to some earlier article where the proof occurred and you got this infinite descending chain of references which in parallel arithmetic you shouldn't be able to get an infinite descending chain. I mean, it was an ascending chain. Yeah, if you kept saying in my next article, it would be one thing. But he kept showing that in a previous article, he had proved it, you know, and since the issues were discreetly ordered and had a first, you know, without declaring a contradiction, he never declared that this was an infinite chain, but in practice, it was an infinite chain. Yeah, I have the honor of being the one who actually introduced Veta to us in Vulcan. I never met previously, I knew each of them slightly. Only slightly. Only slightly. So I thought, oh, it would be quite appropriate to introduce them to each other. But John Mayberry, he was really persuasive on how we have not tested the limits of these induction principles to replace the scheme of induction in quantum arithmetic. Not only have we not tested a statistically significant fraction, I mean, duh, infinite, we've tested only very, very special kinds, only kinds that made sense to us. So that it is thinkable that those principles are too broadly stated. But if they are, the problems are in places we've never been. So even if you're willing to think that, you don't give up on them. These are probably the only categories there, to somehow take this Gödel-Berenice principle, you know, it doesn't exist unless it's a member of something else. But I'm sure, you know, it's a dogma by dogma. Whereas Gödel and Bernice themselves, I don't know if you've ever heard of them, that's just, in the collective works of Gödel, there's correspondence between Gödel and Bernice.

1:05:00 They are discussing two things, I think. One is that the issues raised by MacLean that Pimpsey and I have worked on methods, and also the rumor that somebody is working on a categorical sect here. Now this happened to be the same month that I was writing my thesis in California, and later Dana Scott surmised that Gertle, that Kreisel must have... Who was on the phone with Gödel every day, so I've told him about that without revealing the name of the culprits. But anyway, they were, in their correspondence, they were saying, oh, well, yes, of course, we have to, we have to take finitely the types above the, take those types to be just as, just as real as the itself. The very ones who are taken as the final authorities immediately wanted to consider something more reasonable. It's just that, you know, at least crudely speaking, their original work was related to that from Neumann also, of course. It was a kind of a trick to express everything they wanted to express at the time in a compact way. Yeah, that's right. And it's good to know what trick works and what problem. Of course, of course. Part of it is hidden in what Bernays was telling me, and I think it's published as well. But originally they had two symbols, eta and epsilon. So one stood for membership between two sets, classes. And the other one was the satisfaction relation, essentially. And they saw that within this limited realm they could just identify these. That's the theory that we inherited. Properties, you know, grafted onto sets. The Ida was really the ascription of a property as far as I know. Exactly, right. But you see then, if you get, which of course is part of the legacy of Frege, if you underline, but I mean, you could then imagine, well, you could objectify the idea of property.

1:07:30 Yeah, that maybe you thought it was a formula, but yes, you want to quantify it over each thing, so you also objectify it. Then you can compare them with sets and so forth. So by making one particular choice about how all that was supposed to fit together, we got a completely satisfactory system for talking about real numbers and the power standards and so on and stuff like this, and for large cardinals even and so forth, but not one that permitted you to talk about the class of classes or the class of all functors from sets to sets and stuff like this. There's no reason you shouldn't talk about those things. But they are forbidden by the particular formulation that was arrived at. Hence, in the minds of far too many, absolutely forbidden. Yes, well, of course, in practice, you know, in metatheoretic studies, Cervantes-Frankel's text without classes that's normally used because of the introduction to classes, Presents precisely this difficulty. Well, if we can have classes, why can't we have classes and so on? At least in the Zermelo-Fraenkel case, you know, we don't have them. You're sort of left perhaps in a kind of limbo. I mean, there's this V sort of lurking. Potentially there, but it's not directly referred to, of course, in the theory, so it doesn't have to be mentioned. Whereas in the case of Gödel-Bernays, you know, B is there. It's funny, because that's very convenient for certain purposes. I mean, I often present said theory, it was just really coming from a presentation of David Scott so long ago, where you have objects. You have objects and classes, and any sort of collection, any property of an object determines a class, and then a set is a class which is also an object, and then you throw away, I mean, in order to develop set theory without... Well, yeah, you throw away, yeah. And then you, but this is very convenient. It's essentially, you know, it's good, but it's really a Gemello-Frankel set theory with terms added, if you like, to correspond to arbitrary predicates.

1:10:00 Well, that's how Quine presents it. How does that relate to Scott's idea? It's similar, as far as I remember, but this is a very sort of natural way of starting off, objects and classes, because sets and classes is a curious distinction to make, I mean, it looks odd. Objects and classes, on the other hand, is quite a natural distinction. You know, it's interesting, when we talk about, like Mackay and Reyes, when we talk about theories as categories, we consider that the usual formalism of first-order logic or a positive thing, that they're actually presentations of certain kinds of algebras, but we go a further step and say, well, those algebras are just categories of a certain... I like that word. I don't know. I don't know if I'm stylish or not, but I like it anyway. My interest is... because I like to grasp, you know. But, yeah, so if you leave these categories, in these categories, you know, so that there's an object which is sort of the generic model, really. In other words, typically, it's just a single sort of theory. Then this category has a preferred object, call it V, and the constants of the theory are matched from 1 to that, and the decidable predicates are matched from powers of it into 1 plus 1 and so forth. It's a very convenient way to think about these things. Satisfaction is just composition, and so on. You get a map from 1 to 1 plus 1 as the truth value of a statement in a given predicate is satisfied by a given list of constants. You can also see there why existence doesn't mean existence. Like the theory of totally ordered sets, for example, has no constants whatsoever. So even though there's a true existential statement, there's no map in this category that witnesses it.

1:12:30 Consider models, i.e. funcals, preserving this stuff into some category where epimorphisms are split, or at least simple epimorphisms are split, and it turns into a category where epimorphisms are surjected, in other words. Then it becomes an existential, a real existential state. In other words, there's all sorts of slightly different insights into the idea of theory that you get by looking at it this way. But anyway, having the object B as such is totally innocuous. It applies to all first-order theories. So this raises the question, what kind of a category is it? It's a category that has sums and products and pullbacks and images, and images are just an essential complication, but well, it could be a Cartesian closed category. So there you have, you know, p to the power of b and 2 to the power of b and all that stuff. And then, of course, there's clearly a very nice You see, not so ambiguous as saying take a model and take its power set, but as I point out in my paper on guiding arguments in Cartesian closed categories, the Uneda embedding itself gives you a full embedding into a Cartesian closed category. So you take any such theory construed as a category, just apply the Uneda embedding, then you've got it into a Cartesian closed category. Now, again, you have to worry about what happens to the various existential statements and so forth. You have to take the canonical root and the topology on the theory itself and, you know, immediately all sorts of difficult questions arise, but the fact that it's perfectly consistent to have a Cartesian closed category, which even fully contains, you see, I mean... You might think that, well, by introducing all that stuff, I can define more things at the lower level, but no, it's not, it's not true for maths, it might be true for sub-objects, you can define more sub-objects, but not more maths. I hope that that paper is going to be printed soon as well.

1:15:00 Yeah. Diagonal arguments in Cartesian closed categories. It was kind of a hybrid because, well... I gave a talk at the 1967 Los Angeles meeting on set theory basically, a huge meeting, logicians were there, I met a lot of people, but then I divided up the material that I presented there into three different ones, one was called hypergeography, one was called agonists and foundations, and one was called diagonal arguments, so it's presented as Simply illustrating why the so-called Russell paradox and the Cantor's theorem about power sets being always bigger and Tarski's definition of truth are really all literally the same construction in a single category. You just have to specialize a category and you get, that's how it's presented and that result is there and reproduced in both of the books of conceptual mathematics. But actually, there's a large part which is about Cartesian closed categories, where I point out this obvious, easy fact about the new meta-embedding, that not only does the pre-sheet category always have an intrinsic function space, it's even preserved by the new meta-embedding in case you had one before in the small category, so that in some sense... It is the natural, that these are natural transformations, but it is the natural notion of higher types in all sorts of concrete cases, so like, for example, the St. Kennedy case, it's a natural definition of morphism, of functional, of smooth functions. Analytic case, analytic case was considered by Voltaire and his student. From Topier, Max Dorn, and many other people to be the natural definition of function as for in the analytic context, I mean, to the extent that they needed topology, quote-unquote, they derived it from it, as opposed to, and I also conjecture there that the cursive function theory approached it in this way, and later my student Phil Mohler worked that out in this.

1:17:30 Theses, of course, in which he discovered that the significant fragment of it was always to Erschoff, and in fact that earlier Banach and Mazur used this approach to defining recursive continuity. So the Banach, Mazur, Erschoff point of view crystallized by Mowry is literally a case of this remark. Again, I didn't work it out there. In fact, I think I put it as a rhetorical question. Maybe workers in recursive function theory or smooth analysis would like to consider the ramifications of this. That's where Cohn-Hawley got the idea. So anyway, that article will be read in about seven minutes. So it was simply not published at the time? Excuse me? It was not published for what reason? Oh, no, it was published. I'm saying re-printed. Do you know this electronic journal theory application? Is it just electronic? Yes. It's just called TAC. And it's at Mount Ellison University in Canada. So, mta.ca. It's been published since 1995. In other words, 10 years. There are 10 years of these, you know, in the journal. Referee journal. But they have a special column called reprints. So they're making available all kinds of basic old references. They're hard to get. Like Peter Fry's book on the beauty of categories. They're just today. That's Mike Barr and Charles Wells' book on triples and triples theories, and it's down available. You can get it coming out today.

1:20:00 What, you just download it? I mean, sort of download it from there? Yeah, you just download it. You have your choice. You go to BDF, BDF. And then you can bring something like that in with the publisher. Excuse me? I was looking at Wells with Stringham, I think. Yeah. From various, various publishers. It's strange that they make it available. Mitchell's book was an academic press, that kind of thing? I remember all these green books and autoposites and things like that. Yeah, there was a book library. No, I'm sorry. The academic press. There's a book by Mitchell, isn't there? There's a book by Mitchell, but it's not that real. The first, the very first... Oh, yeah, yeah, uh-huh, uh-huh. No, I thought it was Springer Fairbanks. Yeah, yeah, that is Springer, yeah. Okay, so it's good to know that they are... These are, the very first reprint was my Milan paper in metric space, and it has these categories wherein is contained this observation that... Cauchy completion in the metric space is basically the same construction as the passage from a ring to its finite projective modules for the same sorts of reasons that continuous functions define the rational system. I didn't quite get a rational value, but you have to except in reals. It's why projective modules come up all the time in ring theory. It's the category of projective invariant. So that was the first part. Peter Fragge's book. There's about ten reprints. My paper on set theory, which Colin would prepare, and you can type it in. Which paper? Which one was that? It's the 1964...

1:22:30 Oh, that one! No, not that one. That one's just a five-page summary. Ah, right, right. What, 40 or 50 pages? Learning with all details and more metamathematical remarks and so forth, which has been available all along, provided you go to the University of Chicago Library and ask for it. But Colin thought that I should reprint it with commentary. That's one of the nice things about reprints, because in most cases, the authors have supplied one or two or three pages. What happened, you know, has been intervening for 30 years, and it's quite helpful in some cases. When you were speaking about these matters, I was thinking, I mean, I cannot follow all the details, but I tried to map it in the entire picture of mathematics, and I think it would be an interesting exercise to do the following. Let's say, late 19th century algebra, as represented by a book by Begel, was the main thing. To Van der Werden, there is a change, I already explained, but how the entire, or parts of the mathematical community look at this, at the meaning and the location of this work. So, it's very interesting to see there... You take the Yarbuch, the Proshvita de Matematik, which is the review. You remember the talk I gave at Berkley. How the classifications... What year, which year was it? I gave a talk in... two years ago. I can send you the... No, I mean the Yarbuch. The Yarbuch, well, I take all the... Oh, I see, all the successes. From, let's say, 1895... And I look at how the various articles are classified. So some of them we consider, we look at them, ah, it was an article of the Abbot numbers or on rings, but the concept didn't exist that much at the time of what is the discipline that you are looking at.

1:25:00 You know, the very idea of what is the discipline that these things belong to, it goes to an interesting transformation, and how the articles were classified. Of course, you cannot reclassify. You classify at this point in a certain classification, and then there appears a new classification. Obviously, you could move it, but you're not going to do it. And the question is how this classification appears. And before I came here, I was looking, but I didn't really have the time to do it. Where all these articles are classified, you know, starting, but in the mathematical review, the mathematical review starts in 45. Which is a little bit after, well, I mean, the first article by McLean, I remember, is 42, but the real one is a free announcement, something like that. The first one is 45, so it could be a nice exercise to see how all these matters go in, you know, from 1945 to at least, let's say, 80. And I just look at some titles, I really didn't have the time to do it. There is a section called General Algebraic Systems where all the first articles in categories appear and then it's called General Algebraic System, Homological Algebra. One should see what articles are included there, because possibly some articles that we later see as part of homological algebra were not there at the moment, I don't know. Whereas some that we would see as category theory were considered as homological algebra for a long time. So I really wonder how, you know, all these articles of yours in the 60s and so on, where do they put it? So, possibly logicians, we don't see it at the time, and all... Of course, there is another problem. Who prepares the classification of the mathematical reviews and to what extent it reflects, you know, views of a majority of mathematicians, certain mathematicians, and also how it shapes the views of mathematicians, because if I can, if I am a student... If I'm looking for a topic for a dissertation, then I look at the mathematical reviews and I don't see any of these words because I am looking at category or I am looking at logic and nothing of these appears.

1:27:30 So I don't know where all these things were classified and when, for example, I guess there has been at some point appears some classification for topos. There was a few years ago a total reorganization of these. To me it's apparent that MacLean played a large role in it. I don't know how it looks, but it's very detailed now. Yeah, yeah, yeah, of course. It's sort of ramified, you know. Yeah, yeah, yeah, but there were several before that. Oh, I know. All the time, actually. You know, because it's a real problem. Why is the mathematical field, the mathematical discipline, how do you define the borderline? In fact, I rarely use it that way. I find the name, you see, I look for the name of the author. Maybe you hear, oh, that author might have done something interesting. That's the only way I hear it. I don't even bother with, I mean, I bother. Because you assume that the classification is not according to the... Yeah, that it's not going to be... No, I mean, I should use it more often. No, I mean, classification is always a problem, but I think in the case of all these topics you were mentioning, it's even more problematic to try to... Well, I remember now, you know, there was a time in the last 15 years that NSF hasn't given us any money in category theory generally and in our group in Buffalo in particular, but... But before that, you know, there was quite a big branch on the NSF to, you know, Isbell, Shamwell, Duskin, Bovier groups. Every year we got, we could travel anywhere. We didn't need mics, we could travel. Yes, those were the days. So, but I remember, in particular, there was an extremely friendly official in Washington. He had been a calculus student when I was there, and he apparently really liked his calculus course, and so, well, I mean...

1:30:00 He told me, he told me, that, that Ivanberg had told him, whatever LeVere wants, give him the muttons, because we're trying to look at the information. This is incredibly corrupt, right? Incredibly corrupt. I'm sure he didn't, you know, only follow that instruction, obviously. But he was, he was a very friendly guy. So I remember we were discussing on the phone, Alvin Pollard in Washington and me in Buffalo, precisely this question of the classification and all sorts. We had this report or application, I forget which it was, or maybe we were even discussing refereeing some other, somebody else, I don't remember that content, but it came up, this question of the classification, and I kept telling him things and arguing back and forth, and finally I summed it up by saying, Well, my purpose is actually to totally destroy your classification system. That's my purpose in life. So that all these things are really one thing and they all flow into each other and it's not going to live forever. So they work according to the AMS classification or they have their own? Yeah, they use that. Again, I don't know exactly what they do now. I know in my time in the States, I never ever got anything from here. I got things from quite... I mean, who would... I'm pretty sure Dana Scott must have done it. Dana, sure. You know, NSF made a decision not to support category theory. This was even announced at Missouri by Bill Thurston in 93. What is that? In 93, McLean and McKenzie... We're sort of united, and so we had a joint meeting of the Universal Algebra Academy, funded, of course, by the NSF.

1:32:30 So here we all, we're all assembled, you know, algebraists and categorists of the world, not everybody, of course, but a large portion of this. And the director, who's an NSF official, he comes up and says, well, the NSF has decided never to give you guys any money ever again. You know, words to that effect, literally to that effect. Did they explain why? No. I'll have to explain. When was that? 1993. 1993. Now I knew that something like this was going on already five years earlier because they stopped. They stopped, it wasn't just my, you know, version, Isbril had a big reputation, of course, you know, it's not, I didn't mean to say it was just because it's in the subscription, but it was cut off. The NSS instead started to do massive funding for attention-destroying school education, and even Ardorkel was a by-product of that, a dean of ours wanted to cash in on these million dollars for reforming education, so within that he gave us a little bit of extra money. We hired an assistant. We were committed, you know, two professors teaching one course, and those are the main conceptual ones. But it's within that that we were able to produce the conceptual math. Because he asked us to do something on discrete math. And we said, no, we don't want to do that. Maybe we could do a framework which includes continuing on discrete at the same time. So this was the starting point. That was the concept. And as you know, they've gone on through some incredible things in the field of education. I was advised, I think in fact by Al Saylor, to make an application in the field of computer science.

1:35:00 So I made a big, very detailed, all possible aspects of computer science, everything I had ever done. It was a very impressive application, I thought. It was rejected out of hand. Just because the method seems to be, you see, that the NSF has a stable of reviewers, you see. So you get a number of different people reporting on your application. But if at least one is negative, then it's negative, obviously. This is the way it's done, obviously, in Britain, too. And I mean, it's positively infuriating, because you get to see the reports, of course. And I mean, typically people are so busy, not everyone who's asked sends in a report. But I mean, you can get. And it's the same with the European ones. And this is discouraging a new idea, essentially. Of course. You've met frequently some, I've made up reasons for it, you know, social systems get two excellents and you get one person who might even say very good or something like that, sometimes that can really kill the whole thing. That's what my word application, one thing that strikes me about this, you see, is that, and this sounds like totally, you know, subjective, but I claim it's very objective. The good reports, in the sense of being scientifically analyzing what the hell is going on, those are the ones that are positive, and the bad reports, those are the ones that are just negative, are very, very bad in the sense that they haven't even done anything, they haven't read it carefully. But he doesn't like the field or the approach to begin with or something like that. Or he's told by the NSF, you know, I was turned down so many times. In Canada, I just gave up. Really? Although in Canada, it's actually a situation certainly better for category theory, of course in Montreal, although I don't think it's so good now from what I've heard. I gave up. I mean, it's pointless. You can write a book faster than fill out one of those forms. Even in my own university, I applied for a grant. No, it's amazing. I had a grant for writing this book on the continuous. I was turned down. There it is. I think that, you know, from a historical point of view, these things are more influential than you may think at the beginning because...

1:37:30 You know, if there is a good idea, it will find its way. We spoke about Cantor. And today, more than in the past, you know, you need more than a good idea. You need support, you need professorships at various universities and things like that. And you can make a very detailed analysis of how this good idea was not allowed to flourish. And the matter of classification is related to this, because the other... This is also very prominent in the European funding centers. A lot of it's here. First of all, the bureaucrats use these simple classes to send the... And it's then somehow stuck with them, because if you complain about them, you're accused... I mean, I've been several times accused, you know, of political influence, because I denied that a person who was giving an opinion at the table had any. Since they have never been anywhere near the area, these classifications send them to the massive international tribe. Major figures have gone down because somebody went, say, a physicist from Spain, who will judge an algebraic geometry, or a Greek person who has never been anywhere remotely near mortal theory. I served to a response on a review committee. After I go back to Britain, they sometimes bring foreigners in to evaluate the idea of a review, an extreme example. And it was fascinating. I went there with my first colleagues who had been funded in some other group, so there was five people, we got a look at them. I got a look at one of the first two, excellent reviews, and one, this kind of stuff can never lead to any logic in the remote outside world.

1:40:00 I could never imagine what it really are in those people. What's this like? This guy had killed one person's chances of working efficiently in the subject matter. There's a pretense of expertise. I remember one report. I don't think this was on our... Nobody else had a proposal which involved, among other things, working with certain categories designed to include things like the category fields. So, in particular, these categories didn't necessarily have a terminal object. So the reviewer, he was this big expert on category theory somehow, he says, oh no, that's not the way you do it in category theory. In category theory, if you don't have a terminal object in your category, you first join a terminal object, and then you go on, you see. This is just so crazy! But, you know, with all the arrogance of, I don't know, it should be done. In category theory. From some smattering to personal adherence somewhere. Well, it's the principles in algebraic geometry. You first assume that all equations are true, that there's a terminal field, and then... Yeah, a field with one element. ...with one element, and the notion of a scheme, right, being defined over the field. In principle, one may... But the only approach they have to it, well except for some very viral Japanese approach, I'm still trying to understand, the only approach is via the second group of definitions of TIDs. I mean the first one you can't do it, but the second one you can. If you tinker enough and you get some definition, which then turns out to agree with the special cases that TIDs have found by sort of numerology of matrix groups and finite fields and stuff, that's quite a fascinating piece of... Well, since we seem to have had, from what I can tell, a detail through some of the interesting and relevant sociology, and it's kind of touched on, it's touched on, you know,

1:42:30 over-minimality and tantrapology, and from what I was able to listen to downstairs, I think we've pretty well exhausted the philosophical exposition with which we all began about Nara and Hartford. Would it be a good idea to use the remainder, I'm just going to bring up the water in a second, If you want to go down and get it, I was just going to bring it. I was just thinking, looking at the time, because we're another hour before lunch, would it not be a good idea to go on to the connection between the O-minimality program and same topology and the overall view of the many problems raised and the many wrong turnings taken as a result of this. This expression I want to use, but this choice that was made at the end of the 19th century as to the solution to the initial crisis of geometric intuition, and look again at this whole issue, and then come round to a short discussion about Schengel's conjecture. We've already discussed it. Well, we haven't touched on Schengel's conjecture specifically. Shango's conjecture is supposed to be Shango's library. Shango's conjecture is, of course, a very... Yeah, I think this is another... But it's one more short thing and one more short thing. Well I accept it seems to me to be so intimately connected with the sources of people beyond their construction and how one might see technical ways of reconstructing them in other parts of the world. I mean at least the things that John was saying in his program, he seems to have new proofs of the transcendence of pi and so on.

1:45:00 It may be just too vast and I'd suspect that we'll all unmute in the evening. Well, since I've just spent the last 25 minutes sorting out his bloody hotel room and everything else, if he mutinies, leave it to the general staff to deal with, okay? I see. And I think we should have at least one more session. What is Shandor's conjecture? Shandor's conjecture was, yes, it was directed by Steve Shandor in 1960 when he was working with Sarah Dwyer. It's stated precisely in the second, but it is a very compact statement which basically creates the kind of thing I now call the Shandor machine. In the inductive process, essentially everything we ever imagined can be true in the world of the transcendence, of transensity around E and pi and I. Tremendous effort. The first wave of things around E and pi. Here you have sporadic other things, but one has never been able to prove, for example, this. This is subconjecture. So the algebraic relations are only the obvious ones? Essentially. I mean, the statement doesn't follow me. The idea is, I mean, it is of course very relevant for the formal aspects as well. I mean, basically with the exponential function, the idea is that the only way the algebraic relations can be created between the values of the exponential function is if they are coming from linear relations on the things that are exponential. All of this gives us a means of converting linear relations into multiplicative and multiplicative algebraic relations. Now, Chagnon wrote down the following thing, which may at first look just like numerology.

1:47:30 Let's suppose you assume that these numbers are linearly independent. Square root relation for the exponentials. You get a quadratic multiplicative relation for the exponentials if you had a linear relation over Q. Suppose you make the assumption right at the start that there are no linear relations at all between these n elements you've chosen. Then the channel of the equation says the following. Take your numbers. Now these numbers are rather linearly independent. It may very well be algebraically dependent. Take your numbers and take also their exponentials, the n exponentials you would get. So now you've got two n numbers. The enchanter's conjecture says that the transcendence degree of these two n numbers, the maximum number of independent things you can extract from that sequence of two n elements, is at least n. Never send them back to me. But basically, you can get some things in one line. The transcendence of E in one line gives you linear dimension. So take it, and it's exponential, you've got a transcendence degree at least one, so it's certainly the exponential that's giving you a transcendence degree. You play games with the Euler relation, e to the pi equals minus one. So lambda one has to n are any alphabetic numbers? No, no, complex numbers, arbitrary, totally arbitrary complex numbers.

1:50:00 Complex numbers, okay. Totally arbitrary complex numbers. And, um, so you can, in one line you get e transcendental, and in two lines you get e to the e transcendental, and so on. And you'll get, you'll equally well get that E and E to the E are algebraically independent. That's not known. It's not known that E to the E is independent. Does he say it follows the Hilbert problem number? Well, the Hilbert problem, yes, the one that Gale found, I think it was seven. This is the way to do everything in that way, essentially, insofar as it involves values of the exponential function. It yields, for example, Baker's Fields Medal winning results, not with the numerical bounds, but it meets the basic results Baker obtained in linear forms, linear functions. It yields the most advanced results known, which are due to Nesterenko, and which give things like... This is again cranked out with the Shannon machine in a relatively few lines. Now, it looks like, in a sense it looks like numerology, but the numbers have been got just... So the list 1 already shows that E is transcendental. The list 1E will show that E and E to the E are algebraic and you'll start with, since E is transcendental, you're not going to pair with linear dimension of these two and therefore by taking this E and E very highly, you get transintegrated these two. Out of the three numbers, one e and e to the e, one's certainly not contributing to the other, so you get these independently. So this is the typical kind of cut. You cut out one number in the middle and you count it. Now I, many years ago, got a paper to refer me to. The good side is that I've considered the following decision problems. Suppose I start with just the usual 0, 1, plus and times, minus and like,

1:52:30 And then an exponential. You've got an exo, you know the solution. So start building numbers as the denotations of these terms. You can call these things like the basic exponential. They've got names, and it's unknown absolutely what they tell mechanically. This problem is mechanical. Moreover, something even stronger is true that you can naturally deal with the equational class of what we call E-rings. It's like rings you put with an extra. So we call these e-rings commuted rings with one, let's say. Then basically, when I showed that Shannon's conjecture implies that the e-subring termed... Very briefly, how can this... It's almost the same thing. It's the same thing. You see, in turn, what you can do is you can... At this point, you can even do this for functions, for terms and many variables. You sort of identify the three algebras by an independent structure and you can, if Shannon is true, then it's sort of trivial. It comes down to a problem of checking whether two elements... How did the transcendence and the degree of transcendence translate into an algorithm with certain properties? Well, as follows, I mean... And that has to do with the kernel of this map, the exponential ring. So there's a certain ideal which is the kernel, and so the transcendence measures how big that is. There's a lot of things your calculation doesn't have to rule out, because the theorem ruled them out.

1:55:00 You've got the free thing, and then you've got the map into the real, in the sense that it's the only one that could be. And then the question is, is there a kernel? I mean, is there a kernel in this situation? There is not a kernel, and how do you do it? Well, you need some sort of independent look at the free thing. We, variously, I suspect McLean may have constructed the three things in the 30s from the suggestion that his master's was about exponential algebras, but I've never seen it, I must look at it. No, no, that is what he said. Yeah. I don't think it's that far. I don't know, but Jacobson once said something to me that hinted he might have thought about things like this. Anyway, Van Der Wyssen and I, a long time ago, sort of constructed the three things. And I mean, it's utterly simple-minded. What do you do? You start with the polynomial ring and end variables. You've got a partial exponential defined on that, mainly E of 0 is 1, and that's all you can see. And you want to do some kind of direct limit process which will make, end up with the exponential totally defined at the limit, but it'll be partial at each stage. So the first time around you've got, say, the polynomial ring, or you might just have Z. You've got something free in the ring category, and you've got the exponential defined only on an additive sum angle. So what do you do? You've got this, you've got the thing defined on an additive summand and you've got the other, you've got the complementary summand, but you have not yet succeeded in defining the exponential, but it's an additive group. The exponential, when you do define it, will transform it into a multiplicative group. You can either do it formally, you can do this nice thing, you sort of pretend that you take this additive group, call it gamma, and you form, like, t to the power of gamma. It's like a group knowledge group. It's technically very convenient to think of it as... It was literally, I mean, the polynomial of the ring itself is the monoid gradient. That's right, I mean, all of this comes down to the ring. The monoid, I've often thought that the word monoid probably was abstracted from the term monomial. Yeah, yeah, I think that's what he's saying. And so on and on and on and on and on and on and on and on and on and on and on and on and on and on and

1:57:30 This from the plagued group t to the gamma and make the group algebra. The freest thing you could possibly do. And now you define your exponential for the old elements gamma by e of gamma is t. The most simple kind of thing you could ever do. It will at least give you the function equation. And now again, you have a visible splitting in the in the group algebra. This is sort of augmentation ideal in the rest. Do it again and again and again. And that you can readily prove that this thing is the free exponential E-ring on the set you chose to start with, and you might have chosen to start with nothing, if you did, if you started with Z, you would create, E would be t to the power of one, now you can prove readily, you can prove that it's got lots of extra structure, you can differentiate it, it's the differentiation of an exponential, and then you've got this kind of hierarchy of terms, that is like generalized polynomials, except they're in... In group algebras. And then you can readily prove the kernel in terms of an induction using Shannon's. I think, quite by accident, I have a copy of my paper on it upstairs. It stayed in my briefcase. So if you're interested, you can certainly have it. It's really quite simple in a way. This is an interpretation of this part of Shannon's. So for the numbers directly built, the sort of explicit numbers built up from... That there are no hidden relations between them. Any relation at all can be cranked out from the high school exponential. Then it turns out you can also create classes or maybe Kelly Morse and so on.

2:00:00 But are these surreal numbers that were constructed in a number of different ways by Conway and Kruskal? Conway defined some basic things for them, but Kruskal succeeded in defining the exponential. I mean, it's not easy. He was constantly having to work way beyond Goethe to realize abstract properties as a Burnside rig of something you need to create a new category. Yeah, yeah, exactly. It certainly sounds funny. Anyway, this is the basic thing about Schandl, but it's turned out to be a conjecture of Grotendieck. Well, a motivic version of a conjecture of Grotendieck, which is quite fascinating. I think I alluded to it on our first day, but Grotendieck made a conjecture of a comparison of two cohomologies. Just the singular and this algebraic around about the, and of course these are for varieties defined, say, subfields of C, these are, these cohomologies, I mean, it's a comparison that between them. It certainly drew attention to the issue of what you could say about the matrices, which are absolutely mediating this isomorphism of things, both things have linear structure but slightly different kinds of... He drew attention to the fact that the entries in the middles is generally the field which is independent of it. Now, this was done originally from varieties over Q, and this can be generalized in one mode. So things about the level of the elliptic curves, the bilibrations, they're built up from it, they're formed.

2:02:30 And it turns out that there is a conjecture which is standing up, which on the one hand implies Shannon's conjecture, and also implies analogous things for the periods of the virus-stressed elliptic function. Basically, the only transcendence relations are the ones that are staring you in the face from a function equation of an analytic function. So, Shannon somehow, and the number, you get similar inequalities in many, many other situations in algebraic geometry. Now, I guess what Cartier was telling us, he's looking for new proofs of the transcendence of pi. New proofs of the transcendence of pi, which is somewhat more conceptual, but a little bit less computational than the standard ones. The ones we ever know depend on the basis. And there is this astounding paper, it contains problems which I certainly recognise as logical, but probably not what I imagined was logical, so, where they predict, not just the exponential function, they look at the, they talk about various kinds of important numbers, you've got the algebraic numbers, and then the next important class for them are, you take any rules, or maybe rational functions, or even semi-algebraic functions, or Q, and you... Or you take corresponding forms of this kind and you integrate them round semi-algebraic or over semi-algebraic creatures, you get numbers. And these numbers are what they're called. The classical examples are pi and 2 pi i and things of this kind. And this looks like a pretty... The circumference of a circle is a period. The class is closed under some objects, obviously, so it's not evidently closed under any of them, so I think it's completely unclear whether one of them applies.

2:05:00 But they make a number of conjectures, and this came out the other day from Cathy this morning, that he's using only the form, he's hardly using, the way he's doing things, he's hardly using any topological, geometric properties of the reals in what he's doing, except he's using integration by parts, but in the form of it. And considerations are the explicitly, any equality between periods can be obtained by which correspond exactly to a formal experience in integrating. And they give one example, a highly non-traditional example, which shows what's done. They're related to motives and they also show this thing that Cartier was referring to. He referred to Beukers, this Dutch guy who managed to give an intelligible proof of a period done. So this again is at the same level as Shannon. It's something to do with, in the case of these basic numbers that come up, the qualities between them are somehow coming only from things that we've already observed, maybe integration by tolerance and things like that. But is this matter of Taddei, for example, a peer? I think it has a peer, but it's more readily available off... I usually go to the IHES website and just look at the publication and what they say there. But I think it did appear in something quite important, not too long ago, maybe in the first year for some of the important, but it's, I don't think I brought it with me, but it's readily, I mean, it's like a complete mystery of logic, in the sense that we have this concrete model of complex numbers, the language of addition, multiplication, and exponentiation can obviously be interpreted in there. And the obvious relations are, you know, associativity, exponential law, and so on. So it's just a question, have we completely axiomatized this?

2:07:30 In other words, have we added enough relations that the kernel will therefore collapse to nothing? So this would be the kind of proof that Cartier is looking for. Because, on the other hand, there are many of these things that are being treated very computationally. Oh, God, yes, yes, yes. From computation. Yeah. I mean, the proofs, the typical proofs in transcendence are tremendous and meticulous, and they involve, you know, constructing large numbers of polynomials and auxiliary variables and so on. Sometimes they have, and some of it, some of it is very tricky. You have to use, sometimes, fairly serious complex analysis. I mean, these proofs are... Very difficult and almost completely non-memorable for an outsider. And the kind of conceptual things could come from schemes? Yes, yes, yes, yes, yes. I mean, Konsevich and Zagier relate all of this to Motivik, which, I mean, also related. Bursman and Dyer, for example, another one. The Clay Institute. Going very further with a scheme, something like that, and then calculations or... I think they're going further with the schemes in the sense that they're passing to the mode of such a scene. And that involves a more general notion. Ultimately, it's supposed to be about what underlies all the special cohomology. In some way, it's supposed to be the simplest cohomology. It's not literally a cohomology theory. Assuming these conjectures, standard conjectures, you have the category of particular single varieties together with the... And then you basically forget as far as motives. Anytime you have an idempotent map, you pretend it's a projection, a summand, I mean, formally. That's how the categories are claimed formally. You can say what it is, it's just to get it to have all the nice problems.

2:10:00 The motives themselves have cohomology groups. The focus from this to abelian categories are all the cohomology. This is the intention. This is the intention. In other words, just like David Holm told you, in topology, there's no one to play that. In fact, people who have played it all now, certainly contribute a good deal of the formulas and towards this, and it is some, I think, Professor Horwatzky is having to play both games. But are there hybrids now functioning where somebody uses this to organize the opinion of the group and then calculates the weight? Well, I mean, I'm not entirely sure. Daniel Bertrand, who's come once or twice to Cambridge, had new groups. I haven't seen, et cetera. And you would expect that if that's true, then as the ideas develop, there will also be some vibrance. Oh, yeah, absolutely. I understand just from two words of the Cartier that it has to do with the zeta function. Yes, yes, it's true, for sure. The problems of the zeta function are periods. I mean, this is the kind of thing that's cracked up the English and so on. This is spelled out immediately in conservative psyche. And it's also connected very much to polylogarithms. So it's typically the case that you get logarithms coming up, but you don't get exponentials coming up with speed. They then pass to what they call the exponential period, and you would close the periods under... Well, you might integrate things involved in the exponential function, but that's much less understood. That doesn't activate the series issues, but there is, in fact, experience that won't require it. Quite some time indicates that in these issues, in these problems, very often the logarithm is logically prior to the exponential. I mean, in the sense that sometimes there are quantified eliminations which work with the logarithm as primitive but won't work with the exponential. And it's the case also that the logarithms, you get logarithms appearing naturally. They don't know whether E is appearing.

2:12:30 Yes, that's the point. You see, we don't know whether there's some desperately perverse way in which you could go over something utterly weird, you might get E. It's unlikely that you can, and there may be some kind of subterfuge. But I mean, it can be over something that's tremendously complicated. It's a multidimensional thing. So it's probably quite difficult to exclude. Of course they are. But I mean, you know, you have the similar, I mean, this is also the game, I mean, this is always, you've had, they had to make these big choices. This was the key step forward in the elliptic functions. I mean, at the very beginning, this is how they made the progress. I mean, sadly, my own experience in, in, in mathematics has taught me that the logarithm is, although it's got the singularities, you know, all internal singularities, maybe that's precisely why it's, it's more fundamental. We have theories, definitely, that there's a quantifier elimination partially between one rhythm as a primitive, and there is an exponential. There are other situations that are a bit simpler, where there's a quantifier elimination if you take one upon x as primitive and you insist that it's got a singularity at the origin as it has. And that there's no, there is, there are cases like that. But simply adding that and being prepared to take the consequences of having partial functions gives you one of them, and there's definitely not one. So, I mean, it's hardly surprising. It's certainly brought up, I mean, how the project... ...tremendously big unifying principles of organisation in many levels.

2:15:00 Could you say a little bit more about how it connects with the commonality of the term topology? Well, perhaps only in the following sense at the moment. I mean, I don't know the importance of this. I mean, the first indication showed that this all of Tarski's... We showed that this is, that it's model-complete in the sense that there's warning that you only need existential warning. In the end, it's equivalent to proof is not constructive, and then, Willkie and I showed by essential use of Shannon's proof, it's actually decidable. And subsequently, I've shown that the biostress elliptic problem is again true, that the appropriate elementary theory of these things is decidable if this motif is conjectured. And it seems impossible to get that to be so. These guys have simply got hold of some rigidity property of them. Anytime you have such a rigidity and you like them much, this is the connection at the moment. Beyond that I don't know. There are other issues we've touched on, like well-data function and all that, but we know that we have some inkling about it.

2:17:30 I suspect that there will be generalizations of two more general periods. For example, Konsevich and Zaghi are working on a period to get inside the classical moment. They talk briefly. The corresponding notion of periods you would get if you started working with the exponential function as a primitive as well, which you would then get even. I'm not sure if they've made any motivic congenital. I mean, there is an analytic motivic theory as well as an algebraic one. So, you know, it's early days, I think, in this kind of stuff. I mean, it's connected to motivic integration that was discovered relatively recently. The point being that you could typically integrate two people at the rate of something slightly different. The algebraic, geometric things are the usual considerations of products and so on and he managed to be managed by cunning methods to give a definition of integration into these things which you can then specialize, I mean, to explain certain uniform, I mean, it's really very beautiful. What are the problems that CERN was opposed about the rationality of CERN? Deneff first of all should, for a single prime, using the P-addicts, that these, he confirmed the contributions of Deneff by proving these functions are rational functions of p to the minus s, where p is the prime of the question, s in the complex, and then I, one that should prove the weak uniformity in p, that you always get the same shape of, if you're integrating the same shape of function over the same shape of domain, you always get the same shape of rational function of p to the minus s.

2:20:00 The uniformity is obtained by an ad-hoc logic. And finally, the Neffing reserve shows that the uniformity really is much, much different. You're integrating basically one formula over another, a conseilish, motivic integral, which can then be specialised for each p using the considerations of the... So this formal integral can be specialized into, and then you see the uniformity of these things that we have detected is actually coming from one global formal integral. It's pretty fantastic. And the considerations are very much the same. The first time they did it with a category of Maldives and then later they did it with something less. Issues of positivity and so on come in, it's really a daring start. I mean that's going on now, we hear quite a lot about this at the Newton Institute at the moment. Even further, he's working with Kershtan, so Kershtan has come in from outside of Morlocki, because this does promise to make vigorous or instantiated various intuitions of people about what would say the function. It reminds me of something, in a way, much simpler. Oh yes, I wanted to mention that. Which Morlocki made a major point. Maybe the idea of using an equational theory is not... The best way to get at this, but anyway, more or less, the thing is this, that if you consider what I call finite topos, the object sets, the typical example being finite sets.

2:22:30 There is a certain power, finite, finite pre-sheaves on a finite category, very, very combinatorial, common theory. Well, okay, in most of our work on objective memory, we've always been just looking at the rig aspect. Right, yeah. But actually all these examples have exponentiation. That's right. So we get, we get exponential rigs. Right. Obviously. In all the non-trivial cases of that, the exponentiation is defined by naturality, so that it's not defined point-wise. So the question is, does that make any difference to the equation? In other words, it's wide open. In other words, if we look at the algebraic theory of the exponential rig associated to one of these concrete categories, Are these always the same, more or less, or can they be radically different in terms of how the exponentials are related to addition and multiplication? They can. This is also very much like the conceivage. In other words, this is a huge supply of rather immediately accessible models for Charsky's high school algebra. So we thought, we thought immediately we should try one of these models and maybe this will serve the same sort of purpose as Wilkie's. Which is quite different. Wilkie's thing is not at all objective. You can't construe it as being the abstraction of something. No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, Tarski, you're dealing with the functions on the integers, on the positive integers, with plus sometimes, no minus, and then the binary thing, x to the power of y, and we know from high school some equational rules about these things, and Tarski wanted to know if all the identities true for these functions, so we'll keep giving examples of things which are true, but require cancellation to prove.

2:25:00 And so there are non-equational. They're a bit mysterious, and then I proved it much earlier that the set of true equations is recursive, but by embedding into an analytic situation, it doesn't give very much information. And then Ruben Heredia showed them a very difficult set, and it's communication occasionally. He finally got out and worked with Lee Ruble. He wrote a nice thesis on some aspects of this connected to the linear theory. But he constructed, he constructed some finite models by sort of ad hoc methods. There was models where the original axioms would be true, but some of these wimpy things would not be. It might be limiting to understand these examples. They were just far too ad hoc. But maybe one can get a, you know, get a range of these things by using these models. I would certainly like to. I don't really have any idea how to begin, but this is my question. Yeah, yeah, no. We don't normally... Think about possible, you know, any rules other than the ten high school levels about how a function of physics is related. And the other interesting things come when you take equalizers and push-outs, you see. It's not, you know, not recapturable in any obvious way. It's a miracle. You know, location is one thing, but it's an equatable theory. It's difficult to relate to the categories. Most of the way you understand the categories is through more operations than that. But Shandywell did do one thing. They may prove Chandelier's conjecture, in a sense, in other words, he devised a precise analog of all the ingredients of Chandelier's conjecture, which you can interpret in one of these finite totals.

2:27:30 And it was true. Well, this is interesting. This is also... Just a minute. This was his starting point to try to approach this problem from a Doppler's point of view or he started with, let's say, classical... He started with both. He had a Shandorov's conjecture in mind and we were studying, you know, conjecture of some material. But when the conjecture first arose, I mean, that was years ago. Which conjecture? The Shannon-Wall conjecture. Well, the original. Yeah, the original. Oh, that's the 1960s. That was the 1960s. We found any connection at that point with topo-steam. Oh, of course. It's clearly one. It's clearly one. Purely number theory. Purely number theory. I mean, there are many issues here which I think are more than some very hard mathematical problems. And even if you're just taking iterated exponentials and unvariable ones, x to the x to the x squared plus one, blah, blah, blah, plus that and that, it's true that for these functions, in some sense, there are no hidden, there can't be any hidden identities, unless it's a version of shadow. I mean, at infinity, you can prove it by unwitting means. Scholar, consider the relation of dominance of one function over another at infinity. I mean, these functions are the other. But these functions are ultimately monotonic at the end of the day. So you have the notion of classifying them by way of growth. So you have a linear order. You have a linear order. What is more surprising is you have a well order. And this is related to Kruskal's theorem, things that Harvey eventually got very fussed about. But Scholem's conjecture has never been that they are normal. They are. I mean people can check out, they can take a particular function and look at the things below it. If the function is given in a sufficiently elementary way, you can sort of figure out that the order of things below it is more or less what it should be, and the game is approximately, not exactly, take your x and replace it by omega, and it's an awful lot of things.

2:30:00 Actually, it's very interesting because that ordering, well, this comparison of pager functions was Dubois-Raymonds. I don't know whether he remarks anything about well orderings there, but of course he's really interested in getting infinitesimals, but he does remark on the very strange and intricate order properties of this realm. That's right, in order of infinity. And that's also very interesting from the history of math, because he's trying to set up some kind of algebra of these things, and even to integrate the differentials, and he just can't quite get it to work. It doesn't work because Hardy is not in possession of the motion of valuation, which certainly existed of course at that time, but not for too much before, but it certainly would not have penetrated into the English, I think, at that point. And the Germans, the stronger Germans. Yeah, yeah, yeah. Ushak and Ostrowski, for example. But it's also, isn't it, Borbaki might actually be dealing with this matter. They have a section on Hardy fields somewhere, exactly on the orders of infinity of motion. Because the crucial part is to take functions of a real thing and not have them closed under the trenches. And really the abstract algebraic sense was not worked out properly until the 80s. The axioms eventually depended on something like Mockatel's rule where they didn't equate much. This is an interesting line of development. I don't think that Marnell could have done anything in the well-ordered brain because it does depend on slightly non-trivial things. I guess Graham Hickman first discovered some of them and then Kruskal. I think it's very interesting. Well, Dubois would have known, but maybe not at the time he did this. No, well, 1880s, he would have...

2:32:30 He probably would have known because he had these arguments with Cando. Cando didn't like it, of course. I'm not sure. I mean, often mathematicians were much more appreciative of what Cantor had done than vice versa. One other connection here, which I think is surprising, but this is so... It can involve the trade exponentials. Say I take one and define it over the rational. So I start with, I always just use rational operations and the exponentials and create a term. So this will give a function of infinity. So when I write down two terms, and I want to, and I know because of Harvey and Guarana work, that one of these functions dominates the other infinitely. How do I decide it? The only known method is... I mean, this is rather surprising that something original in transcendental number theory is needed to be studied. How is it that the complex numbers turn around into the real numbers? Yeah, well, yeah. You used it. Yeah, yeah, sure, sure, sure. Is there an easy explanation of how the complex theory can affect the real theory? Yeah, well, it certainly affects it very much. ...complexes and do some quite serious winding number calculations and we don't know any other way to do it and the corresponding thing happens in the proofs I do for the biostress functions. At some point you have to come back here. The other thing, the connection between e and pi, both real numbers, is mediated by the number square root of minus one. I got an Italian student recently to check out, it's not trivial, she just finished it. The only relation between them, which is unconditionally, but assuming Schanuel, the only relation between them that you need to get all the others is Euler's. The only what? The only relation between...

2:35:00 Exponential algebraic relation, the only exponential algebraic relation there is between e and pi is e to the i pi is minus one. Well, don't call your own shadow exists. I mean, I think this is rather beautiful. I mean, I knew it must be true, but I couldn't prove it properly. And I got her to do it. She's done it. So, they tell me, Giuseppina Terzo. Isn't it still open and published? It's available, I think, now. She's coming back to see me in Cambridge in about two weeks' time. I'll get her to send what she's got. I've checked it. It's certainly OK. I think it's beautiful. Sorry, isn't it still open? No, this is an assuming channel. It's an assuming channel. But you've no hope otherwise. For all we know, we don't know that your reply is not rational. We don't know that e over pi is not rational. What about pi? Isn't the other one that says pi to the e? Is it pi to the e? e to the pi is transcendental. Neskalenko proved that, I think, that pi and e to the pi are algebraically independent, was it? Probably that. But pi to the e, no, but you can prove that pi to the e is transcendental using Shannon. It's not so easy because you end up doing things there like i to the power i and stuff. It's a bit tricky. I did that in my Shannon machine too. That was the most difficult. I want to ask you something about this. I once heard Barry Mazur speaking about the role of conjecture in modern mathematics. And he was speaking specifically about number theory. And he said the following. He said, it was before Andrew White's proof of truth. We are now in a position that didn't exist before, and I agree that it didn't exist before. I don't know how much it really holds today. We are willing to take in some conjectures. He calls them architectural conjectures. He was thinking especially of the Taniyama Shimbura, but also based on conjectures. And also the Riemann conjecture. We take them, we assume them. So, I mean, we know it has not been proven, but we are willing to build a very great deal of mathematics that is based on this. This is what you are saying about Chanueli.

2:37:30 Yes, of course. I mean, Chanueli saw that there is some very simple, I mean, it may turn out to be false, although I can hardly imagine how one would ever prove it false. I mean, that's something we might discuss also. But he just told us some... Fundamental architectural principle here. I mean, this one little inequality that's written down there just explains everything that we could ever want to know about E and Y, for example. I mean, whether it's true or not, it's a fantastic thing, because it's not artificial. You know, it's not artificial. In other words, there's an object... I mean, Maser is confusing the objective and the subject a bit, right? There's always been a logic, the deductions there. To prove A implies B, you assume A, right? Well that's all it's got, but objectively one is proving that one kernel is contained in another, isn't it? Yeah, yeah, yeah. No, but look, at the lowest level, a person wants to build a mathematical career, okay, this student. She's willing to put a lot of effort into a proof. ...of something that perhaps is wrong. I mean, but people will accept it as good mathematical work anyway. I believe so. This is, I mean, for many reasons apart from everything else, I mean, the Euler thing is widely regarded as, you know, one of the most beautiful equations in mathematics. If this is the only thing that one needs to know about everything else but quantum mechanics, one can deduce that this is certainly a major piece of mathematics. I mean, of course it can be false. I mean, on the other hand, I often wonder, I often wondered about, if I know, I mean, there's some sort of fucking, we know that it must be terribly difficult to prove, shall we ask. What would constitute a proof that it's false? I keep asking myself, what if it's an example? Well, but you see, it's not so easy. I mean, you have to find... You'll have to find two different ways of representing these numbers. Is there any case known, that's really my question, is there any case known in mathematics where, well this is far circular, but people somehow entitled to an opinion, people with extensive experience in the area, had suggested that certain numbers were algebraic independent? I don't think there's ever been a case.

2:40:00 Why don't you ask him the more general question, whether people entitled to believe had a strong important conjecture that proved to be false? So, I think, yes, sure, I understand, yeah, I mean, it's just that in the case of this one seems to be a bit more, it's constrained by, it's wholly related to, I mean, that's about the only way we can ever imagine it. If these numbers are periods, for example, and that, you know, you do, you find out that there is some hidden geometrical connection that will reveal that two integrals are dependent, I mean, one can vaguely imagine that happening, but... I don't think it ever has happened. But I'll give you an example. For example, a human believed that non-regular primes would be something very rare. Yes, yes, that's true. Of course, he didn't believe they were infinite of them. So, of course, it didn't change much because, you know, whatever is proved for regular primes remains so and people keep looking for non-regular primes. So, this was an expert who failed in the... He didn't put it as a conjecture, and to a certain extent, people took it as, yeah, well, the proofs of the Fermat theory will follow this, because it will be very easy just to take out the non-regular case. So it turned out to be different, so... Yeah, no, I mean, it's tricky, but I mean, I did somehow deliberately want to use this here, but people who have... Right. I mean, Comer at that time perhaps didn't really have a right to an opinion. He had discovered things, but there was no... But better to say, he didn't offer it as an opinion. He offered it as a hope or something. Yeah. I mean...