Morning Discussions, incl. FW Lawvere, Colin McLarty, Leo Corry, Angus MacIntyre, John L Bell

0:00 Okay, and it is the 16th of June, a week of fencing our discussions in what will be the last full day and going into the final furlong. I think it would be helpful, since the time is now short, if we tried to stay within, as tight as possible, within a prescribed... 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 oppositions that drive the conceptual organisation. We should just look a little bit more closely at one or two loose ends left over from the And that might naturally lead us on to a discussion of this extraordinary memoir, as indeed left with Jack Duskin in 1972, which I know Anders would like to hear a bit about, and that in turn would lead into the broader philosophical topic, which might naturally lead on to the kind of thing John would like us to be discussing, 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 of, I think, We could hear more about as to how logical structures do in fact fit within this overarching view of geometrical structure as the right framework in which to achieve a unifying conceptual organization of mathematics as a whole, particularly the way that the principles of principle of choice gets expressed in terms of a condition and so on, projective and injective objects and categories.

2:30 The kind of limits and co-limits that are involved in understanding those, and indeed the way that choice and extensionality principles reflect the way that, and in varying different versions, reflect the way that constancy, constancy of the objects in the topos, and what's involved in both the variation of that. I know those are big subjects, but they lead from the specific and the technical through the more conceptually, well, the more issues which are directed towards the overall conceptual organisation of logic and geometry and into all the philosophical themes, so I think they might be a good way to take the first half of today's proceedings. Now, do you want to go ahead and say a little bit about Croton-Deacon's schemes? A little bit about all that. Okay. So first of all, the TOAST and TOFO series is not based on logic. It's a form of algebra and a form of logic flows out of it so that its axioms are that you have the function space or map space construction. Right adjoint to Cartesian product, and then you have the power set function, which represents arbitrary monomorphic sub-objects, or equivalently, you could just ask for truth value objects, so formulating it, but in any case, it flows out just of those two axioms, that the sub-objects must form a lattice, and indeed a hiding lattice, with an implication operation.

5:00 And in a natural way, that is, that those operations are preserved by arbitrary substitution, otherwise known as pullback or composition, and also the quantifier. 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 lead one to. But then, on the other hand, there's another 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 preserved in these operations. Rather, they reserve them up to uniquely defined comparison maps. As I said, this was a phenomenon which I think was perhaps first formalized by Leon Henkin in his proof of the so-called computing hysterics of higher-order logic. There are so-called logical morphisms of toposism as well, and Peter Fry is perhaps the leading expert on those. You know, it's said that Virgil and I wouldn't believe that the continuum is out of two, but that sort of condition is easily expressed, I'm not saying, but there is such a thing as the free topos having an object, the 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 happening that... That the set pairs would rather squeeze, squeeze way down to constancy in order to get a sort of non-trivial result, but at least that.

7:30 But anyway, that has to do with logical morphisms, and these only arise in geometry, I guess, in sort of two ways. The one thing, if you're just looking at representations of groupoids and sets, then all the geometric morphisms automatically have this property that... That's the higher order structure, i.e. the function space and the powers that are in order. But that's just for group voice. The other one is a local homeomorphism, of course, is a very special kind of continuous map. And it did essentially describe, I think, by the fact that the inverse image bunker strictly preserves this logic once again. This is a very special case. Between local homomorphism and general continuous math, of course, there is the open math. And again, this has a special relation with logic because it means that the logical fugation, that is to say, universal quantification and implication, are preserved, even though the higher types are not preserved. But the portion of the logic that is preserved by the general geometric morphisms is... In some sense, already visible in the very notion of a topological space, the arbitrary units, finite intersections, well, if you move outside the lattice of sub-objects and look at the Cartesian products and so forth, what that means is essentially a very special kind of formalized logic. 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. This, I think, is in a way misleading because precisely we are not, you know, the universal quantifier and implication are precisely 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 the fundamental idea that, you know, you can deduce, if you can deduce truth, you know, then you've proved something, but that's a special case of, of, you know, arrows, the truth, truth arrowing something, of course, proves that something, but the whole, so instead of arrows, we use...

10:00 There are various kinds of arrows, but these are, what do 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 morphism. It's called by meaning positive because it has this positive character. It's called coherent, completely erroneously in some sense, because coherent had to do with finiteness. Yeah. And then I was generalized to alpha coherent and then that's the point. But the coherent came from finance, which originally came from car jobs in a union situation or something. An important name is dynamic logic, and we only run across this. I only came across it recently, but it's our friends down the road here in Wren who like this. I mean, it's nice, right? Thank you for your attention. All of these things are quite positive in that they involve existential quantification, which only makes a difference if it's on the right. If it's on the left, it can be eliminated. This could easily... Yeah, there's an actual... 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, you know, sound, beat, and so on. They're working in precise systems of this size, of this type. It doesn't seem to have been done for that. There's not, in other words, there is no implication as an operation, although, you know, you can assert an entailment, you know, this is, in other words, this, what you studied logic, this may have looked like some strange sort of fiddling, but in fact it comes out in an objective way in this context between entailment and implication as an operation, a binary operation on formulas as opposed to...

12:30 Entailment, which you only use entailment when you actually assert it, basically. And its meaning, I mean, its geometric meaning is clear. I mean, one sub-object is included in the other. The goal of the formulas is to define the sub-objects. There are lots of logical systems without implication. I mean, you know, you have to add additional conditions in order to introduce it. So it's very natural to set it up at some point. But they do have this binary... Of course, of course. It's a deductive system. Oh, I know. Yeah, it's good. Okay, so notice that the and and or, true and false, because false is a constant, so a lot of, if you actually want to assert a negative statement, you can do it, you just, it's just that you can't have, not a pre-floating thing, if you see what I mean, so. You can't consider a negative statement, but you can assert one. You can't put it in an andesite, but you can assert it. Right, so for example, the idea of a field, I mean there are at least three different ways to formulate a field, but one of them which is... Paradoxically, it's 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 up as general as the fibers or something like that. Because it just says that something entails false. So basically what you can't do is to have a formula for all and apply it to the right-hand side. But you do have the false in 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 the 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, axioms, in that context, you can just introduce new primitives.

15:00 Whenever the axioms that you want to state call for negation, but then these negations are stated in a Boolean way, the union is one and the intersection is zero, so that's additional axioms. So any classical theory is basically encoded, but not every intuitionistic theory. Every topos over a given base u can be interpreted as the classifying topos for some theory of some kind of structure described in that language. So that the classifying topos, for example, if we consider the pre-sheaves on the category of finite partially ordered sets, then that's the classifying topos for distributed lattices. In the sense that if you take any tokos at all over the same base and any distributed lattice object in it, then there's a unique geometric morphism whose inverse image, such as the inverse image of the generic versus generic element of that pre-sheathed tokos, is two to the power of blank, or two to the power of blank, in order of preserving math from the variable when I post that into two. And then in that way you see that the various stronger theories in this positive logic correspond to the sub-tokos associated with it. The simplicial sets, obviously, are sub-tokos because linearly ordered sets are special posets, classifies, it turns out, the totally ordered, or in general, totally ordered objects with distinct endpoints and so on, and they're arbitrary. Or, if you consider just the discrete postets, then there's the subcategory, which is the subtoposis, which is the self-depreciation on finite sets, and that classifies the Fungian algebras in any topo, so in that way the classes of models turn into subtoposis.

17:30 So I've described these structures in a semi-traditional way and logic, namely accepting these various changes, the binary relation of deduction instead of merely a class of true statements, only the positive operators, and so forth, but Rotenbeek 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 principal classifiable. You might have to check something. 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 what I was looking up to because Mike mentioned this amazing sheet of paper in Jack Deskett's file that some day, presumably in 1973... Grotendieck was in Buffalo, and he started off on the theme of the classifying topos, and it was well known and standardly used that the notion of local ring is classifiable by a subtopos of the ring classifier, which was used very extensively in algebraic geometry, as well as the classifying topos for, what, pencelian local rings with separable... Those are long conditions, but there's classified topos to that, complicated though it may sound. And that, again, played a central role as they grow topos and, you know, break down. So he got into this mood of, well, let's see, what kinds of rings are classifiable? And so he wrote in very fine handwriting, you know, a list of maybe a hundred, you see.

20:00 And then he turned it around like this. He thought of some more. Like this. And then he wrote the terms again, you know, like this. A piece of paper which is freezing in Jack Duskin's file at the University of Buffalo. So we have to send it to regular amounts. Yeah, who has the technology for reading these things? Yeah, it's the only copywriter. The palimpsest. So often it's mostly illegible, but I think I've seen the original one. By the way, you know that... It could be dejected, but I don't know. It's like Civil War letters or something. You know, it's an interesting thing because this is a historian of mathematics with reading all the Archimedes palimpsests, you know? And the people who do the technical things are at Buffalo. Yeah, yeah, 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, very nice person. And possibly with very, very sophisticated techniques of reading all kinds of palettes. So you can say... That's terrible. No, I mean, the lack of communication between different parts of the universe is amazing. Another instance is that, you know, I discovered a few years ago that there was a group of graduate students in computer science who were studying topos theory. They had no idea that I was at that university. And as soon as I was, you know... It was only after quite a while that I realized this. It must have been before the internet was quite... But one thing about Google, it has to occur to you to try it. They could easily be reading about topos theory and even reading about Le Verre's role in it without it ever occurring to them. If you look at the individual now, 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 through an attractive sounding lecture on a new version of the data zone.

22:30 In any way, I have to decide. So yeah, that's one thing that I wanted to mention with this sheet of paper. And then the other aspect is that... We were discussing with Pierre Cartier, actually, Pierre Cartier with this article in the book, AMS, about Brodendy, and he mentions somewhere along the way the contributions or logic that we made, I forget exactly what words, but somehow that Brodendy had never dreamed of, which is true enough, but Brodendy told me himself that he was really floored by the fact that there was this... Sub-object classifiers, 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, I told you, in response to Pierre Cartier, I wrote, you know, the slogan, 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? Of course, there are technical reasons. Well, any math can be factored. You have to get mono, and any math has a kernel pair which is in turn, you know, comes with 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 the code limits or limits in it, you don't necessarily reduce everything to the discussion of sub-objects. You know, I can understand some of these examples in the list. Many of them are so technical that you'd have to figure out what he's talking about and never deal with a Japanese.

25:00 I was just about to ask you if Japanese are excellent students. Excellent, yeah, excellent, certainly. Seeming to 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, you just see directly from the definition of the thing that the only things involved are the inverse limits and therefore must be classified. So in that sense, the Japanese has to do just with decent properties of the completion process, the analytic completion process. It might not be classified. I think the strategy around Encelion and separable conditions at all is if you can put this in terms of solutions of some equations, then you can show that this is equivalent to certain kinds of equations having solutions, then it's... We'll say algebra is algebraizable problems and it's exactly that that you don't lose but I mean you can you can better understand this but keep doing it by doing the completion in the sense that you can retain the solubility well I mean right I should have pointed out of course that it's precisely this class of structures which have been shown already in logic and model theory to have a special role in spite of the claim you know Arbitrary automations of quantifiers. In practice, it's these positive morphisms and so forth that are really, like, you know, Kiesler's characterization of properties preserved under homomorph languages, Morley's work on, you know, the compact states of models and stuff, I mean, it's sort of, it's sort of worked out that these positive, oh, do Robinson, what does Robinson call it?

27:30 There's a whole lot of stuff around this issue. If you've got an universal existential prefix, that this is directly preserved under direct limits from the map, from the impairs of astronomy. It comes down to this Robinson test, which is that if you want to detect that there are twice as many models of physics, all of us are elementary. But the memory, for I would be only interested, will suffice to show that this thing is not much of an existential problem, and that's enormous, the power of it, and it's always seemed to me that that should be done in the world, that one should look at it, and without going through all this ghastly stuff about the organization, I wonder if that's the way it's usually done, and much of the theory can be based on that. Much of the useful models can be based on these concepts, like the Karski limit, the enthrallment of the mass, and then how you can detect at all maps, and so on and so forth. Right. In other words, there are reasons coming both from geometry and new practice and model theory to be used to see that, in a certain sense, this is really the central... The concept of first-order logic. And then the other things are, you know, the strengthenings in different directions, but this is what's sort of the variant. I think even quantifier eliminations, really. I've often thought that, well, I mean, for example, the real closed fields, 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 the only quantifier around is there exists, it's just one hair's step beyond the equation. In fact, it's the problem in the vast majority of the theories where it has been useful to make it possible. I mean, you're just a hair's breadth away from being an equation. You'll typically have something that you've got. The important basic sets are just projections of equations and a treatment from one other dimension, and that's modality, but that's how it was. I mean, apropos of this, I mean, this is a slight thing, but it's not to say, for the record, I mean, the Grudenbeck technique generally, say, in cohomology...

30:00 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 1950s 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 this one would be positive in the basic plan. It's not quite a universal phenomenon, but in practically all the cases that they have it built and so on, you know, it's very interesting when you understand something you have a thing for. I mean, it's much crueler than one of these, but it's almost the same thing. Yeah, in other words, it seems to me, I don't know whether this would be a correct description in the general case of quantified elimination, but what you certainly can do in certain cases is... What it really means is that you can define universal quantification in terms of an existential quantification, 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, you have an effective definition of logical negation right there. And so you can define higher or higher alternations of quantifiers just in terms of this positive theory. So I think that sort of thing is true of more and more examples, but I don't know if it would really be public. General characterization of what's meant by, not by quantifying, not total elimination, but reduction to the case where you have only the existentials. Again, if you say AE for it, 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 XA as a... I mean, I don't think, I mean, it's the business of if.

32:30 I mean, the physics won't completely only collapse in the theory of elementary and so on. Right. There has to be very positive logic in the fact that there are all of them. So it's sort of the division of labor between the general and the particular logic. Right. In other words, the general logic should be this more general one. And it's particularly on the basis of the mathematical content rather than pure logic that you get out the external expressibility that we're all in. All maps are elementary. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Thank you for your attention. That's what Jerry Sacks does at his school for Maury, he has pronounced that bad on him, too, and it's nice. And when I read McKay and Reyes, 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 actionatize a complement to a given predicate. Yeah. Well, yeah. Yeah, that's a point about it. Some papers might report a little bit more. Oh, yeah, yeah, yeah, but I'm going to take it in a little better even now. That's all that's going to matter, yeah. I do work on these fields, and it's still a nice book, I think. Oh, yeah, yeah. By the way, yeah.

35:00 Right, okay, so does that take care of the logic part? Maybe. I'm trying to follow Mike's outline here. So, yeah. So, yeah, about the points and about the, yeah, okay, Grubnick's, why did Grubnick consist in 1973? All the general stuff about schemes previously done was wrong, and that everyone should immediately adopt his new improved version, which practically none of them did, except me, and I was already converted, because combining what Gabriel had taught me, you know, with my own speculations, I had arrived more or less at the same point of view that he was promoting, of course, I didn't. I didn't have the faith that it would all really work out, and even if it did work out, because if 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, you know, much emboldened to say this, no matter what I'd already hardly thought of because of this lecture, and... Well, I think I had already said this before, and I don't remember if I really explicitly mentioned this in the lecture, but one of the crucial points is that if you consider the idea of scheme as a topological space plus something, chief of ranks or whatever, then you're led to the underlying topological space function. But this underlying topological space function does not preserve Cartesian products, so therefore I remember already in graduate school one professor of algebraic geometry telling a student, you know, I wasn't in that, I didn't know what they were talking about, but he was saying, you've got to realize that an algebraic group is not a topological group.

37:30 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 on which multiplication must be defined and yet that doesn't correspond to the Cartesian product that's basic and so there's no, so the underlying set function is not, I mean the underlying space function is not. Much less the underlying set of, does preserve products. And on the other hand, neither does the, neither does the space set of components function. Components, it's a crucial property for category spaces that the connected components function should preserve Cartesian products. These are two different functions usually to the same category of less cohesive spaces. Everything becomes incredibly complicated. So essentially, and there are many other things, like one slogan that Groben Dieck told me on the side after his lecture there was, don't forget that the points have a lot of morphism. Points in a topological space don't have a lot of morphism. What could that possibly mean? But the points of a scheme should have, namely the Galois groups. Basically, points are associated with fields of definition, and those fields have analog loops over the ground field and so forth. This is the algebraic origin of the remark, 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, Underlined is the Galois topos, rather than abstract sets, rather than topological spaces. It's a topos inside which you could consider topological spaces, and then it would make sense, you see,

40:00 because essentially the prime ideal, this is to get it back to the community of algebra, if you have a community of ring A, then you can consider, for each field K, you consider the homomorphisms from A to K. So those are the K-valued points. Of course, those homomorphisms may not be surjective, so the kernel is not a maximal ideal, it's a prime ideal. Now, if you consider any field extension, then of course a point of fact A in one will be turned into a point of fact A in the other one. And so you have a, for a given ring, you have this diagram. There are a number of sets parameterized by the category of field extensions. All field extensions, all homomorphisms of fields are actually monomorphisms of course. So for each of these you have the prime ideals. But 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. That's where the unnecessary discontinuity is introduced. So the more geometrically conceptual... Yes, indeed, we have these slightly more general points. Both the set of components and the points in the space are really objects in this category of all functions on field extensions to sets, rather than just sets. Actually, they are always achieved with respect to a very strong topology in which every map is a covering.

42:30 Logically speaking, the meaning of existential quantifiers and disjunctions is modulo-arbitrary field extensions, so they're sheaves with respect to that. Now, if you want, you can consider topological spaces that you can associate to a scheme or a topological space object in this topos. It's a Boolean atomic topos. We call it the bar topology because Barr pointed out this fact that if you have a category that consists entirely of epimorphisms and has a certain amount of information property, then you can just take every map in sight as a covering and the topos and sheaves that you get will always be atomic boolean, that is, the lattice of subalphaic could be atomic boolean algebra. 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, 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. Because the key thing is the action of the Galois groups, 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, you know, the thing to do is to learn to link in that topos. Yes, but the Axiom of Choice doesn't hold. So it's really something like a symmetric model. Yeah, yeah, yeah, because it's, you know, the Yawa group is... And some guys is the main thing, but I'm saying the most natural guys is not a group, it's this little category of, you know, all possible finite field extents in this.

45:00 Well, this is, again, this is very interesting, because with the way that Dr. Jason and I and the channel have a lot in common with respect to my community, the convention of work, it's been a false, you know, principle in axiom class and the interest in classes. Really, and if you don't know the experts in advance, you will detect them by reflecting on the nature of the finite extensions and what the Galois groups look like. But one thing you cannot do, if you first started watching, nothing would be understood, would be to attempt to go all the way to the core and try to quantify over its elements. I mean, it's far too complicated, you can't code it right. But each element, each of these finite extensions you're talking about is basically... Coded back in the films, who point also much of the class field to you, you can detect any little bit you want, if you get all the group back in that field, you certainly can detect the whole thing, I mean, the whole thing is a convenient reason for studying these things, because sometimes you can do, I don't know, like analysis or something like that, and some of it doesn't point to it, but it seems to correspond perfectly with the needs of all of you. You know what I mean? In other words, somehow, I think, just speaking sort of psychologically, there's the idea that, well, this category of all-finance field extensions is impossibly complicated, so we can't contemplate that. Well, it is complicated. But then to go and, you know, think, oh, if we have just one group, then we've simplified it. But in fact, we've complicated it. We've stressed this, and you'd better keep this. The system in front of you, and just do a little bit softer quantification, that way you'll get decent information. The moment you actually do a completion and try to do logic in there, forget it. This one topos, 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 Shadywell topos, which is a much simpler thing. The classifying topos for infinite decidable objects, which is basically the category of coherent functions, the sets from the category of finite sets and monomorphisms,

47:30 equipped with the topology suggested by Myhill's observation about finite intersections. Uh-huh, yes, that's right. Because what happened was that when Chachanuel heard from Maillot about certain constructions, I think involving isols, who had certain kinds of constructions, he noted that this condition, Maillot already had this condition, that a functor from finite sensing monomorphisms to sess could preserve intersections. But this actually had all the properties of being a sheath with respect to a certain notion of cover. And then we then might have had this unique expansion. Objects are uniquely expressed as coproducts in certain ways. It's a very concrete sort of expansion, I mean, isol-like. You know, just a few days before that, I had been telling Shannon about Shabar'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 Gangwell topos, which has this very, if you think about this category of finite sets and bottom morphisms and the role that it's playing and why intersection preserving is equivalent to the sheet condition and so forth. Just a little bit. Then it becomes very intuitive. What happens then is that the machos who do all these pro-finite things, they describe this classifier for the decidable class system as the continuous actions of an infinite group immediately obscuring its elementary character. It's the same totalism. But they're incredibly mysterious. It's very elementary and intuitive character. It plays a key role in its independence results. There are various interpretations of it.

50:00 So yeah, it's only, you know, actually it's taken me years to... Convince people, again, what it was, again, it's still the same book, but they always describe it in terms of pro-finite. It has this elementary meaning, intersection preserving, the operation of intersection on finite sets. There's a subtlety, you see, that it may not be the pullback in the category. Get rid of the sums of products, but it happens when you restrict the maps in the category that already has the co-products of products. You'll still have a functor corresponding to addition and multiplication, but it may no longer have the universal property of it. Because, you know, universal property requires that you get a unique map every time you have a test situation. That unique map may not preserve the property it needs, say, a different one. 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 house sets are finite. So that implies, of course, that the endomorphisms, all the epis of an endo-epis of something funny that can necessarily be invertible. So the fact that there are all these Galois groups just follows from that extra axiom on the syllabus, I mean, coming from this general vantage point. There are a number of atomic toposes generated by finite objects. You still don't have the key features of galwalking. You have to notice that there are properties of these toposes. Basically, you have these groups.

52:30 The endomorphisms of any object are invariable. But if you have two maps, you can transport one to the other. This idea of having that kind of an atomic topos rather than abstract sets as a base for algebraic geometry, in principle, Roderick already in 1960 had proposed a similar thing. For analytic geometry, that is, for the study of holomorphic functions in 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 Grotto post. He doesn't call it that at that time, for the analytic situation. And again, these, in both of these cases, the, the, uh, the appropriate, the appropriate base, I'm struggling to figure out what is the appropriate base, appropriate point and component functions do preserve finite products. Something which is not true for most hippos and for shoes on topological space will essentially never satisfy, uh, these type, that type of condition, so that, so the fact that you have qualitatively different type of... When you look at a model of all spaces of a certain yoke, Max Talley once tried to teach me that you don't say the same yoke, but the yoke is already the same yoke.

55:00 Is that right? Yes, yes. You could pass the equivalence class that is the yoke, but it's not normally done. It's subtle. What do you think is the translation of ilk? Well, to be of that ilk is to share the relevant property. So to be of the same ilk would be to share the same relevant property, I suppose. Which isn't what one is trying to say. The two comparisons would be of the same ilk. This would be in respect of the same property as this one. But yes, this thing is of the ilk of that one, because it shares a property. Yeah, not that it's of the same ilk. So this led to some of my faiths in computer science and so on, qualitative distinctions. Always having this qualitative distinction between the two talks. I think about a drug deep in 1960 in the analytic case and in 73 in Butler in the algebraic case, although, of course, the 73 talk was, as I say, quite consistent with what I learned from Gabriel in 1965, really, as it appears in Demetre Gabriel's book on algebraic groups, at least they start with that, with a certain point of view, basically that... Basically that the general spaces are just covariance functions from the category of algebras of the type that you want to intersect. So that you have figures of the type, circular figures or cubical, all 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.

57:30 So you may as well just say so right up front, reduce the set value puncture, from which the figures and incidence relations can be calculated to enhance and fixable all properties. And this pulls us closer 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 Ariella Frankl said, or that one. They could be dedicating cuts or equivalents. 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-valued functors? Completely different kinds of things. And Groteek wants to say, at least. That it doesn't matter which they are, and is favoring the set-value-funkture approach. Because it's from that that the certain properties like those I mentioned in the other 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 the multiple reduction in ZF, or trying to get Deligny to talk about it, or Deligny or Twain to talk about it. Because he doesn't do that kind of thing. He doesn't do ontology. So, I mean, here's this very 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 Grotendieck stuff, 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 frame. We don't say these are this kind of frame. We just know there are as many of them as there are of, say, this kind of frame. But, I mean, the picture, the geometric picture that the functor directly describes, the geometric picture is there's this ensemble of figures. And then the incidence relations, you know, I think that's the basic definition of geometry, you know, and its vision in general sense to include this figure and the incidence relations. So you even get a picture, and sometimes a very low-dimensional picture of curves and surfaces, sometimes a very geometrical picture of what doesn't have an ontology at all. This isn't a picture of something.

1:00:00 This is a picture of the relations that hold them in this thing that we haven't even said there is. We just know what relations it has. Well, I should have pointed out, there's a third picture, which is the discrete vibration. I mean, the susceptibility function is equivalent to the discrete vibration, but what does that mean? It means that this, the generic... Big figure shape, figure types and the generic incidence relation, they form a small category and the typical space is really just a category, again a category, but fibered over that. So in other words, you have the... So if you take 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 fibration is. It's that labeling outcome. 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, but this one, and you bundle it into a... So instead of looking at it one way down here, you look at it a sub-fibered over this. Yeah, so it's fibered over the, you know, there's a lot of, you know, situations where people talk about labeling, label 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, with many different particular ways, over a fixed, you know, there's a fixed category 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, each label of them, and the fact that you have this forgiveness labeling functor that has the property of being a discrete vibration is completely equivalent to having a set value functor, and sometimes that's a more appropriate picture. There are a lot of situations which I normally thought of as geometry, where you have this labeling process.

1:02:30 I mean, there are situations where the arrows are thought of as processes, and processes can be labeled by, for example, how long did it take, the duration of the process, is again a labeling sort of functor like that. And this gets to the principle that you talked about in Florence, and you mentioned there also that, and it's already, it's sort of mellow, 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't. We don't want to say, oh, that's too big, that's not a set, that's a proper class. But they know perfectly well I'm talking about proper classes, so they're there. Then we know I'm talking about classes of them. Well, there's a simple matter of model theoretics. If you have a consistent first-order theory of these things, then there'll be models. Models will have power sets, put up, you know, literally, and the fine point is, well, there are several different ways that you could interpret what means definable and power set and all this. So there's not a unique model of whatever. But certainly the consistency of the higher-order thing where you've collected these things itself follows from just a low-level set theory. I guess that's more or less compromise. Well, it's a kind of a reflection principle. I mean, it doesn't follow from ZF that there are models of ZF. But people who use ZF take it that there are models. And so since there are those models, there really are... Yeah, if you sort of take the fact that consistent theories have models as a matter principle, which is what the reflection principles are. And you take it that you know that the theories you use are consistent. We know CF is consistent. Well, we don't quite, but we're pretty happy to take it that way. And if it's not consistent, something like it is that we could have been using. Yeah. Either that or, unless his name is right, that already the objectifying the natural wonders was inconsistent.

1:05:00 No, sir. Anything else? No, no, no, long before that. A lot of people have asked, yeah, but... The ones who claim to be truly inconsistent are half respectable figures. Oh, yeah, that's right. You mean Venta? Yeah, Venta. Was that the International Logic Review? I'm not sure he's half-respectable now, but the International Logic Review gave an interesting proof that pan-arithmetic is inconsistent, in 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 pan-arithmetic you couldn't be able to get an infinite descending chain. I mean, if it asked him to change, yeah, if he kept saying in my next article, I'll prove it, it'd 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, it began out of first, you know, without declaring a contradiction, he never declared that this was an infinite chain, but in practice, it was an infinite chain. I have the honor of being the one who actually introduced Vento to us and we loved him. I never met him previously, Matt. I knew each of them slightly. But John Mayberry, he was really persuasive on how we have not tested the limits of these induction principles, the replacement scheme, induction power, not only have we not tested a statistically significant fraction, I mean, 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. There's some. So even if you're willing to think about it, you don't give up on the project, is there? Who doesn't think so? I mean, who would refuse to think about it?

1:07:30 The only category theorists would somehow take this urinal-burn-eyes principle. It doesn't exist unless it's a member of something else. It's a dogma, whereas Goethe and Bernays themselves, I don't know if you were here when we mentioned this, in the collected works of Goethe, there's correspondence between Goethe and Bernays, and they're discussing two things, I think. One is the issues raised by MacLean at the 59 Warsaw conference on entheotistic methods. And also the rumor that somebody is working on a categorical sect here. This happened to be the same month that I was writing my thesis in California, and later things got surmised at Gödel that Kreisel must have was on the phone with Gödel every day and have told him about that or not. But anyway, in their correspondence they were saying, oh, well, yes, of course, we have to take finitely the types above B, take those types to be just as real as B itself. So the very ones who are taken as the final authorities immediately wanted to consider something more reasonable. It's such that, you know, at least critically speaking, their original work, which is related to that of von Neumann also, of course, it was a kind of a trick to express everything they wanted to express at the time in a kind of, in a compact way. Yeah, that's right. And it's good to know what trick works for what problem. Of course, of course. And part of it is hidden, and Baranais was telling me, and I think it's published as well, I didn't know that, but he told me that originally they had two symbols, eta and epsilon, so one stood for the membership between two sets, you see, in the classes, and the other one was the satisfaction relation, essentially, and they saw that within this limited realm they could just

1:10:00 The theory of properties, you know, grafted onto sets. The Ida was really the ascription of a property as far as R&D is concerned. Exactly, right, right, right. But you see then, if you get, which of course is part of the legacy of Frege, I don't know what I'm going to do. You could then imagine, well, you could objectify the idea of property, and that maybe you thought it was a formula, but yes, you want to quantify over these things, so you also objectify that, and you get something like classes, and then you compare them with sets, and so forth, so by making one particular choice about how all that was supposed to fit together, we've got a completely satisfactory system for talking about real numbers and the power standard of reals and so on, 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, which there's no reason you shouldn't talk about this thing. 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 sort of studies, Sir Bela Frankl said, without classes, that's normally used because the introduction to classes sort of... It presents precisely this difficulty. Well, if we could have classes, why can't we have classes? And so on. And this is the Zermelo-Fraenkel case. You know, you 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 that it doesn't have to be mentioned. Well, that's the case of Gertrude Bernays. You know, V is there. It's funny, because that's very convenient for certain purposes. I often present set theory, it was just really coming from a presentation of Dana Scott's long ago, where you have objects, you have objects and classes, and any sort of collection, any property of objects determines a class, and then a set is an object, is a class, which is also an object, and then you throw away, I mean, you know, in order to develop set theory.

1:12:30 Without, well yeah, you throw away, yeah, and then you, but this is very convenient, it's essentially, it's good, it's really a Gemello-Fraenkel set theory with terms added, if you like, to correspond to arbitrary predicates. Well, that's how Quine presented it. How does that relate to Skock's idea? It's similar, in terms of data. But this is a very sort of natural way of starting off objects and classes. Sets and classes is a curious distinction to make. It looks odd. Objects and classes, on the other hand, is quite a natural distinction. Once again, curl lining words. Yeah, exactly. It's interesting when we talk about Macaulay and Reyes, we talk about theories as categories. So in other words, we consider that the usual formalism of... First order logic, order of positives, fragment of that, and all these various things. There are 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. I don't know if I'm Scottish or not, but I like that anyway. My Sanchez series is merchant, but they do. It's all like the grass, you know. Yeah, so if you, 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 if there's a single source of theory, then this category has a preferred object, called a d, 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.

1:15:00 Satisfaction is just composition. You get a map from 1 to 1 plus 1. That's the truth value of a statement that in any given predicate is satisfied by a given list of constants. And you can also see there why existence doesn't mean existence because... 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, it's only picked up when you consider models, i.e. functions, preserving the stuff into some category where epimorphisms are split, or at least as simple epimorphisms are split, and it turns into a category where epimorphisms are surjective. But anyway, having the object V as such is totally innocuous because 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 songs and products and pullbacks and images. Images are just things of sensual complication, but, well, it could be a Cartesian closed category. So there you have, you know, B to the power of B and 2 to the power of B and all that stuff without any as of. Then, of course, there's clearly a very nice You see, not so ambiguous as, say, take a model and take its power set, but as I point out in my paper on guiding arguments in Cartesian closed categories, the UNAIDA 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 UNAIDA embedding, then you guide it into a Cartesian closed category. Now, again, you have to worry about what happened to the various existential statements and so forth, you have to be able to take canonically the topology on the theory itself and, you know, immediately all sorts of difficult or very difficult seemingly questions arise, but the fact that it's perfectly consistent to have a Cartesian closed category which even fully contains, you see, I mean...

1:17:30 You might think that by introducing all that stuff, I can define more things at the lower level, but no, it's not true for maths, it might be true for sub-objects, but you can define more sub-objects, but not more maths. And in the Yameda model, I hope that that paper is going to be printed soon as well. Yeah, which one? It's kind of a hybrid because, well, I gave a talk at the 1967 Los Angeles meeting on set theory, basically, a huge meeting with Yishun for a minute, I met a lot of them. But then I divided up the material that I presented there into three different ones. One was called Hyperdoctrine, one was called Atonism 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 the Tarsian definition of truth are really all literally the same. There's a large part which is about Cartesian closed categories where I point out this. Obvious and easy fact about the United embedding that not only does a pre-sheathed category always have an intrinsic function space, it's even preserved by the United embedding in case you had one before in the small category.

1:20:00 So that in some sense... It is the natural, because these are natural transformations, but it is the natural notion of higher types in all sorts of concrete cases, so like, for example, the C-infinity case, it's the natural definition of morphism, of functional, smooth functions, analytic case. Analytic case was considered by Volterra and his students. From Tapier, Max Zorn, and many other people to be the natural definition of function as for in the analytic context to the extent that they needed topology, quote unquote, to derive it from this rather than imposing it on it. And I also conjecture there that recursive function theory could be the approach in this way. The thesis, of course, in which you discovered the significant third way of it was earlier used, I don't get it, to Erschoff. And in fact, earlier, Bonnach and Mazur used this approach to defining recursive consciousness. So the Bonnach, Mazur, Erschoff point of view crystallized by Mowry is literally the case of this remark. I think I put it as a rhetorical question. Maybe workers in recursive function theory or smooth analysis would like to consider the implications of this. That's where Phil Homer got the idea. So anyway, that article... It was simply not published at the time. Excuse me? It wasn't published for what reason? Oh, no, it was published. I'm saying reprinted. To make it available. Reprinted. You know, do you know this electronic journal, Theory and Application of Categories? No. Is it just electronic? It's just called TAC.

1:22:30 No, they're not updated. It's at Mount Ellison University in Canada. So, mta.ca. It's been published since 1995. I believe it's 10 years, or 10 years, I believe so, you know, during the journal, refereed journal, but they have a special column of reprints, so they're making available all kinds of basic old references that are hard to get, like Peter Fry's book on the beauty of categories, and they're just, they're just set up today. Thank you for your attention. Well, it was Springer, I think. Yeah. I think there have become various, various publications. It's strange that they make it available. Barry Mitchell's book was, was, was an academic book. Is that coming to you? There's a book by... We may wear all these green boots on the top of... Yeah, it's, it's like... No, there was a book by Barry Mitchell. But that, no, I'm, I'm... They're the academic press. There was a book by Mitchell, wasn't it? There's a book by Mitchell once more. By her. The first, the very first... Oh, yeah, we know. No, I thought it was Springer. Yeah, yeah, that is Springer. Okay, so it's good to know that they are... The very first reprint was my... The one paper in Metric Spaces has these categories, wherein it's contained this observation that... The quotient completion of 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.

1:25:00 The same sorts of reasons that continuous functions define the rational suit. I didn't quite get a rational value. You have to extend to the reals. It's why projective modules come up all the time in ring theory. It's the category of projected models which is really the invariant to motion. So that was the first part, and then there's Peter Fry's book. Oh, there's about ten reprints in that same case, not one. Peter Fry's book. And my picture on set theory, which Colin helped a lot to prepare. I mean, you can type it in both sides. Which paper, which, uh, which one was that? It's the, um, the, the, the 1964, um, it's the... Oh, the percentage? Oh, that one. No, not that one. That one's just a five-page summary. Ah, right, right. Of a, what, forty or fifty-page typewritten version with all, 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. Colin thought that it should be reprinted with commentary. That's one of the nice things about the reprints. Because in most cases, the authors have supplied one or two or three pages of what happened in the intervening 30 years. 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. I am seeing the move from, let's say, late 19th century algebra. As represented by a book by Weber, what's the name of the book? To Van der Werden, there is a change I already explained, but how the entire mathematical community look at this, at the meaning and the location of this work, so it's very interesting to see there, to take the Yarduch, the Foshrit of the Mathematik, which is the review, you remember the talk I gave at Berkeley.

1:27:30 How the classification schemes... What year was it? I gave a talk in... two years ago. I can send you the... No, I mean the Yabuch. The Yabuch? Well, I think all the... Oh, I see. All the successes. From, let's say, 1895 to 1930. And I look at how various articles are classified. On rings, but, you know, the concept didn't exist that much at the time of... As you see, it goes on an interesting transformation, and how the article at this point in a certain classification, and then there appears a new classification, and the question is how this classification appears. Before I came here, I was looking, but I didn't really have the time, which is the first article by McLean and Heidelberg is 42, but the real one is a pre-announcement. The first one is 45, so it could be a nice exercise. 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, but there is a section called General Algebraic Systems,

1:30:00 ...that we later see as part of... ...some that we would see as category theory... Exactly. ...were considered homological algebra for a long time. Exactly. So I really wonder about, you know, all these articles of yours in the 60s, and so where do they put it? Okay, it's categorical, but on the other hand, only logicians did not see it at the time. And all views that it does, to what extent it's of a majority of mathematicians, and also how it shapes the... If I am a student for a topic for a dissertation, then I look at the reviews and I don't see any of these words in that category, or I am looking at lots of things that have disappeared. So, are there schemes or do they appear in that category? There was a few years ago an organization of these classes, which 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. Oh, because it's a real problem. What is the discipline? How do you define the borderline? In fact, I rarely use it that way. 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 it.

1:32:30 Because you assume that the classification is not according to the... Yeah, that it's not going to be... 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 for the last 15 years that Knesset hasn't given us any money in category theory generally and in our group in Buffalo in particular. But before that, you know, they gave, it was quite a big grant from the NSF to, you know, Ismael, Shanuel, Dustin, Mabir group. It's very young. We could travel. We didn't need bikes. We could travel. What's his name? Taller. Owen Taller. Oh yeah, yeah. He had been, he had been… Taylor. Taylor. He's still a bogeyman. I think he's got to be… He's elevated. He came down from the heights to see it. It was in November or something. Ah, yeah. For some logic things. Yeah, yeah. He was, he had been a calculus student at Highland Berry. Thank you.