Conversations on Set Theory as Fragment of Algebraic Geometry (contd.)

0:00 The connectivity of the truth value object, the point you made about the connectedness of the sub-object classifier, the distinction between the typical cases in algebraic geometry and the specific cases in set theory, I understood the second too. Thank you very much for your time. All of these factors are actually the equivalent of what they would all be, and so therefore invariantly defined things like truth value would be preserved. So pi of p lower shriek, the component factor, would be preserving omega, but it's totally, you know, not preserving omega. It's killing it off instead. This is the positive way of guaranteeing the total difference. Except in the case where both categories are true. Yes, in the case, as you say, where it's correct. Similar in consistency. Yes, yes, yes. I see that point. There's also a point about, again, facing this whole discussion of the way one should think of set theory as against the background of algebraic geometry or in this framework as a special instance. So, we've taken a lot of these constructions in algebraic geometry from the category theory viewpoint into this broader understanding of the relationship between categories of space and categories of quantity, particularly, obviously, the behavior of covariant, well, covariant and contravariant from respect to intensives and extensives and the... Distribution of product or co-product in the case of categories of space. Spaces as, obviously spaces have many roles. One of their roles is, one of their roles obviously is parameterizers of motion.

2:30 The other role as you say is domains of variation of quantity. Now as domains of variation of quantity, you made a very interesting remark I remember. I think it was back in the Eilenberg Festschrift. or Alex. It's been a guiding insight for me ever since, especially when arguing with benighted logicians, that there are two ways of thinking of the structural domain of variation, the space as the domain of variation, for quantum, and the more general, more general way, definitely more general inclusive way, is in terms of the existence of lattice homomorphism from paths. The first is to think of the space or parts of the domain into parts of the quantity which is varying with respect to the domain. And the second way, which is the typical logician's way, which is to think of the whole domain as consisting of points. Or, let's say, Phrygian objects in the true sense. Whatever we want to call them. Abstract decidables. Anyway, points will do perfectly well. Well, that's spot on, of course. I don't need to be reconvinced of that. But again, is this not a useful way of introducing the game to logicians and to the benighted? This whole issue of how one can see set theory as naturally falling into an algebraic geometric framework, I mean, by focusing on the lattice homomorphism, yes, okay, well, okay, I just wanted to think a bit more about that, with points being the very restrictive case of that lattice homomorphism. It seems to lead to the distributive lattice classifier, rather than abstract sets, to be very important.

5:00 Sorry, say that again? No, no, I have to think about it. Yeah. All of the break-downs are going to see infinity too. Then actually you start with the Drury notion of quantity, and then you construct the tuples of spaces, which is somehow... Act as domains of variation for that, one or two. As at least for the, yeah. And typically you get more general things, like projective spaces coming from the... But yeah, so that's what I was sort of emphasizing here, was that one needs to, one shouldn't start with the most general notion of quantity, as you might imagine, because this extensivity is a small step toward... Geometric property for the opposite of the category of algebra. The opposite of the category of algebra in success is not really geometric unless the category of algebra has a special feature like this extensivity. The first point made in the notes of Grotendieck's 1973 lectures, I don't know if you have any of them. No, no. Alas, I won't. Well, I think this can tell me that in general they are a very poor reflection of Grotendieck's. I'm sure they are. Why do you think some of us believe in recording things, Comrade? And even, yeah, even... If only, if only I could have been around with a tape recorder. Grotendieck was in full flow. Unfortunately, they were a bit more intrusive in those days, but even so, one could try to make. Yes, indeed. What a missed opportunity, however. But continue about the point of Keita's record of this. Anyway, so whether Grotendieck introduced this or Keita, I don't know. He starts off by saying. You know, what is the special about rings that makes it appropriate for the community of rings? And by the way, this doesn't work at all for non-community rings. No, no. There's nothing like this distributivity for non-community rings.

7:30 Well, you made the point yourself that the first example, actually in ring theory, of a ring is in fact an example of a two-ring. Yeah, that's right, that's right. By the way, am I right in thinking that a quantile is a two-wraith? Well, typically they're non-communicative, aren't they? I tend to use the word rig only for the commutative case, because I have this property for the supremum, you know, the finite events, yeah, distributes, yeah, just, yes, but the multiplication is not commutative, no, but yeah, in other words, the commutative content would be the two rigs. The commotor control is central to it right now. The commotor control is the additional feature of that. So the real numbers, you see, they're by metric space, co-domain. It's a closed category, but it's also arrayed, and it's also quantile. So that's why people who like quantiles, that's the way they generalize. So this generalized metric space is the field of control. I have this sort of prejudice, you see, it's an oversimplification, but it is that there do not exist non-community brains. Meaning that, you see, there is one example. This is a very important example, namely the Quaternions of Hamilton, which is a non-community range which exists for its own reason, namely to be the universal cover of the rotation group and three-threes and all that sort of thing. That's to perform all those tricks with bells. So it's a clear example. But any other example, this is just a sort of general one. And of course it even applies to the Quaternions. It's really the endomorphisms in an additive category. Fix an object in an additive category with the endomorphisms in a non-commutative ring. Non-commutative rig, if you want. But you see, so you've got square matrices, but you also have rectangular matrices. But it's a very computational term.

10:00 You have all the rectangular matrices. And this additive category determines the multiplication, of course, of its composition, but it also determines the addition because of the universal property of the product and product agreeing. Yes, as in any category of quantity. As McLean showed, that implies a unique definition of addition as well. So this category is somehow a fuller picture. And then just to get an idea of the number of different lectures. In a typical example, you know, if you say, I mean, imagine, imagine, and then you sort of dialogue, I make this crazy assertion, and someone says, well, what about two by two matrices? And I say, well, what about three by three or two by three? They automatically go together as different objects, just one. So in that sense, you're not likely to find a non-communicative range. It really exists in its own right. Yes. As opposed to the quaternions being one example. It's very interesting that those are entirely, as I say, have an entirely sort of concrete meaning in terms, as you say, of rotations in three spaces. Which doesn't tell us anything mysterious about the non-communicativity of the geometry of the space. No, not at all. But this is what a lot of people have wanted to play. I'm afraid this is a trap into which Basil Hiley has fallen. There's business about shadow manifolds and, yeah. Anyway, so, on the other hand, the commutative rings, or more generally the rigs, they have a somewhat different function because they define... All K-modules is an example of an additive category, so they, they, they're K, they're, well in particular K-algebras, the notion of algebra, the notion of algebra only makes sense for a communicative base, you know, complex, you know, complex, linear algebra over the complex over the real.

12:30 Over any rig, this makes sense, but if the base is non-commutative, then it doesn't really work in theory. In particular, it's the idea of partitions. That's sort of the concrete, you know, internalized expression of the extensivity in the case of k-rigs and related categories. And I really know... See, really the origin of this paper is I'm giving a bunch of examples because I don't know... The general theory, that is to say, you can ask, given an algebraic category, meaning all our product presenting a set value after it on some small category of a product, a multi-sorted theory or whatever, any such category, might be that its opposite is extensive. So for which theories is that the case? You quickly find that there must be two distinguished nullary operations, zero and one, and two things like that. If K is the initial algebra, then K cross K, it duly is 1 plus 1, you see. It's the sum of one point with itself, geometrically translates into K times K. And so the extensivity has a lot to do with K cross K algorithms, if you like. That is to say... There are maps from K to K cross K. But if you want to explain how to implement this by ejection, it becomes sort of short. My friend Aurelio Carboni actually proved the characterization theorem and some other kind of exactness on the theory, which is equivalent to the category of algebra for the theory, for extension.

15:00 And I'm afraid I offended him, because after his beautiful lecture, I said, well, okay, then how can I use this? And he said, well, you didn't say in order to use it, you just wanted a characterization. In other words, the point being that this categorical condition is particularly hard, or at least I find it particularly hard to verify in a particular case. So instead, I'm commonbraving classes of examples in the hopes of understanding. It all has to do with whether or not you can see from one point of view. This has nothing to do with cohesion because the topos itself, topos always have extensive sites, you think of the natural sites, subcategories that generate, well of course they're extensive because even the whole topos is, so as long as they're closed under sums, those sums will behave properly. It's rather when you want to construct the topos anew, starting with the notion of constant sets and the notion of varying quantity and putting it together. And what I'm saying is, well, generally you get a pretty poor match, actually. Therefore, my quip, maybe algebraic geometry would be better. Well, yes. I knew you weren't seriously going down that road, but I can see where the... You know, and see where the sound bite comes from, because it's clearly very, very, very difficult. And part of the answer must lie with this behavior of the, you know, this feature that you draw attention to in the behavior of the, you know, the directed idempotent. ...addition when you've got well precisely these two rigs yeah yeah the two rigs um which they 1 plus 1 equals 1 being an example, just an example. Because, as I understand it, part of the interest of this is, and part of the reason for wanting to do algebraic geometry over ordered structures like lattices rather than over number fields, You do have this, but it does become clearer what role the idempotents are.

17:30 You want arithmetics which are, in some sense, finitely generated, don't you? That's the aim. This idea of the numbers being born from real space and variation. Yes, so therefore finitely generated arithmetics in which the two rings are clearly punishing examples is a natural error investigation. Yes, I agree. The multiplication is sort of arbitrary because, as I say, any commutative monoid, for example, the finitely generated one, the generalized polynomial rig, is just a finite subset. And so the addition is really just union. And indeed all the two rigs are quotients of those things. I find it convenient to think of a general monoid, the actual free ones, where the monoid is natural numbers, times natural numbers, times natural numbers, as exponents, as exponents on variables. The piano rejection of the idea that a number could be thought of as the expedited operation seems to me to be one of the greatest kind of wrong turnings that was made in the history of... I'm sorry to say I once had a tremendous row with John Mabry about this. Well, it did come quite heated actually. And I pointed out, precisely because I'd read your correspondence with Alberto De Ruzzi. ...that one should precisely think of this notion of natural numbers in terms of monoid operations. And it's a very natural way to do that. He'll say, oh, no, no, this is a terrible idea. This is to go back to all that massive confusion that numbers do with counting and to get away from temporality and all that massive kind of Kantian confusion.

20:00 And you'll be saying next that you think the idea that numbers is an exponent of an operation... ...was, of course, completely laid to rest forever by Frege, and, you know, the stake was driven through its heart and its coffin, and, you know, and the only person who's ever tried, and you tell Bill O'Fair if he thinks he's going to revive that, the only other person who's ever tried to revive that was Wittgenstein. I'm sure he doesn't want to... Well, it's all right. I mean, he calmed down after a while, and I got him to see that it's just that he's very, very wedded to the Dedekind definition of simply infinite systems of the natural numbers, which, curiously, is the kernel of your own definition of the natural number object in a topos. So there is no contradiction here. This is what I was trying to make him see. There's no contradiction between this Dedekindian viewpoint and the viewpoint of natural numbers as a... On the contrary, they're deeply connected. But he, because he still will not go out there and learn category theory, has wedded to the idea that you've either got the true deep structural characterisation ala Dedekind, or you've got some sort of, what do you call them, Wittgen-Schweiner-Eye, as a confused nonsense about things being mixed purposes of operations. You can't see that these are not in contradiction at all. Although people like Wittgenstein may very well have tried to make them so, and of course he would, but... Actually, so he sees Wittgenstein as the slayer of philosophy. Well, yes, I... I'm sorry, Frege, Frege was the result. I think I've managed to sort of straighten him out a little bit on that score, although only a little bit. Oh, I saw recently... Incredible quotations. Russell. Russell says... Another man who did so much damage that... Yeah. He was sort of the link between Frege and Hitchcock. He was indeed. No, but where is it now? Some journal that I received, I just glanced at it before I left the country, and then I said, let's look at this again, just for the sake of wallowing in its ugliness for a moment. Russell says something like the following.

22:30 The question consists of being an A and then a B. This is absolutely all there is to it and those who prattle on about how there's something like motion are wrong and should forget, you know, go home and shut up. Go home and shut up. Have you understood this, Vladimir Ilyich Lenin? Well, look at the rubbish that he wrote about. Oh, no, I know exactly the passage you're talking about. I think it's actually in the philosophy of mathematics. Yes, and it's his attack on infinitesimals and on, as you say, on any kind of objective motion. Weierstrass has shown us that the arrow in its flight is actually at rest. I think that's part of the same quotation. Weierstrass has shown us that the arrow in its flight is actually at rest. And the whole notion of... The idea of motion as consisting of anything other than the occupation of a body at one place at one time and another is a massive confusion. So it follows that velocity is a purely subjective construction of what's real, right? Well, it obviously follows that all real becoming is an illusion. Which rapidly, of course, takes us in the direction of which even Russell I would not have wanted to go in because of kind of extreme kind of Indian mysticism and monism. And the whole point about Zeno's was that he was trying to know he was trying to He was trying to show that the predicament of Parmenides and Heraclitus between them had set up. This was a genuine and Indeed, it not only was, but still is, in many respects, one of the deepest of all problems in philosophy and natural science. To just say that it was resolved by Wierstrass, with arithmetization and analysis, is just crazy. But then, of course, Russell clearly bought into this, well, somebody's christened this kind of platonic atomist metaphysics, which was very much, I think, under the influence of Leibniz, of his early study of Leibniz. Which was that the characteristica universalis was also arithmetica universalis, that there was a purely arithmetic ontology, but of a very Pythagorean, Platonistic sort, that the decidables were just everything, that all qualitative distinctions or becoming were just...

25:00 Purely illusory and could be reconstructed. I had a thought when I woke up the other day about the monads, you see, and that actually this construction and the axiomatic cohesion of taking those objects where pi zero equals twice. What you get there is very much like a swarm of Leibnizian monads. That's the idea, you see, that they don't have anything to do with each other, but each one has only one point, so it's very, yet it has a rich internal structure even though it has only one point. Again, the typical example is among directed graphs with those that have only loops. So they can have plenty of loops, you see, and even different kinds of loops. Yes, they can have loads and loads. They can be as loopy as you like. So they have, you know, they have sort of quantum numbers and so forth, and yet, in this approximation, they don't interact with each other at all. And it can be treated as just as... Externally, just as points, yeah. In other words, they... Which means that you're left with nothing except pre-established harmony and the mind of God in order to make the world for you. If you think that this is all there is, but if you think of it as just one of the many partial pictures of the world. Then of course it's definitely alright. So I was thinking, well, maybe Leibniz wasn't so crazy after all. No, I mean, Leibniz with the strong injection of Hegel to make clear that this is just one aspect of it. Yeah. There is a very... Yeah, yeah, yeah. He was a great and deep philosopher. Yeah. Yeah. So, you know, I started to read, I didn't really... In many ways, he is the bridge between Aristotle and Hegel. Yeah. But... Oh, yeah. That's the thought, yeah. Yeah, I think... There's certainly a far more serious philosopher than Russell, isn't there, wasn't there? Oh, I don't think so. I mean, setting the bar terribly high. But you see, I mean, I... OK. I mean, I tried to read some passages from the monadology, and they all seemed so incredibly dogmatic, you know.

27:30 You couldn't possibly accept, and then I got to thinking again, that this dogmatic presentation could be... That he actually had a mathematical theory, and he's trying his best to explain it in ordinary language. Yes, and so a lot of what Leibniz wrote and published in his lifetime particularly was written very much exoterically. Yeah, I mean just try to imagine if I take this paper, take my paper on axiomatic cohesion, let's say, and I've related it to his papers. I, you know, with great effort, you know, I could think for months and figure out all the nice words and so forth, explain it all in ordinary language. What would happen? No one would possibly understand it correctly. No, exactly, you'd be ignored. This is exactly what you did do. It's the categories of space and quantity paper. Don't you remember? You published in the San Sebastian Proceedings. You did the whole thing without introducing any notation at all. Well, no notation, but a certain amount of mathematics. I still think that was a relatively good paper. I think it's a great paper. In the sense that... It's an absolutely great paper. In the sense that you can get some idea from it, even if you're not... Well, I hope I managed to get quite a few good ideas from it, but it took me years of toil and study. Anyway that's relatively simple compared to this. So what I'm saying is that any attempt to explain it without actually giving the mathematics is doomed to all kinds of misunderstandings. My personal conjecture, for example, may interpret Newton this way. When Newton talks about absolute space and absolute time, He's not saying necessarily this is the way the world is. He's saying, look, I've got this very fine theory, and that's how you should think of it. And that is how you should think of it, because it does have a timeline and so on, right? As an adjunct, if you explain it that way, if you have a mathematical theory as an adjunct, you can give little hints as to how you might picture this and follow it better, then it's fine. Yes, sure. It's no... No, there's no bar at all to giving useful hints and, you know, crutches to the understanding, and there are some of us who need them more than others, but the great danger, of course, is that people are too lazy to do anything more than...

30:00 Just grab at those footholds of understanding and then not to go through the hard business of learning the math and understand exactly the constructions which these are supposed to provide a useful heuristic insight into. So I mentioned this to Audi. He said, oh yes, there is a theory. Leibniz actually had a mathematical theory which he was explaining in the moment. In this way. And this theory was by a French person around 1900. Around 1700. Cotterock. Cotterock? Yeah, Cotterock. Oh, sorry, but you can't mean Cotterock. You can't mean... How could Leibniz explain something that somebody did in 1900? I haven't understood. No, no, no, no. I misunderstood what you said. It's a... The philosopher's historical textual interpretation of Leibniz... Ah, I see. Somebody is claiming that Leibniz, somebody, Kutera, claimed that Leibniz had precisely such a mathematical theory, but has not been able... Well, of course, in 1900, a great many of Leibniz's texts were still... I don't know about this person at all, but apparently he's one who made it, made it, better known. I thought he even said that that's where Russell learned his Leibniz. Well, that's exactly what I was going to say. Russell certainly read Couturard, and his book on Leibniz is said to be, to lean very heavily on Couturard. I'm not enough of a Russell scholar, that's not an oxymoron, to know the details. Well, there are people who devote their lives, though. Oh, I know, I know. But I have seen it written that Russell has lent heavily on Couturard. His book came out about a year after Couturard's. Well, it might be about a year after, I see. Yeah, I think Couturard published in 1899 and Russell's book was published in 1900. I went out to buy some tobacco, and I feel I didn't do it, so I decided that this was right, but then I decided he was wrong too, you know, completely unserious.

32:30 Oh yes, the story about how you can describe the validity of the ontological proof, that's right. And then decided that... Completely shallow personality. Well, yes. And also the way how he decided he didn't really love his wife and just have all... Exactly, yeah. And he just... And there was no lottery of artists. Oh, yes. That's right. And how Lenin was the most wicked, but also the most imperturbable man I ever met. I think he would have had to remain pretty imperturbable to have had to endure Russell's condescension. Anyway, maybe it'd be worth... Well, of course, we all know that, but of course, I had, then of course, I had demonstrated to him that he was completely confused about the objectivity of motion. Given that, he should realize he has to give up right away. He should really give up all this bullshit and nonsense right away. Oh, no. Well, as you say, a deeply unserious streak in Russell's mind, very deeply unserious. It just comes so much from his class background. Yeah. But maybe it will be worthwhile to look at this guttural. I can't pretend to have read Couturard, but he wrote quite a lot of works on philosophy and mathematics at that time around the late 18th century. Poincaré had a correspondence with him, I believe. You may be able to find some of his letters in the complete works of Poincaré. I suppose we better make a move back to the class. Thanks very much. Do you teach? Yeah. As always. What, of Leibniz's understanding of the monodontic? Well, it's not unbridled speculation. Several of Leibniz's other writings were, precisely, more popular than what he had. He wrote a lot of work on mathematical theories, didn't he?

35:00 Yes, he was here a long time ago. But certainly true of his dynamics is some of the things he wrote in things like the Louvre, etc. Well, you've worked out an extremely interesting theory of the moment, whether it would actually be like this is, I think, has been less interesting than the theory you've worked out. I think so. Sorry. No, I don't think so. Oh, hang on, we're going out the wrong way. This is off. We have to go round there. But it's really not all men can take a nap. Good idea. It has the right smell. There should have been a detailed mathematical theory of the windowlessness.

37:30 And yet the fact that they clearly have this in terms of structure, they are just points for each other. And I can see how that would be very suggestive of thinking. I say it on a very deacon theme, the bad points never being really points, which typically have remotely analogous going on in life. There's an awful lot of medieval thing that are like this. I mean, not all of it, you know, just to be dismissed, the idea that... Medieval philosophers, and there are even female species. He was speaking of that. And many other things which were, of course they were, wedded to the field of gender, position of the church. Medieval society were swept away in the, or quite the official stories that they were swept away in by the Renaissance. Renacanthus, you certainly know very well the kind of one-sided intellectual liberation that Some of the small textbooks of my childhood used to tell us that it was, that it was a huge leap backwards into that realm. Reviving things which have been dead since antiquity.

40:00 Reviving a very aggressive form of objective idealism. Distorting or killing off a great deal of... These are the sophisticated, proto-scientific ideas that have come out of the RSVP by just decision-seekers today to rubbish their opponents. Congratulations on your meeting, gentlemen. That's very kind of you. Long overdue. Well, that's a slightly strange thing. No, no, I don't mean to say you shouldn't. We should have been 60 years ago. We should have been 60 a long time ago. No, no, that's it. I didn't mean the occasion was overdue. I meant simply that there should be such a meeting. I knew as soon as it was around my mouth that that would be it. It's very nice of you. This is interpretative. Are you really only 60? I thought you were quite a bit older. You're not dead. It says here in the Daily Telegraph that you're dead. It's a very respectable newspaper. Are you really sure? I think I shan't be enquiring when I get back to Cambridge about how much the department got, or just where the money is coming from. Just where it's coming from. I'd much better not to enquire. If I don't know, then obviously I didn't. As it were, say it could happen if I thought. No, no. He's a very excellent administrator. There's something quite English about him. In the kind of sense that actually... I doubt the interests of the department.

42:30 Heads of department come and go. Yes, that's right. That's exactly the right thing to say. They're of no significance whatsoever. Well, it's a bit like that in France sometimes. The French Civil Service runs the country. Of course. Even Sarkozy, there's only so much he can do. Interesting cases, isn't it? No, but you're right, the French particularly have a tradition there. But this is something, as it were, even better, more interesting, is that it's the departmental secretary. The analogy is more with a really good individual. Of course, it's actually the RSA who runs the regiment. The officers may have the deluge, and even call the lowest little subordinate sir, but everybody knows perfectly well who runs the regiment. They're even in a tight corner who they would actually turn to for decisions and leadership. People like that do come in very useful. I maintain that I'm an academic delegate. I wouldn't be taking any of these. Great. Ronald Reagan's paper calculated that... It's sort of a denial of responsibility, isn't it? But in his case, it was jolly useful. First of all, he was a great delegator, never a man of detail, and then of course when the Iran-Contra came up, we claimed that he was already suffering, and he was sick with Alzheimer's, so he had at least two shots in his locker. The great thing about being a born delegator is that it helps to have somebody good at their job to delegate too. There may be problems at the higher level, but we wouldn't be surviving if there weren't a reasonable amount of people just getting on and doing what's necessary.

45:00 Of course, like all well run institutions. I will particularly try to understand a bit more about what you were saying the other night about the as this was brought to you, do you think specifically about Leibniz's monad? No, I didn't, you know, I don't understand it quite right. I have to thank them for returning this. No, it's fine. Somebody lost it or something. I thought it was you who had it. Thank you very much. Yes, I don't want to hide anything. Thank you very much. This is the list of those first two talks. I'm just going to go to telling the science talk and then continue with my talk.

47:30 I hope you enjoyed it. I hope you finished the idea and I hope I gave you a chance to talk about it. It would be absolutely fantastic. Have a great time and a nice rest of the week. Thank you. Thank you. That's a great joke, as I said. Thank you for your attention.