FW Lawvere — Tame topology program (contd.)

0:00 Funny though, they never address these ethical concerns towards the organization of the economy or anything like that, which might actually question where they get their funders, yes. I was going to say, it must be a very unethical society. As far as these kind of people can do, especially for ethics. I'm being against abortion. That's very ethical. Covering up. There's a nice story told about Samuel Alexander, the, uh, the, um... British, early 20th century British idealist philosopher, who was in fact, I think possibly one of the founders of the Gifford, well Gifford obviously was the founder, the first Gifford lecturers, but, and who wrote a curious book in 1918, which was in fact based on the Gifford lectures you go, called Space, Time and Day. However, setting aside that aspect of his record, there is one very good story told about him, which was that when he was an old man, In the 1930s, he was at a reception when he'd become very deaf, and he was introduced to a guy visiting from New York who was a professor of business ethics, and a guy who was the chair of the department introduced, and this is Professor Alexander, Professor Emeritus of Philosophy, and who did you say, well, I'm, you know, I'm Sam Goodwin, the professor of business ethics at Columbia. Sorry, I didn't quite catch what you said with Professor Hoffs. Look, I'm very sorry, I'm a bit old and deaf. Could you say again, business ethics? And then at the top he says, I'm really sorry, I just don't, I get it. It sounded for all the world as if you said business ethics.

2:30 The way I look at it, if you want to buy some clothes, you can go to any store and buy some clothes off of them. But if you want clothes that suit you in particular, you go to an ethicist. You go to a professional. It's the same with ethics. You can get ethics off the rack if that's okay, but if you want an ethical decision that suits you in particular, you go to a professional. If you're the Human Genome Project, unfortunately, you don't have that kind of pull with you. Yes, yes, you're not, you're not Jim Watson. And that's what did it. The Human Genome Project, Watson said 3% of our budget will go to ethics. This is hundreds of millions of dollars, and what you do when you're an academic student is you decide that there are some serious issues that need further explanation, and he has decided that this is going to support the math of the event, but of course he can't support everybody, and I don't support him, but still he didn't have to decide math was going to be the test case for this theory of mathematics. There's a pity because he's a seriously smart guy and he shouldn't be wasting his talent on this kind of thing. But anyway, that's the Wellcome Foundation and they provide, and I mean they do do some good things. They're original medical research grants and still do. They fund a hell of a lot of cancer research. But Thatcher did everything she could to privatize the funding of university research, and so these people were the big corporate interests that stepped into the gap. Now what Bill's going to ask me, I suspect, is who are the people behind the, well, who effectively controls the Wellcome Foundation? It's a very good question because nobody knows. Because it's a trust with kind of interlocking clusters associated with, and I don't think anybody knows, I mean people know who sit on the grant committees, but nobody actually knows who, well officially of course it is a trust so there is no beneficiary other than the people who... The benefit from the grants, but obviously the foundation was established and is still effectively controlled by in somebody's interests.

5:00 Wellcome itself was actually absorbed in a whole series of mergers in the 60s into other drug companies and I think it's now effectively became a division of Zeneca. I think they were absorbed by ICI which in turn was absorbed and became, which split and became Zeneca and there were a whole lot of corporate. Mergers and takeovers. So, you know, it's a wilderness of mirrors. Nobody really knows who does control, welcome, but they are the people who effectively fund. I don't know what the percentage is, but it must be getting on something like 30% of all the civil research in the UK. Although they're not exactly A, B, 4, or 5, I think. So say again what the definition is that's different from the... Sorry, just right past me again there. Abelian categories themselves, it's just that you can add the maps. In fact, I'm not even sure you can subtract the maps. You may only be able to add them and there is a zero. Because the only representation theorem it comes up with is that every abelian category can be represented as a category of monoids. Okay, but then when you get to the abelian bi-categories, kernels and co-kernels added, he still doesn't talk about subtraction, but in fact it's definable, I mean, he just doesn't notice that, because he does have byproducts, at least by that stage, byproducts, kernels and co-kernels, and I think... He may even say that infinite chains of sub-objects have soups, and these soups are preserved by direct image. Or it may be in, because I forget which now, right? But I do know that preservation is direct image, not infinite. So essentially most of the issues... ...without the goal of derived punctures and spectral sequences, and in particular without the focus on injective objects.

7:30 Because Grotendieck's proof that the categories of features are topological, people have thought for a long time about features, but not proven. And I just talked to Brooksbaum. He said when Grotendieck's paper came out, he was kicking himself. Because it is, simply bears proof, for a billion- A million groups applied over topological spaces. How could any of us not have known to simply do this? Well, because it's a lot longer trans-finite induction and you don't have data. Okay. You're okay? I'm fine, yes. I just thought there was a school somewhere lying there. But yeah, not... If you want something, just say so. We can get another class. Okay. Let's get another class. Good idea. See if we can catch his eye. Don't forget the details, the method of palming into the surface. Thank you very much for your attention. Does anybody want a dessert? Or are you guys okay? Just hands up if everyone wants a dessert. Just one. No, no. Bring the wine. Okay, that's fine. We might decide to have one, or we might pass. Well, that's interesting about it. We might not have options. I've been talking about linear categories. I explicitly don't assume... I use the word linear just because the additive was already taken. I mean, if I write the word additive, it should mean that, I think, but it's so clearly established. If you've got byproducts, you have to use the exact ones. I first remember seeing this in Freud's thesis. And I remember I came up with this construction a couple weeks ago, but I'm not remembering it either. How do you get so fast?

10:00 I don't remember exactly. It does use the exact line. What's the significance of subtraction for you? Well, it's very easy. So I'm trying to categorify. I'm trying to find for all my favorite sets, categories, set of isomorphism classes, where the structure on the category gives rise to the structure on the set, and so... So for the natural numbers, the classic answer to that puzzle is the category of finite sets, the cardinality of finite sets is a natural number, and addition is a co-product. But if you try to do that, say, for the integers, you would need to invent sets with negative numbers of elements, and you can't have a category with co-products. It gives you the piece of this, the set of isomorphism classes, because the sum of two things is zero, the initial object, and they both have to be zero. So, in other words, there's a nice categorical analog of addition, but there's not a very good categorical analog of addition, except for subtraction of something inside the big thing, which is a big co-component. So the question is whether something like negative 5 is just a formal tool that people invented. To make calculations for the positive numbers of mathematical numbers or whether there really is a kind of thing out there that deserves to be called a set of negative five dollars. I did completely wrong here, but wasn't Tom Leinster and Michelle Peer doing something? Right, they are, yeah.

12:30 He's here tomorrow. He's here, by the way, Tom Leinster. Yeah, so you can make up, so you can make up interesting categories that are like, that are more like these things, or like Gaussian, or other things like that. So you can make a distributive category with an object such that x is isomorphic to x squared plus x plus 1, and that equation is satisfied by the number i. They do refer sometimes to Chanuel, but actually Chanuel worked out and published about this 20 years ago. It's in Como 19. Finger lecture notes, 1488. Isn't that the complex stuff? Well, other people did that, though. About 15 or 17 years ago, Giancarlo Maloney of Milan worked on, even worked on a computer program. Is that the stuff I covered? Yeah, but there's a very important paper of Andres Blas. Oh, if you will. You know that paper? Yeah, that's... If you understand that one, then the I and other things like that are more or less clear. They were perfect. There's a general theory due to Robbie Gates in Australia. Thanks very much. Most of the pages are from excerpts from mathematics, but the rest are from cohomology. I haven't read that paper.

15:00 So they give all due credit. I mean, the crucial idea of Shannonwell that none of these people seem to mention is simply... There are two canonical simplifications of the rig. One is to join negatives, getting a ring, which he calls the Euler characteristic of the object, but the other one is to tensor it with 2 instead. 2 means 1 plus 1 equals 1. So you get you get a rig with some coefficients in two but the multiplication is not in fact the multiplication records the dimensions of objects so there's kind of inherent dimension as well as inherent or other characteristics yes and in many cases for example there's negative stress but all the cases treated by gain t these two invariants are sufficient either one Now, these kind of categories are more and more complicated, object-wise, but also math-wise. It kind of balances out, you see. The isomorphism classes are still the same, namely precisely what Shannon calls the rig of geometric quantity, which you get by adjoining to natural numbers. And elements that satisfy x equals 1 plus 2x. So you can represent those by the degree of the polynomial becomes the dimension, whereas the value of the polynomial minus 1 becomes the paragraph. Distributed category, where you take, say, 100 sets and then go on X to 6. X to 6 sets, there's one way to do it. So, yeah. So that's... As I say, much more complicated categories have the same result, because the varied objects compensated by the... there are many more maps, you see, that's richer, but still get the same number.

17:30 That was his simplest realization of the piecewise-linear thing, because, I mean, they're piecewise-linear, but not even continuous, that's true, but essentially, in other words, this equation, x equals 1 plus 2x, describes an interval. You can take a point out of the middle of the interval, what have you got? You've got two copies of the interval plus one, so that's why x equals one plus two x, and if you iterate that, you see you're dividing the interval into a sort of dyadic rationalization, but that's where the, that's where the... I mean that's that's the fundamental thing you see is it occurs for a general negative but you also have this dimension concept which is recently much more refined than their natural numbers What should the dimension of the disjoint sum of two spaces be? So you have an idempotent operation corresponding to sum, namely the max. On the other hand, when you take the product of two spaces, we usually say that the dimensions add, but that's really just the logarithm of the actual. So the product in this rig corresponds logarithmically to dimensions. But precisely what dimension is may or may not be just math. In fact, already for the unbounded polyhedral sets, you get a very detailed analysis of the unbounded... the bounded ones you get for the intuitive dimensions, but for the unbounded ones, you get very much more. I've read that paper, but it was too long ago, so I think we should read it again. Yeah, if you're interested in these things, I suggest you read it again. But what's very important in getting me interested in it?

20:00 Now, the thing is with these, if you have an x equals f of x, where f is a polynomial a degree greater than 1, then the whole dimension rig tends to collapse into sort of minus infinity 0 and 1. No, so minus infinity 0 and infinity. In other words, there are no... There are finite dimensional objects, but it's still very important to keep track of which of these three it is. And with those three, together with usually part of the complex numbers, the Bernstein ring is simply part of the complex numbers, whereas the dimension rig is always these three elements. The whole rig is somehow three copies of the complex numbers glued together in a certain way. But even though that's so simple, these three elements, you won't understand how a grid works unless you keep it in the chat. He really had a good idea there. What he discovered later, which I don't think has been published, but we already knew about it, But there is sort of an infinite zero and an infinite minus one and so forth. We tend to have the natural numbers and so on, and then all the infinite, all the things of infinite dimension. The ordinary individual natural numbers have dimension zero, and then all these things that have infinite dimension. And of course the empty set has, well, it's only the empty set. At least these infinite things themselves form an actual ring. Quite remarkable. There's a unique point that you could call the infinite zero, the infinite minus one, the infinite one. It's a whole... And that is isomorphic. They are just a quotient on the one hand, but on the other hand, you fill in minus one, but that thing remarkably has a section. Not an additive section, but one that preserves multiplication.

22:30 And it maps precisely onto all the infinite elements, so there's sort of a one-to-one correspondence between the infinite elements in the thing itself and the things in the quotients it brings. Yeah, I was just actually thinking about that. It's quite remarkable, this infinite thing. It's something that, you know, it's like Dirac's electrons or something. The world is an infinite number of electrons, but if you create one... You've made a difference. It's still infinite, but it's really different. And very precise. There's no ambiguity. It's precisely different. I dream, you see, that someone will be able to incorporate that into their training, things like that. It's that sort of thing. The things are infinite, but the infinite is not all smushed together like we normally think from set theory. It should be. Do you know if any of these people have thought about... Something like Galois theory for these things. So like if you throw in, so maybe you already know what I mean, but like so the Gaussian integers has automorphisms and this 7, 3, and 1, that's really how it works. That was my example, that was the first example after the linear one. Category that extends from x to x to x to x to x to x to x to x to x to x to x to x to x to x to x to x to x to the 7th is x, but x to the 6th is not 1. It couldn't be again for the reasons we said before. Right, yeah, sorry. I thought I took two things at the same time. No, no, but I mean, it's not impossible that there's some way to go something around. I was just sort of getting interested in how physics works. Thank you for your attention.

25:00 Well, certainly you can do algebraic extension, although, yeah, well, still, it's still not, that's what I say, the reason for that whole paper on seven trees, but it's still a non-trivial theorem, in a way, see, because intuitively, you might think, well, Why should the isomorphism types correspond exactly to what you get by mere abstract rig operations? Couldn't there be more isomorphisms than those that you can prove just starting with the axioms of a rig in a particular presentation? I don't think that is still... I must refute it in general, but Brock did it for that case, and Gates did it for a vast number of other cases, but it's just somehow, except for the whole question of adjoining, you can adjoin an element to a rig, with certain equations, you can also imagine adjoining an object to some nice morphism. None of these will always coincide, but if you could prove they'll always coincide, it would magically make everything much easier. Otherwise, we'd sort of have to beat our heads against each example. So, I remember how you got to the top. Good. Right. The absolute you get is negative. Take a spike product of this. Take another spike product of itself. That has a co-diagon, which is really, you saw what was important, you can't take its curl. Its curl between them is the one that's the negative of the other. And of course, the more that the one is the identity, so the other is the one. And you subtract that. Can you just go back? I'm sorry to, you know, go right back to the...

27:30 Well, we went off into this extremely interesting exchange, but can you just remind me again that the 1948-1950, the McLean definition of abelian and how it differed from what? There are two levels here. Abelian categories per se simply have addition of functions on them, but the abelian bi-categories also have zero odds of byproducts in kernels and prokernels. So those are the abelian category axioms, plus they're required to have some exactness assumptions about the lattices of sub-optics that he came up with, and these are a little complicated and none of them are exactly Grotendieck's AB 3, 4, or 5. They would both extend into the infinite, but not quite the same. They're not in the same way. The basic finite thing is in the end the same, even though the names are very different. And they have in... Kratonik has said he wasn't careful about sources in those days, and of course he wasn't. I'm sure he never said, here's an idea Saunders had, I won't tell people so. He said... You know, I think you could have categories like this, forgetting that Serre had told him such things before, and I'm sure that Serre didn't tell him exactly how to do it. He did figure out for himself how to do it, but only after somebody said he could do it. I don't think the 1950 paper goes as far as that. It was presented as a talk to the AMF here. Yeah, yeah, yeah. In fact, I thought there was an exact reference in my book. But why did the, why did the terminology abelian category not stick? Why does it have to be reinvented? Okay, well, sure, sure, sure, okay, reinvented was completely the wrong term, but I, but was the definition of exact categories as it was used, you know, by, yeah, was, how different was that from, This definition of McLean, the 1950 McLean definition of abelian categories. I don't know what the definition of abelian categories is.

30:00 Yeah, it is. And look at the current abelian category, so there's not the kind of stuff on top of that. Thank you for your attention. And then he looks a little bit at injectiveness and things like injectiveness. Actually, it also came a couple of days ago. So who is he? Sorry, this is... My thing about Huxbaum, though, is he says he doesn't know how to define the products of matter. He's really very strange. He's already... 1945 paper, they had this. Well, maybe in some way, but they define limits and co-limits as only productive systems. They don't define products? No, they may use the products in a few places in remarks, but they don't. It's really strange that you recognize universal property with an infinite number of entities. I mean, I think that's probably one reason why his paper wasn't read very much, because anybody could read it, so I wanted you to try it a little later on, but it wasn't stupid. It's just an oversight, of course. The other thing is, I mean, at least the version is authentic. You've got to be a bit of a hero to make it to the end of Clarkson Island, right? Meanwhile, a good newspaper has appeared, and everybody in Paris is saying, and I think quite rightly, well, this is really what it should have been anyway. And Grotendieck, you see from the correspondence with Sarah, he was aware that he was doing what Cartan-Eilenberg should have done, he hadn't seen Cartan-Eilenberg yet, but he is doing it the right way, and certainly by the time it appears, he feels, yeah, this is what they should have done.

32:30 And in some sense, they thought so too, I assume. Remind me again what it was that Grokendieck... I'm sorry. Remind me again what it was that Grokendieck did that was... And stable... Stable intersection of sub-objects. Stable intersection of sub-objects, yeah. The natural level of generality is over. It's open-arbitrary, really a category that I'm supposed to study. And the story goes that he had trouble finding a publisher, but what he says in a letter to Sander in 1954 is that Sammy says he wants it for the annals if I'll do expensive editing, which I'm only willing to do if they'll pay for it like this. Yeah, I'm going to be telling you this story. It's a different era now. Yeah, yeah. Well, in hindsight, I have to think he's right. I don't think it would be better if he had it, the way I think he's right. Uh, Eilenberg has a very different style from Goethe. Eilenberg gets to the point where he doesn't need the hard information, for example, the heat. And he explains the reasons that I can read it, you know, whatever he wants me to do. Though you have to be very smart, as it were, to get the reasons, because it involves, as they say, the immersion of the knot in a great deal of liquid, and you want to see as to where the liquid is. You know what you want to say and you dogmatically say it. You don't equivocate. You don't suggest that there might be some other idea. Well, that certainly is very different from Grotendieck, isn't it? There's a certain logic to it. But Grotendieck's absolutely confident every time that he's doing it the only right way. You couldn't do it otherwise if you knew what you were doing. But he does explain why he's doing it that way. What I've never understood is why one can't do both. Why you can't have in the same book both the austere presentation or, you know, maximization and that kind of thing, and also the explanation of why you're doing it, maybe in another page. I can never understand. People think it has to be one or the other.

35:00 Yeah, I cannot read the SBA, the SBA. I'll call them out. I do have to give it in a shorter compass. And so far, maybe I'll give it in a shorter compass, but I don't think so. And so far I haven't really understood the subject yet, so I don't know what to say about it. I can't say it works for me yet, but it's a good theme for me. But finally, you do have the essay. What is this about, and why is it about this? Because then it takes forever to get to a theme. Yeah, you once made the remark that... I mean, you attribute it, you don't have to attribute it. Space is anything that has full power unless it's the main... I shouldn't have, I don't want to... I attribute it, I interpret it. No, but I mean, anyways, whatever, I mean, how do you... That seems like it's hard to growl because thermology is a technical construction. So how do we see directly that that has anything to do with any concept of space? Well, what he does say is that the way you know... He says that toposes are the proper objects of topology, and this is because they have cohomology. Now, slightly reading into that, what he's saying is that topology is a subject for the study of things that have cohomology, and should have been studying toposes because those are the things that have cohomology. But they have a lot more too. Well, in particular, as I was saying to Michael, one of my weak points, they have homotopics. Yeah. And which we don't understand well at all, at least, well, I don't understand anything much at all, but even really smart people don't. They also have internal functional analysis of these things, you see. But that's not his argument. To the point where he went... I mean, I think those things belong to space. They might have to say that cosmology is a mathematical place. You know, unless you're trying to talk about everything with cohomology... In Brogan's bubble, everything with cohomology is a purpose, and purposes do have this. So all you have to require is that we should have a cohomology that would imply it. So did you look at the 1960 seminar that I suggested?

37:30 Everybody looked at it at home, yeah. I mean my basic point there is we're actually talking about a growth phase, not a space, so this argument wouldn't apply to that. So the category of space is the category of analytic space. The things are different from others, if not being generalized. Even though, as I heard, the first example was really about chemistry. Yeah, yeah, yeah. And even though they're different from others, that's why you want the other part to be the next one. Alright, alright. Thank you. But certainly, you can recall that the summary of this theory, a telepost is a spas-nouveau-steele, a neutral space. That's what a telepost is. I'm trying to think if I know any phrase where there's an A because it would tell me, but I don't know. No, no, no. My dad knew. That's a wonderful. Wow. Wow. So we could go back and offer him therapy, whatever. I'm sorry, I'm just really not very interested in what Plotinus thought, I have to be honest. There's now a Cambridge Companion, so you can just go to the secondary sources, believe me.

40:00 A little bit, a little bit superior to that to be the objective, but yeah. Professional, yes, the professional, yes. Well said. That's a spot on. The professional dummies, exactly. I'm glad to have read the Cambridge Companion to Plato, but not that I agree with it. Can I just ask one question, which is going back to a very simple-minded point about the components factor. And the points, you know, just going back to a very simple point that you made in the lecture notes on SDG in one of the appendices, which you've made further length elsewhere in the Cambridge lectures, about the way that one can see set theory as, as it were, derived within the geometric picture in terms of the... What was the condition about there being no, on the definition of being a connected component, about there being no non-trivial co-product decomposition? That's simply the definition of connected. Yeah I guess I see that. Yeah, yeah. That's the definition of a connective. Oh, yeah, okay. So, do you have, I mean... Although it could be, that's a basic idea, there could be some little refinements on that. No, no, no, they hide it away in advance. So, okay, so in that context where you define within spaces something called discrete spaces in some way, and you require that the inclusion has a left adjoint. That should be pi zero for this situation. Right, yeah. Now, that may or may not correspond exactly to...

42:30 Say the structure of light in space gives you one. Is that true if and only if the space does not have any non-trivial cobaltic components? Well, I think there are some extra conditions. Okay, that's... You've expressed the question I was struggling to formulate. It's not always that way. No, no, that's what I was a little confused about. And so on and so forth. So at least for the finite number of components, these refinements about exactly what the speed spaces are won't make too much difference. But could there be kind of, you know, spaces with higher connectivity properties where they do make it? Well, higher connectivity, that's my... We have extensive categories. We have this simple, finite book college with a certain property. This, in a way, basically means that you can explain what are the trivials and what are the disconnected spaces. But that provides the setting which you can contrast the other objects and study their connectivity by mapping other objects into these trivial objects, you basically split them up, so that's the basic action of extensivity, that if you map into a sum, then the domain splits as a sum in a certain way, and the two maps... But it's not just connectivity, what's called higher connectivity is... You can maybe give some more complicated definition, but it's usually measurable, simply by saying, you know, you have a fixed space A, like it might be a three-dimensional sphere or something, for any space X, you take X to the power of A, in terms of constant space. Now you take pi zero of that. That's the eighth homotopy group. You measure how connected X is in the higher sense by looking at all possible exponents and seeing how connected they are.

45:00 That's where the higher levels of various dimensions come in. Yeah, yeah, I think I get the last point. You have this function space, you see, so can these, or think of them as maybe as impregnants of the other, or can you deform one into the other or not? Now I understand the specialness of the case where... The components just do reduce to the point, so you go from point to point, yeah, yeah. Yeah, because x might be connected, so you get x to the a naught. Yes, yes, yes. No, but simply connected just means that for a certain particular a, you get a circle. X to the A still has only one component, or non-symptom connected and non-symptom connected means the components of X is one, but the components of X to the A is not one. Well, it is when you have a good teacher to explain it to you, but it's difficult. Space, I call, Dayton disagrees with me about the word, but I call contractible the space with the property that all of its powers, x and a, all of them are connected. Now, you see, that means exactly that x in the homotopy category The homotopy category is defined by a new hom set, or a new hom discrete object. You take pi zero of the function, you see that if you have, again, a and a and a and a and a A to the power of B and other ones. You can compose these. It's a category, although you reduce the onset drastically by taking their components rather than the actual things.

47:30 That's called the homogen. That's where A was constructed, the homogen. Exactly, using the exponentiation of pi zero. And the fact that pi zero preserves finite products is crucial for making this thing into a category. Now I do understand what you're saying. The actual function spaces, y to the x of z into y to the z, that's just tautology almost. Now if you apply the phi zero function to that arrow... You get the new homotopy homosets in the three places, but because Paisio presented kata, you also get kata in the middle, so you again have a composition problem. So in order for an object to become terminal, it means that for every a... Well, there's only one left, from A to X. It becomes permanent if the components of X to the A go on. Yes, yeah, yeah. Well, it turns out that that's equivalent to saying that X to the X is connected. Yeah. You don't have to believe in X to the A, it's equivalent. Because X to the X sort of acts like, sort of acts like a... Not to mean literally, but it's a normal word with a zero. And so, if it acts on something, that something must be connected. If it's connected, then it's acting on the other way, for sure. You can have these rheumatography-like things without having a specific interval. If you use x to the power x, where x has a given point, you use it like, you know, you prove that everything that it acts on is connected. Well, x to the x acts on any space that you can make. Yeah, so? What happens in good cases is that every space, no matter how complicated, can be invented into a contractual space. Sort of like intuitively you fill in the Witten space, but you can fill in...

50:00 Yes, just make one, yes, yeah, sure, that you can kind of move stuff around without having any... That is absolutely beautiful idea. Does that even connect with the explanation of... What one thinks of the absolute choice in this context, the absence of any obstruction? This is wonderful. I'm, you know, I'm really beginning to understand stuff that I've struggled to understand for a long time because I started too late, that's my problem. What do you mean by the alleged default category? I'm sorry, I haven't understood that. What people call the category of topological space is a continuous span. Oh, right, okay. It's a default in the sense that if anybody trained in 20th century thinks he has to express cohesion somehow, he has to define a frontier from his things into that category. But he's wrong. Sure, but that's what the so-called general topology deals with. Yeah, I know. Sorry. Sorry, Bill. Don't. Sorry. Hey. Yeah. No, no. Bill, you rest your voice. Okay. Wait for the sky. So I'll still be listening when he's paused, don't worry. Yes, I now understand what you meant by, you know, doing topology in the Horavitch spirit.

52:30 But listen, you need to rest your voice. I'm serious. You have five days and there's a lot to come. But thank you. Thank you, chief and comrade. I think we're going to... Yeah, let's call this quits and get the ball. We can always go. Oh, absolutely, except you're not going to give me that, that'd be ridiculous. You can give me the rest. Okay, right, it's probably going to be, okay. We'll go through them. Yeah, well even then it's, oh, more than enough. Yeah, for sure, okay. I'm going to go back to this photo there. Yeah, exactly. We'll see you in the morning. 8.30 is when Francesca is coming, and don't worry, I'll check that she's got the mic and everything. As I say, when I make a decision, I think it's better anyway. No, that's taken care of already. We already have a taxi for Dana anyway, so, you know, you can share. That's the plan, yeah, yeah. Guys, look, sorry for, I mean, that's very bad manners, I mean, you know, just ignoring you guys and monopolizing Bill, but you just don't get the chance that often and you do... Topology was really about, and I got a much clearer idea after tonight. Lisa, I'm sorry, I was just saying, just atrociously bad manners, I feel awful, I just did. A theory from their own point of view, and it didn't really have an impact on the real development of the theory, because it didn't connect with it. It was a logician, yeah, internalized. That's, after listening to him tonight, which was on the whole, it's only half true.

55:00 Of course a mathematician would see it, that Bill Witt. He was happy at one foot and that's why he went. Bill, Bill, you know, years ago, you know, Hi guys, you're still in there hanging out. I'm glad you're here. Very glad. I'm glad you're here. Both of you, very glad. It does, it feels longer, but it was actually about half that time, but it... No, no, no, no, no, it was just before the, the first time I, I invited you here just after the, not this year, but the previous year, the 2002, just before that, so it's about 18 months, about a year and a half, not three, it's about a year and a half, but you're quite right, it feels longer. Thank you very much for your time, and I look forward to seeing you again soon. I'd really like to get him down to the chair for a couple of days. He's a smart guy, very interesting. Well, that will take care of itself. We don't really have to study that. You'll only start. I must say, I was going to have a bad feeling for you. What a good idea! Oh, wow. Somehow, I'll... Oh, aren't you nice? Thank you very much. Thanks. If it's well expressed, it's like it's the world in itself. It was absolutely fascinating. I really remember... I remember that very well indeed. It was absolutely fascinating. You could accept that, that they're just maybe a little colorblind or tone deaf, you know. Mathematicians actually are quite good at music. It goes back a long way. Scratch a mathematician, you find a musician.

57:30 Except in certain cases. Except in certain cases, which Bill has certainly won. Yes, he doesn't like me. Totally totally. But no, he told me. Goethe had absolutely no musical sense, whatever. Nor did Hilbert, nor did Goethe. So it's not a universal. When Bill, we had this nice dinner in my place for about six weeks, you know, when he came to this hotel. Incredible, that guy. He's got a study at the beach. But anyway, had a very nice evening, came to my place for dinner. And then I put it to him about music. It's too much. He couldn't deal with it. He's not simultaneous. Well, that could be with Bill. That could be. That could be. That could be. That explains the geometry being at the base of everything. Certainly, I hope it explains my trivial contributions. No, I know, but what I mean is... No, that's a very interesting observation. No, you're not! You're not! You have to be. No, John Astartu has enormous breadth and refined judgment. I'm kidding. I'm kidding. Ask him. The most you can do is pretend to be a monomaniac on a particular topic. But I don't believe that you could ever actually be a monomaniac. No, I was really happy. But listen, I'm surprised. Because most people would be. I couldn't do it. After all these years, 30 years, dinner six weeks ago at our place. He said, it bothers me. That I can understand. It distracts me. Yes, it distracts you, exactly. Dare I say, John, to be completely crass and vulgar, if you were driving around the fucking bed by playing bloody fucking Hindemith 2, as you did to me till about 4 o'clock in the morning, when I used to come to you in the hope of getting lessons on top or spirit, I'm not surprised it said it distracted him and drove him mad. So you were the other examiner? Yeah, yeah. Well, I'll fucking see you then. So this is kind of a spilted jury here to ask the... No, no, to be fair, it didn't work, but the idea was to logicize the whole thing. In other words, there ought to be some, uh, you know, some, some, uh... There is a way to logic, that's the real thing.

1:00:00 At the moment, John's talking about a logic book by Chapman and Rowbottom, the two brisk little pitchers in the 1970s, who wrote a book about model theory. You take a category of geometric morphism from E to F. You factor out the image. Now you have a surjection and an inclusion. Now the inclusion part... That's a sheaf subtropos of the codomain. So that's logical, right? Because you've got the codomain, and the sheaf subtropos is a modal operator on the codomain. And the other part, the surjection part, is a... ...is a left exact co-monad on the domain. So that's a modal operator on the type theory. So how is it represented in terms of calculus? You have two calculi. One on the domain, one on the co-domain. That's what they were trying to do. You have the internal logic of the co-domain, the internal logic on the domain. On the one, there's a modal operator. On the propositions, that's the Veritiernik's Apology on the propositions, on the other one there's a modal operator on the types, and that needs to be, the syntax of that logic needs to be spelled out. It's a left-exact comonet on a... So it could be written down. It could be written down as a type theory. Now, here's the question. What's the connection now between the two? You need one further step, right? Because now you've got this domain and a logic on it, a co-domain and a logic on it, and you need to connect up the two somehow. So you need to say that somehow the co-algebras for the lexikomonad here are the same or equivalent. So, it has to look something like the original insight of the, you know, for quantifiers. No, no, I mean, in other words, it simply has to be spelled out in some very simple way, immediately obvious to the eye. Listen, listen, it remains to be done. Well, no, I mean, I think that was what... Certainly, certainly, Fred was trying to do, because he had this note, he didn't do it. But he had this great, because he came from Scepter, he did this brilliant work, he's always wearing a blossom, one of these washed up guys, you know, he did brilliant work, and he had one paper, he proved, he was the one who created this stuff, you know, large cardinals, measuring, influencing the size of the continuum, no guy could go but a...

1:02:30 Hein Geithman is a professor at Columbia now in philosophy. I knew him years ago. He was a clever guy. Jack Silver is nowhere. He nearly remained in Berkeley. He's done absolutely nothing. He's one of the biggest names in the period. He was. Absolutely. If it hadn't been for Cohen, Roboton's work would have been the hot topic of the program. Oh yes, but the thing was, no, Cohen latched on. You say that, but I don't think that's quite the case. None of them, as Dane admitted, even in the introduction to that book I wrote years ago, which is mostly his work. A fascinating book, a fascinating introduction. Actually, it turned out to be a mainstream idea, but it was a sort of, you know, like the flowers. I've got to tell you, John, the guy that did, that un-ramified the poem was Fred. I know, I know, but that was all, but that was all later. I was the inspiration. To be fair, those are the next songs. You know, I was giving a course on forcing. And I was talking to Frederick, and I said, look, this grammatical language stuff is too complicated. It's got to be wrong. And the two of us, I would come with questions, and Fred would produce the answers. Yes, of course. But Fred's really, he was the kind of guy that could, that's true, but his really original work was marginalized. No, no, no. No, it was a large cardinal work. The large cardinal work had nothing to do with forcing. The original work of large cardinals is called the theorem, right? You know the theorem that measurable cardinals are inconsistent with an axiom and had nothing to do with... Sorry, are inconsistent with... No, yes. You know what that means? It's an ultraproduct. Oh, really? The proof is an ultraproduct. Don't you remember the original thing? All that had absolutely nothing to do initially with forcing. Of course, later on, sure. A few years later, well, how long was it? Well, Bill saw that it was all the same.

1:05:00 Yeah, everything fits into one framework. And now, of course, all the adequacy and co-adequacy conditions are... But the point is, Cohen, I don't think, ever noticed that Cohen's genius... That's what's really going on. Cohen, I don't know, did he have any contact with Gordon Deak except for that field's medal ceremony? Robottom got rid of the... This is the fact. Robottom got... I saw it happen. It was done... Under my pressure, because I was dissatisfied with this stuff, and Fred was producing it, but the stuff that Schoenfield later, you know, the unrundified forcing paper, which was quite influential. Oh yes, but Schoenfield, but the difference was, Schoenfield, he would never have claimed that he cleaned it up. Fred's really original work was in large cardinals and all of that stuff, but actually he's the source of the Woodin program. If you want to know where Scepter is, you want to latch on to somebody, you know, a completely different thing. Latch on to Woodin. It's completely opposite to what's going on in category. Woodin is what they think is, you know, they're exploring Everest. Yes, I know. I remember I went to Sicily. I spent a week with Woodin and with Angus MacIntyre and with Angus MacIntyre. And see me, I ended up believing in Angus MacIntyre and in the fact that all this stuff actually lives inside algebraic geometry. And once you understand what aliquistic condition or the right map space is, you'll see where the whole delusion of... But the whole philosophy is different. The whole philosophy is different. Large cardinal theory is just based on conceptual delusion. It's a realism. It's the idea of light with physics. You're just trying to deal with something that's already there and you really can't change it. Wooden dustbin. He's very, very clever. Oh, he's incredible. And you know what's more, he looks amazing. Like Zachary Scott. Oh, I'm really good looking. The girls, believe me, the girls who did the Francescas of this conference, a lot of pretty young girls have been all in the kind of process. They actually did, at the very end of the meeting, they did like a kind of 10-minute routine for Woodin. It was hilarious, absolutely hilarious.

1:07:30 They would have kind of they would have they would they would have kind of done a strip tease on him in public if they could have done it. I want to have another beer. John, I think it's a good idea not to be too good looking. In our case, do we have a choice? Well, one person around this table certainly does, but it's far from... Can we have another drink? And I'm not talking about you, Bob. Who's speaking tomorrow? Bill, Bill, Cartier, Scott, and where am I? And Angus. What time do we start? You've got to be there. In the lobby, when if we are twenty past eight, bamma, yeah, the, uh, look, all that's going to happen when we get down, we'll listen, no, let me, yeah, one more, no, let me tell you what's actually going to happen, no, we're talking real world now, and who saved him, that's why he looks so young, listen, I don't want to embarrass you, but can you, can you believe that this guy, can I tell them, that this guy is 35? Well, he's a kid, from my point of view. Yeah, I know, but he looks about 27. It's ridiculous. 25, 27, what's the difference? The perspective is 60. He retired, he retired early, about 50 when he retired. And I said, Craig, what are you going to do? He said, I'm tired of all this abstract stuff. He said, I'm going to work on dry fantasy now. And he is doing it. That's what I'm going to do when I retire. What Curran's saying is going to put the ream in my pocket. So what Fred said was, okay, so I proved a few theorems in spoiler already used. So what? So what? No, no, no. Fred is the greatest underachieved mathematician in his time. He played the panel quite well, but it wasn't... But Curran is... Come on! You want to see people achieve... Where are the compositions? I'm waiting for them, I need some serious confidence. Well, look, Grotendieck had just a glimmer of... I don't think we're going to get there. I could be wrong. It could be that somebody is going to do a...

1:10:00 I think he's done a lot more than just that. He's done a lot more than just spin. He's done a lot more than just absolute first class exposition. He's actually had a lot of key ideas. He was the guy who really brought functorial, functoriality into logic. The guy who really brought functoriality into logic. And so category theory is the way... Yeah, yeah, yeah, that was really important. That was really important. Would you not have heard of it? Well, maybe... One could hope, at least. I mean, it hasn't happened yet that, in general, the logic community has taken... To the extent that it has happened, it's been very largely due to him. But, on the other hand... He's an old-fashioned guy who had the foot in the church. It was churches he had the foot in both, which is precisely what made him so strong. He was extremely brilliant in both directions. Nevertheless, I think he never produced... I wish he had. Produced what? A book? A theorem? We can't never produce a fucking theorem. What was he supposed to produce? Come on, tell us, tell us. We can't never produce a theorem. To be quite brutal, Bill LaVere's never produced a theorem. Why is this regarded as the absolute fundamental test of any mathematician? It's quite clear that the conceptual organization of the subject is at least as important as... We could probably find some theorems that we could call LaVere's theorems if we wanted to. Yeah, sure, that's my point. It just pisses me off. And yet, data is not as important as Bill. Which is not enough, I'm not talking in for a second, but it just pisses me off like fuck when people regard this as the sole test of thanks very much indeed.

1:12:30 Daniel worked with a trivial, you see, what we ultimately regard as trivial. Ah, cheers guys. I think that's probably yours. Well, if it is, this is one that is mine. Oh, this is very nice. Here, here. And the many metaphors of me. Here. And drop some friends. Put it away. Formerly in philosophy. Yes, you went into philosophy. He suffered in the math department. We're all philosophers. We certainly are. It's an interesting question as to which of the people around this table is more a philosopher than any others. You're writing novels. I always liked them. I could test them. I could prove a few things. They weren't all that good. Yeah, I'd rather have played them. Oh, they were not bad. I was a... Yeah, but I'd rather have done something else. Surely. Sure. Play the violin. Like Heifetz. But you can't. Listen, I played the violin in my youth. I was... Very good. Terrible. But then... The violin is hard to make up for. John, I have to tell you sometime about, you know, the two greatest... You know, teenage violinists in my life and what happened the night they met, but we'll go into that another time. But they'll always agree on the Usher Heisman. Yes, you will. The only thing that you would study is the three-arm law. No, no, that was the great escape. You see, you had to have human bodies for which all these anxieties, you know, in other words, where you didn't have that competition. Now, when you actually practice something, then, oh yes. And then, you know, that's what I'm going to talk about. Have I ever read... Have I read Tolstoy? Yes, of course, but given the lectures on Tolstoy, I... It's interesting, because that's very much Italian, because I'm a conservative. Oh, sure, sure. Yeah, I certainly can accept that. That, of course, does play... That's very much the R.S. Murdoch view, of course. Have you ever read that wonderful dialogue, the mock dialogues, the Akastos, isn't it? Yeah, they're wonderful. Yeah, I have, but years ago, I should go back and read them.

1:15:00 I mean, they're ethicist dialogues, they're not, but they are good. But if Kant's whole philosophy turns on his philosophy of beauty, who knows? Now, this is the point at which we've got to be on our toes, because if we did, whatever. I don't know, it was the first time. Every man for himself, because if we try to take care of somebody, they're like, no. Late Greek skeptic, you know, could have turned around and... 230 B.C. and said, listen, nothing is ever going to matter at all. We're all so near the beginning. It's a beauty that you can see and I've long since ceased to see any connection with the outside world. But I don't think you're... I think you're lying. I'm sorry, it's a strong term to use. I don't mean to be offensive. But it just simply flies in the face. You fly in the face of everything that you've ever really written or thought. You know perfectly well that you think it connects with... What the hell are you talking about? When you talk about the continuum, why do you think, why the hell do you think it matters whether people think about the continuum on the basis of arithmetical... ...locality that I really have no fundamental understanding of, by the way. Mathematics, the thing I could really do, provides no explanation whatsoever. The illusion of total, fundamental, global understanding is something which has to be given up. It's just part of the maturity of human intellect. Yes, in that sense you are being six years old. It's not disgusting, because Aristotle is also 60 years old, and so is Spinoza. Not liveness, not Newton. Being told that you're more like Aristotle and Spinoza is hardly an insult.

1:17:30 But the methodological and foundational issues are so integrated, and it's... The point at which we're going to disentangle them in some framework, which even then will just be a staging post on the road to development, is a long way off, it might be. Will you use it quickly, because I'm more desperate than you. Oh, you join me. Well, no, I won't actually go that far, but do please hurry.