Conversations on Set Theory as Fragment of Algebraic Geometry

5:00 I've been doing this now for about 40 years. The title of the archive is The Archive of Mathematical Sciences and it was started 25 years ago in England.

7:30 I took over running it about seven years ago and it moved to France. You are the editor-in-chief? Yes, yes, I am. Well, I'm actually more or less the chief cook and butler. I only have one person to help full-time, or two other people who are part-time. So we're hoping that may change because we've now been formed into a foundation. This will now result in us getting some additional funding, mainly from Stanford. The main project we have is to digitize all of the interviews and all of the other recordings that have been made over the years of other material when the mission has been given and to put them on hold for a while, because that's its main value, to be able to make it available. They could go back to 1971 and they include a lot of interviews and discussions with Girard, Degner, Grouillard, and possibly Kennedy-Pivot. They could go back to the first ever quantum gravity meeting, which was held in Chapel Hill. It's a huge amount of material. On the maths side they include the 1974 International Logic Collection, which is a huge amount of material, and all of the older ones for about 2000 were done.

10:00 I don't know what you think. Nowadays, of course, they're quite different, but unfortunately, they have to, of course, happen as well, because of the shortage of space for these things. Until recently, now, of course, they have very good memories of B. We used to have to burn CDs, almost straight away. So we've got all stuff recorded on CD, but we then have to put it back, re-record it into... This is a hard drive. That's kind of straightforward. The big problem is we're re-recording all the stuff we've recorded on audio tape. And digitizing data. And that is a huge task. It can be done. It can be done. You can look at some of the alternatives, because there is a kind of software now for computer and art studio restoration, which could be made up of 32 videos, through digital samples, so in theory... It would only require about 1,000 students, which is a 3-year project, but the practice is going to be longer, because a lot of the older students are not going to be here about this, and you would need an engineer to actually design the digital samples and listen to them in real time. So you have to make a selection. The stuff which is, if you go to shape, which is just really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, The manufacturers of the recording state lowered the standard of the manufacturer. There was a big problem with the thickness and durability of magnetic coating and other things, just in order to save money, because in places like Taiwan, in the old days, the 1960s and 70s was about there, and companies like that made all the tapes. They made them for the European market, and they were made with pretty high standards.

12:30 We've got lots of cheap, ancient imports, which were being produced at a much lower price than the standard manufacturers, so they dropped the actual standards of, as I say, the European figures, which I think don't be very sharp, as a result of that, a lot of deterioration. So it's a big project. We need to put properly about 150 euros in Europe to carry that. So we're hoping that we'll be able to carry those whole pieces out in the chair of the Trump, the Stanford, the PI. It's going to be pretty impressive. In the meantime, I'm just chugging along, as I say, recording stuff myself and cataloguing it and putting it online whenever we get a chance. It's coming forward, I'll say. Sure. Oh, yeah, sure. No, because I was just going to go and talk to Bill. I was going to take an interview. Well, if you... Seriously, not... Well, you're extremely kind, but are you quite sure you don't want to... That's okay. I mean, one of the invited speakers arrived yesterday night. Uh-huh. And you haven't seen him, right? Oh, okay. Why don't I keep this in case you... No, no, no, use it, use it. Okay. Thanks. Thank you for your attention. Thank you for watching this video, if you liked it, please leave a like and subscribe to the channel, and don't forget to share it with your friends on social media.

15:00 And I just need to sit down for a moment to put the right arrows in the right places to make sure that I get things in the right direction. But because everything is in general, I can't focus. Thank you for your attention. Thank you for watching. Thank you for your attention and see you in the next lecture. Thank you for your attention.

17:30 Thank you for your attention. Perhaps I'll just have a look at that after you've eaten. I saw from my email yesterday, which I was able to access, that I got an email from Mark to say that our friend Rodin... ...has taken on himself to invite Baez to give, surprisingly, a series of lectures in Paris. There's an old principle, birds of feather. Well, Mark was extremely annoyed because he'd done so without consulting Mark. And Mark had made it clear that he didn't want them to take place in his Categories en Physique seminar.

20:00 And so he's now arranged for him to talk at the Ecole Normale instead, which is even more... I'm sure that a loony will parrot at his feet. Impressing. This is apparently all part of his big propaganda sweep through Europe before he comes to give these lectures in Glasgow that I was telling you about. Oh, yes, yes, yes, yes, yes, I mean, he's actually the official. Yes, that's exactly what I was saying. This was why, it was one of several reasons why Mark was extremely annoyed that Rodin had taken on himself to do this without consulting him. Mark is one of the people leading the battle against Templeton's involvement in, attempted involvement in the new funding system for research groups, especially in cosmology and astrophysics. It's being brought in under the guise of this Sarkozy reform of the CNRS, which in fact is nothing more than the destruction of the CNRS and its replacement by a series of effectively semi-privatized consortia. It may be that the Constitutional Court will hold that they have acted beyond their powers in doing this without primary legislation because the CNRS was set up by the Senate, was set up as by primary legislation, it's not clear whether they, although of course the French politicians tend to believe in the droit administratif, they think they have the right to do anything they want to do simply because they're in office. But even in France there is at least some sort of legal restraint, and it's just possible that the Constitutional Court may declare that the Minister's action in suspending the CNRS, if they haven't actually gone so far as to say it's been abolished, that it's functions are suspended indefinitely, pending the reform, but they may rule that they've acted beyond their powers in doing that.

22:30 The strategy is very clear. The strategy is to break up the CNRS, to have a series of more or less ad hoc sectoral research consortia, which will be expected to go out and find a mixture of public and private funding. The public side. Mainly from a closer liaison with the universities than the research groups in the CNRS have had before, and the private side, of course, from, you guess where, I mean, obviously Temple is one of the people already, six months ago, as Mark was saying when you were in Paris last year, they had actually sent somebody to his laboratory to approach them, to soften them up. I'm rather surprised that he had such a powerful, principled, and fixed objection. Of course, he's only one guy, and he can be pun manoeuvred or overridden, especially with the kind of power they've got. But for that reason, partly also because I think he shares your estimate of Mize's work, having studied it after meeting you. He was outraged that Rodin had done this thing, just inviting Byers to give a talk at the seminar at the astrophysics lab without apologizing to someone. You mentioned it to me that there was a conference about beauty. It was called Deep Beauty and it was in honor of Ben Limerick.

25:00 Dias was the main speaker. That's interesting. Do you know who that was in the physics department? It would be interesting to find out. And interesting to know what they said. Some blogger was saying, well, he was characterizing, he's from the history of ushering in the Templeton Prize, you know, so he, I forget how he phrased it, but he had to express his opinion about this, and so during, he entered into this discussion. Oh, this is terrible. This is... To suggest that I actually might have... This would be a... But I think you may recall on occasion when we first met, George Rousseau, my father, advised me... There's nothing wrong with a good ad hominem attack. You've taken the words right out of my mouth. I'd forgotten that George Rousseau said that. But those would be my sentiments too. Sometimes, you know, there's nothing wrong with an ad hominem attack. We have two scientific organizations. People proclaim the goal is to infiltrate science so that we can use science as a subgrade to anti-scientific propaganda, and is that worth saying? A person who did that? Is that ad hominem? Well, then so be it. Sure.

27:30 Thank you for watching. So this was, somebody was giving the opinion that I said. Well, he is Chris Hysham's student, so I'm only going to make allowances for loyalty taste. Loyalty, yeah. I missed his talk, but I must admit I, well, because I had to go back to the hospital just to have some more. I'm going to take off the neck brace that they've given me instead. I would be very interested in your reaction to Ross Street's talk of a couple of days ago, the one that just preceded your own talk on extensivity. I didn't really have much of a reaction, I guess. Maybe I was just worried about my own talk. Did you not like it? I was very interested by it. It seemed a very interesting piece of work. He certainly seems to have focused on some of the right issues as to the role of Frobenius in construction. I'm not quite sure he's actually got to the heart of it. But there were a couple of things I wanted to ask you about the, um, I don't know if I'm going to have time now, um, one was about the extensivity of the talk.

30:00 No, not a mismatch at all, just something that just left me, I'm sure entirely through my own lack of understanding, a little less clear than I thought I had been after studying the three papers on adjoint characterisation of quality that you sent me a while ago, from the JOIL meeting and the, which... Well, it is five years, of course that's astonishing, and I learnt an enormous amount from these, also particularly about, well, all sorts of subjects, but the extensive quality and the characterisation of homotopy through the coalescence of two of these four adjoints. Thank you for your attention. Very, very illuminating. And also clarifying enormously this insight of Grassman's about the characterization of intensives as almost equal and extensives as, you know, retreatably unequal, which was immensely clarifying. I want you to place within the setting of algebraic geometry in relation to those three papers that I wasn't sure whether they any longer made sense after hearing the remarks that you made in your talk two days ago, but let's… There are so many questions about that talk, particularly to do with the Schaniel Dimension Rate, which I would like to understand better. The shanty dimension they bring to lie in the core variety is that all the objects be separable, and in the case where the objects are actually decidable then that's a...

32:30 Separable, decidable means the same thing. Exactly, in this case it means the same thing. Exactly, it's the same, it's actually the same thing. But it's part and parcel of that point about the unity of conditions to be separable, decidable and unramified. I'm not going to be able to formulate any really penetrating or useful questions like this, but just looking at this issue of the disjoint covering topology, the existence, for instance, of an appropriate size. I don't want to particularly question the side of the science. Can you say something a little bit more about how all of this relates to the general program for, let's say, reconceiving set theory as a fragment or rather as it's assumed within algebraic geometry? That was the thing I mainly wanted to ask you about in the little time that I've got left. I mean, here at the conference. I don't mean... You know, I tried to kind of check out, you know... If I said that, it might mean something different. No, no, no, no. Well, I hope you're not. No, I mean, it's a little time that I have left today to be around. I'm actually going to stay in here and talk about the continuum hypothesis this afternoon, I think. Yeah, actually, I hope I can stay awake because I'll make several of these talks by young people. Yes, I want to stay for those as well. Oh, I'm sorry, keep taking that. There's another bottle up there if you need. One of the good things about putting your neck in a brace is that if you do fall asleep in the talk, you... You know, it doesn't show because you get like this. Your head doesn't roll over like this. It's crazy. I had quite a good snooze a couple of times this morning. Anyway, Scepterian algebraic geometry. Well, by Scepterian, okay. Well, I mean, I don't know what you mean by this. One point is that the role of the truth value object could be thought as a kind of quantity. Yeah. In that sense, you have maps from x to omega functions that actually form a grid, even in distributive lattice and so on.

35:00 Sometimes this relation to distributive lattice is generated. So, in that sense, it could be a good thing. I'm not quite sure what I'm going for there. In other words, Boolean algebra, for example, is something like commuter algebra. Yeah. Maybe commuter algebra. Yeah. In that way. Centrally, Boolean algebra is something like that. So, yes, obviously there's more to centrally than just explaining Boolean algebra. There are many aspects, that's one aspect simply, that you can carve out, just as you can have varieties defined by all the equations and see if one of these are real or not. You surely can have some objects defined by gradient equations or hiding equations. And in some cases, these kind of figures, you can go back and have figures of shape omega. Right, right, right, right. Sub-object. Let's say, just to give a rough idea, it might be that all the sub-objects that find the powers of omega actually generate interpolated figures. Ah, now I understand. So that would certainly be a fragment of algebraic geometry. So it would be all from the same set theory, all the same. The other thing is maybe what you mean by this is this idea that the analysis of cohesion is really the dialectics, cohesion and not cohesion, as expressed in the axiomatic collision. I talk about topos, but it's always really about the topos and ovaries are the base. I mean, the one action which really guarantees that it's less cohesive than the other one is the fact that the components of the truth value object is one.

37:30 The truth value object is connected. It couldn't possibly be two of its own isomorphism. It really makes them different. Theoretical theory is phrased in terms of a gibbous adjoint there. Perhaps a strong axiom is quite a bit different, but there's also the question, can you construct, as, you know, in principle Cantor did, the idea of discrete is sort of extracted from existing cohesive concepts. I guess it's the same thing that maybe you would call the arithmetical. Yes, yes. Well, it's the case, it's the extreme case. There's a complete absence of pedigree. Yeah, yeah, yeah, yeah. But then, you see, we always... One object can become another in a completely trivial way. We always relativize, you see, so it can be qualitative, perhaps, which is not total absence. Yeah, sure. And this is another way in which algebraic geometry comes in, because algebraic geometry over a field is not algebraically closed. I say it should not be thought of as being over Cantoria abstracts. Yes, I was going to say, where of course the Galois construction reveals the point that points are hardly ever, I know you actually said never, but there is the extreme case which Cantor did extract where they are, but hardly ever merely points. It almost always is this Galois, which is the whole thing which Galois theory gets at. The fields are qualitatively more discrete than the general rig. So, you know, intuitively it really is a big qualitative difference, even if you don't put the total in that section. So that's the second remark.

40:00 But then, yeah, how do we define it? So why, why, given the classifying purpose of rings, why should the fields... So, this is another part of the equation, which is that there is this equation, s equals s to the power d, s equals s to the power d, s to the power d, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, Given any Cartesian closed category, you can define this relation between two objects, S and D, that the inclusion of S as the constant of S from D to S is actually a nice one, that the space of the general maps from D to S is the same as the space of C. All maps are constant. So, intuitively, this says in a relative way that D is very cohesive as opposed to S, where they're anti-cohesive. Yes, non-cohesive in relation to S. But it's very cohesive. Yes, yes, yes. So... One also, of course, has a whole aspect of what you call the screwing up and down of the various levels of determination of cohesion or lack of cohesion, which is in there in the... Use that to... But this is one way, given the right conditions on D, even a single object D determines the whole so-called discrete category. Well, it's clearly going to be a reflective subcategory when the reflection preserves products. Yes, because it's a... Because the intrusion is obviously preserving external changes. Yeah. S to the power x to the power d is equal to x to the d to the x to the x to the x to the y. Yeah, so it fits very naturally into the framework of world relationships between categories of space and categories of quantity in terms of...

42:30 Yeah. There are a number of different types of products, and I could distribute over all of our isomorphic tools. It's going to be like a subtopo. Well, for example, if V is one of these atoms, in fact, that's how I write it. That's the notion of an ATOM. Yeah, well, once you discover that there could be such a... Outrageous thing isn't it? The thing is, this extra right adjointity, so the maps from x to the d into y are the same as the maps from x itself into y to the 1 over d, which is the actual exponentiation. So this, what I just said, this 1 to 1 correspondence, is an abstract adjointness, but you'd like it to be a strong adjointness, so that the space of these maps, the isomorphism of the space of these maps, can be defined. If that's only true, if it can't be true, absolutely. So the space has to be a discrete space. This defines the notion of discrete. In other words, that adjunction, that adjunction is strong relative to some object s, if and only if s is discrete and not the previous discrete space. That brings to mind a very interesting characterisation of Grassman's discussion of continuous and discrete that you had at one point in those qualitative papers on extensive and intensive quality. I think you used the example of trees in a forest. Oh, yeah, yeah, yeah, yeah, yeah. Well, it's a... Perky liked that one. Yes, well, I did too. Yes, that's very interesting. How do you think of measuring qualitatively? How strong is the adjoining this? This right here is a bijection.

45:00 Given that in categories of space, one typically has this construction which allows you to measure things in precise... Well, no, this just generates the way of measuring. In other words, to say it was really strong. You could say that y to the power of x to the d is a nominal isomorphic to y to the 1 over d to the x. Well, this is not true. But then you could say this would be total strength. How strong is it? Well, there's a comparison map between these two things. We have a bijection here for every one of these D's. And if you just calculate, the answer is that S must equal S over D. So, in other words, if this is taking place in a certain category, then you define S this way. So then, if you have any questions... Given the grounding, you define the enrichment, the E and S, as a, you know, it's just a programming order of the actual E and S, but in this case, this is the, though this is always the maps from the individual S's. So by this definition, POM, we get that the strength is stronger than that, but weaker than that, because it's saying that indeed E of X to the D is Y, which is isomorphic to E of X Y to the Y to the D, but this of course is in S only, not in E. So that's the answer, how strong is it. So, in the process you've defined this thing, which is going to be useful for all sorts of other problems.

47:30 Yeah. Maybe. Somehow, in other words, somehow this is not as smoothly parameterized as it would be a phasor. Now to show that this is a topos, maybe it is a topos. There are some small things to be proved, but in good cases this will be, if we have this next year, this will again be a topos and this a topos, in fact one that has all four of these adjoints. Right, I was going to, yes, you say it will have the whole range of points, components, discrete and... Right, right, right. The further. The trees, the wood, yes. And then the, go to speed, the power. The unity and identity between discrete and co-discrete, which is trivial, but it always impressively is. Oh, it's tremendously impressive. Two copies of abstract sets are actually totally different. That was the thing which really first got me excited, the incredible power of the notion of adjoint functor, because it completely clears away all of that massive confusion. There's a confusion that led Zermelo to, well, effectively to distort Cantor's saying, to actually suppress his text because he couldn't make sense of it. That's right, that's right. In the same way as Grassman's editors suppressed the bits that they couldn't make sense of. Yeah. It's absolutely fascinating. Yeah, the same with Piano, who was acting as Grassman there. Well, Studi appealed to Bjarne Didi for clarification, I don't know the story. Piano claimed to be the messenger of the gospel in Italy, so he made it completely incomprehensible, because he couldn't understand it.

50:00 Piano is, I think, someone who definitely believes in St. John's. Well, in the beginning was the logos. drink is the logos yeah yeah when they translated that word yeah exactly yeah so they really believe this piano was why they really believed it so it was actually in the introduction to his position geometry that he introduced the piano axioms i never realized that was in that that's extraordinary irony isn't it given that given the grasp i said explicitly arithmetic He does not need axioms, because he viewed the natural numbers in modern language as the natural operations on endomorphisms of objects in arbitrary categories, in arbitrary, you know, totally, you know, the idea of iterating an endomorphism in a particular situation has a lot of structure. There are thousands of objects in all possible categories. You can only get a natural number, but you see this is an objective definition. You're not building it up at all. Yes, so particularly by making it a kind of categorical invariant in that way, or not making it well. I mean, Grassman's whole philosophy is completely different from Piano's. I don't know what his position was, whether Grassman actually wrote anything directly about the infinite, but he certainly had absorbed Hegel's distinction between the... The good and bad infinities, which Piano never seems to have got his head around at all, and certainly he wouldn't have readily accepted a discreet and complete infinity in the way that Piano did it. I think he would have been more concerned to distinguish the wood and the trees, dare I say, precisely focusing on the continuous and the discrete.

52:30 When they were actually measured before metrication, they were known as hopos feet. A Hopper's foot, H-O-P-O-S, within one letter, Hopper's foot was the unit of volume, it was the cubic, a cubic foot of timber was known as a Hopper's foot. I don't know what the origin expression is, but it would be so lovely if they could just have changed the...