MW notes on topics for discussion / breakfast conversation (crucial) (contd.)

0:00 And then a little example here is the real number, the rational number, or the polynomial. So you turn the conceptual forever upside down, until the end of the 19th century, algebra is based on the numbers. You build polynomials, and you build a graph theory or whatever on that. You say whatever you want to say here, you go the other way around. You understand the numbers because you understand the structure that builds it up. This is, I call it, an informal, because you don't define what is a structure. You just do mathematics in this way. You have, let's say, the prime field or whatever, and the extension, and you ask what's inherited from here. For example, if you build the ring of polynomials of a certain domain, the properties that were here are transmitted there, these are kept, and so this is a kind of structural question. So what I say is that people in that very new paradigm, because he does it very completely, so they have to instill in a lot of the ideas. They build the filters, and lots of filters, and all these different, these are all kinds of concepts introduced by Cartan, I think.

2:30 If you go back to here, they continue with the idea into a broader view of mathematics, about numbers. Numbers are not that special. So, the question comes, if there is, because here what you're doing is, I, I, I, I, I, I, I, I, I, I, I, I, Is there a concept that can be compared to an underlying general support for all of these? And then they came with the idea of a structure, which is nothing. It's just a way to define what you call a Judaic operation or whatever. It's very ad hoc. The data find the structure and they don't use it at all because, you know, it's the first chapter, two or three chapters in the book on algebra, and one section in the book on topology, but it's just, you know, to connect. It's not really strong. And the next step is the... I have this impression, and I thought some others, but Cromer got more than that, that people say, like, possibly the younger people, Fair, or something else, why don't we try to put here category, because a kind of real basis would be more than in the basics, perhaps it can go up a few minutes.

5:00 The idea of overarching theory of structure crystallized out from all these specific concrete mathematical developments in the way that they set up for that theory of structure. They were attempting to, perhaps too soon, crystallize when the developments within the mathematical specific fields themselves had not perceived that risk. They do have the unifying of the realm of punctuality and punctuality. I mean, they do have, one shouldn't lose some with the point short, I think. The way it's just piled up in complete disregard of code. There are, for example, the theory, I think they call it the derived structure, which is basically definable cultures between definable categories, in some sense. They recognize this as a significant thing. Yes, this is a good example, because I think they recognize it by using categorical concepts. Well, they cannot just say it in the book. They know about the categories, so what do you do? They know about categories. This is a lot. They all talked about categories. They had already set this framework. Yeah, but nothing to do with it. Nothing to do with it. You could explain the kind of contingency of accidents involved in that. And the people who have written about what the people who've played throughout their life have been through. Right from the beginning.

7:30 We do everything using categories. Well, I think that would have been difficult not only for practical reasons, no, because they would have had to use, since they did have this idea of providing some kind of foundation for the whole thing, which is in the Cheverges Ensemble, the formal language in the West, it's a bit difficult to see what they would have used as a substitute, like for the first chapter of the Cheverges Ensemble, which is because of the kind of logic that comes out of it. Well, there's various levels they could have done it on. It depends how thorough a foundation they want. If they could really just talk about large categories without explaining what categories were, then they could have done it in a very simple way. Just explain how to make these various universes. I mean, there is this problem... You know, where these successive things live. Another would be to give up theory of universe. I think they might collect things as they were. Because if you have a book on algebra, or a book on topology, you don't really need, in order to study theory, a comprehensive theory of everything. You can just have the answer, like in Van der Berghe's book, right? Yeah. You know, Tom, a general idea, a really stable, final answer to the question of where do large categories live. Well, it hasn't been answered yet. But again, they have another ideological thing here, which was... Here is the book, here is the book that will stay for generations to come. So if I already wrote the book for generations to come, and now comes a new reader, is this a book to stay for generations to come?

10:00 They were all the time claiming that we are the French mathematicians. Now we know everything. Everything is clear. The axiomatic method that they considered was perfectly clear. So this is how mathematics works. It's not just how you write the O.N., but also how mathematics should be presented. Very clear in discussion, in fact, the very morning before you arrived and the previous evening, that that was not into your vacuum of the axiomatic language. Yes, absolutely not into the axiomatic language. But the B.S. fought against that. Yes, I know. That was very much into your axiomatic language. As for the thematic thing, you know, as a game of chess or something like that. Thank you for your attention. Thank you for your attention.

12:30 Thank you for your attention. These were the chief engineers of the project that was on the project. Thank you for your attention. There are several places in the book, for example, because I don't really think that MacLean is the writer of history work, but I think I've done it, and so I don't think I've done anything like that. It depends on the autobiography, but I don't recall that. Did he take a look? I don't know. Thank you for your attention.

15:00 Mathematical foundations of science and beauty have got a lot of content there. I get the feel that he's very much at the back of church. He's gone back? That's from the scene of... Yeah. Oh, but he's got that, isn't he?