Further discussions, incl. FW Lawvere & Colin McLarty

0:00 It's quite active, you know.

10:00 Oh, yes it is. So, Suleiman, and... Well, you know, he's still advertised to be prime minister. Well, that was a little later admitted, but... To be one of the two chiefs of... And yet the curious thing is that in his prominent career, he had the reputation of... Okay. So... So... So... They should volunteer as colonists in India, in Israel, in Palestine, and the only reason Salisbury undertook, apparently, to try to do this was because the Sultan wouldn't agree.

12:30 Yeah, of course, because it was still a prefecture that was nominally held as a regional prison. So, Oliphant actually retired. I must learn more about all of that. I think he's very significant. Yes, he's really only a name for me. I must learn more. He traveled with Lord Hogan to Japan and many places. Ah, is that the old man that destroyed the Summer Palace in Peking, that looted and that sank Peking and destroyed the Great Sun? Yes, I think so. He was the grandson of the one who stole the old marbles from the harbor. Yeah, he's a very... So he went to China, Japan, and many places on this voyage, and I have to, I might really wish to complete the book that I started some years ago, and then he set up this thing, you know, and this, with an American This religious snuff community on the shores of the Atlantic. That's right, that's right. But then, you know, he was still active enough to get this idea. He set up a U.S. colony to protect the current compass and travel there. And, of course, he was already a friend of Salisbury. How does that happen? Some religious snuff, so-called religious snuff, is a friend of Salisbury. A friend of the chief of British and British. Exactly. The practice of the British Empire when it was actually at the zenith of its power and territorial empire. And of course, they went on to reveal at the time about, of course, this is really a very beneficial project because it's all about globalization and the completion of the world market. In fact, even some so-called progressives like George Belichon were sucked in, believing that it would be the nucleus of a humane world government. The Webbs do, and people like Adam Fabian sort of. And Russell, of course, the young Bert Russell, I can go to the corner of a slide at ease.

15:00 John says it's an OSCE. Oh, that's right. They figured that out. There wasn't another name. It existed, but under a slightly different name. I think it became the OSCE in about 1900. And there was certainly Russell's lecture there in the 1990s. Yeah, yeah. It wasn't wrong. No, no, it wasn't wrong at all. It was just a minor point that it was then called something different. But it was the same institution in the same location. Thank you. Oh, yeah, they were very keen on Bernstein. Yeah, Bernstein even was there. Oh, yeah. Bernstein, look. Oh, I've got the sales report on my own, so I never realized that. Yes, yes, yes, yes. Bernstein was there. Bernstein was there in the mid-90s. I don't know how long. Part of his political capital in the continent was that he was allegedly a friend of Engels. ...mankind was the victim of this conspiracy to write the history of socialism without consulting him. You know, he had to cut wind of it and protest it with, you know, what's his name, you know, the various division of state. No, there were several of them involved anyway. We won't get off to the next topic. And we're going to write the history of socialism. They weren't going to, they didn't consult Engler at all. Obviously, the leader of the world social movement was not consulted. They went around with him. So, I mean, revisionism wasn't just an accident. And of course, Russia's got its own principles of the other social movement. And all those people that you're asking about, all the callers, the socialists, the economists, the philosophers, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web, the web,

17:30 They all look like one well-known joker. It might or might not be a good thing. I don't know how to say. Having an international category meeting in South Africa in itself would probably be a good thing, but it was organized by then. When you say the others, you mean some of the other Georgian categories? No, no, no. Are we ready? Yeah, I am. You are. Go ahead. For me, the first entrance, if you please, is to take your... but after the lobby, is that possible? Two entrances. Two entrances for me, not one. That's okay. That's okay. Thank you. What was the second one? The one, the second one I was going to have was the lobby of this very thick bean soup, but you probably want that as a starter. Oh, soup, yeah. No, not as a starter. Do I bring them all together or one after the other? No, one after the other. You might want something a bit more filling than that, and that's just two starters, but see what you feel like after you've had a good one. Would you be okay just with a bichet of the house wine, or do you want something a bit... I don't think you guys are Rosé fans, but you have some kind of. We've enjoyed it. But you probably feel it, right?

20:00 Now, I'm very impressed with this, what they call Galois theory, I can't believe it's called Galois theory. I still haven't figured out, I've got that book, I've tried so many times to read it. I don't know where it's going, I can read, say, the first two chapters, which is stunning. I can read the last chapter, which is about topos. This isn't my particular sort of construction, but then the part in between supposedly explains what this Galois theory quote-unquote is and how I just somehow my eyes glaze over whenever I try to read that part. I still haven't got it. If you haven't, I sure as hell won't. But this has nothing to do with the so-called geometric Galois theory. Well, there's a big book called Geometric Galois Theory which contains a lot of references to and also some of the aspects of what Angus was talking about. Well, it certainly doesn't take any references at all that I could see. Well, I think that's the issue about the absolute Galois group, which is to understand the absolute Galois group using the central formula among other things. And I have never understood what results they've got in their line. I've read some introductions of how they get the results, but I never got far enough. But they claim to be following out the program that Grotendieck laid out. And it's a very clever fact, and it depends on a deep theorem of analysis. But what it tells you about the absolute Galois group I have not yet discovered.

22:30 But it has no connection as far as you know with what these... Sorry, I can't tell what they're doing. No, well I said they can't. I'm not going to say what it has to do with it. Oh, okay. Observations about foundations, about what he thinks is the significance of category theory for the formalization of mathematics or the understanding of the world. Considering he has such a totally weird ideological gender in every other direction, maybe that's a blessing. I guess he's going to explain how Groton wants to represent Galois theory in terms of categories of syntacton by the Galois theory. Which, if he does, that would be good, because then I'll understand it. Yeah, except it's not that, or maybe something similar to that is another final result. Okay, well, that's the best part. Other than that, too, I don't understand. Yeah. But Deligny had comments on that in SGA 4 1⁄2, so there must be something to it on Grotenbeek representing, doing Galois theory in terms of somehow representation for the Galois group. Oh, yeah. Yeah, and I would like to understand more exactly how that's done. Well, the first three or four chapters of this book, which is called The Inventor of Galois Theory, which is in London Mathematical Society, it's all about Groten laying out this program and where it stands now. Well, I thought it was called the Eschustan Program. I thought it was about Desquiesc and Pagal, for the paper, but it's not about that. That's about something else altogether. Yeah, no, this is something from the 60s. The sketches. The Desquiesc is something he wrote in the 70s about a lot of other things just not on the phone. This Galois theory in terms of root group representations, which is standard, for example, in elliptic curve theory now, wrote in the 60s. Okay, but what is the subject matter of this, of this, of this, of this, of this, of this, of this, of this, of this, of this, of this,

25:00 I guess it's more or less about automorphisms, about algebraic automorphisms of Riemann surfaces, well, automorphisms, somehow the automorphisms are classified by finite paths on the surfaces, so then this is all algebraic, this becomes something about the Galois group of some field extension. Related to the periods on this surface or something. I have not understood it. But by now I can say I've looked and looked for the results. I understand how important it would be to understand the absolute power group. And this bears on the absolute power group. But whether it has yet given a result, I don't know. Well, I don't understand any of that. Is there any chance at all of giving me a glimpse of why it would be so important to understand the absolute Galois group at once? Well, because it's all the Galois groups at once. It's all the algebras. So it's all the algebras. Okay, so we're just going to use all the algebraic numbers in some way. Or is he studying that atomic boolean topo? That is the object. It consists of representations of the gawa. The gawa group by E. Simon Pennington is the base of the gawa group, the actual gawa group. So, representing the group, say you want a linear representation, well that's just an abelian group in this topo. The element of that, there couldn't be categories. So I don't see... Well, I mean, the new part is this functional analysis and classifying them by graphs and Riemann surfaces. Yeah, oh yeah. But the fact is that a graph on a Riemann surface doesn't look a lot like a Galois group. No, but then we define an extension and we associate with an extension, you know, the rational.

27:30 So with luck this book will excel. Oh, that's your... I can tell us if you do like it or not. And we have a... We have a... Do you eat this with a fork, it looks like? No, no, well, you could. You could, but I would recommend normally I eat it with a spoon. Then you're right, you can eat it with a fork. Good hang on. Maybe it's like Chomsky. It gives you away the total winters you're trying to get with us. So if this book will go through how you actually do Galois theory in terms of category representations, well then good, I'll look at it from the front. I mean, I understand the categories encode the same information, right, so I don't see classical Galois theory in those terms. There's a paper by Speer in Masterich that we could put some kind of scale of q and quant of q.

30:00 Well, I mean, I know the phrase, but I don't know what it is. That totally blows your mind. Incredibly, incredibly deep stuff. Yeah, yeah. Of course, fine. What is the, what is the... I didn't quite catch what you said that appears in this paper by the staff. It's not one group of irrational numbers. Well, that's the title. Dozens and dozens, not hundreds of pages. Very, very deep and detailed stuff. And somehow it's a lot more than if you just knew the classic Oahu theory of... Each algebraic number of fields, and the relations between them do a lot too. Well, to be honest, I think I'll just stay home and try and get my head thoroughly around your evolution of naturality. And try and understand a little bit more about... Curvature then. Yeah, that'd be good. That'd be a good starting point. I'm not going to try running before I can walk. I'm afraid this is way, way beyond me. But I think I can see why it's important. I don't know. Unpublished, rejected subjects that result in complex analysis that Belli grew. It's Belli's theorem that links the graphs to the Galois representations. And so Belli, I guess, proved that Grotendieck had complexed it. I'm not sure anybody knew that, but Grotendieck saw the link to Galois theory.

32:30 And it has something to do with O-minimality kind of stuff. It has something to do with... Only getting, well, only being able to find graphs on your Riemann surfaces because of the thinness of your means of description, which then matches the problem of algebraic extensions, even though you thought you were working with continuous Riemann surfaces, but you really didn't have arithmetic in there. It's really, it's really a way to continue some of these theories, you know, do you understand why Fry introduces this cubic curve, you know, given a solution to the Paramount Hill, and you use that solution with the coefficient, you know, anyway. I've never seen any explanation of why that's a good, why that's something you want to do. No, and see, and this is something I think even Eligwar is right. Try does it because Eligwar did it. Eligwar did it because he'd been working for decades on a list of curves. And he got thinking about the Fermat equations and for some reason tried looking at this curve. He looked at the curve and he immediately saw it had very odd properties. Properties so odd that he thought there couldn't be such a curve. Then Frey crystallized that as, it's not modular. Again, a conjecture. It's not modular. But Eliguark, just based on decades of experience with elliptic curves, tried putting the coefficients that way and could see that this would be a weird elliptic curve. Huh. I mean, you know, there's no explanation for this move other than that it would work. I'm nowhere near expert enough on elliptic curves to follow it into detail. I don't know whether he's somewhere given an explanation for people who really do know that much of exactly what makes him think it's a theory or not.

35:00 But this book he wrote, it starts with pendulums, goes into contra-integrals, goes into number theory, and he's completely immersed in the whole subject of the elliptic curve. And I think he's considered something of an outsider, someone who paid too much attention to elementary properties before it occurred, and so to be considered a serious mathematician, you know, he was all, knowing all these things about what occurred, that anybody could figure out what's in here, but nobody did. But it led him to this observation. Isn't that something to be said, though? A lot of work can be done, perhaps should be done, without using the concepts of the general level of abstraction, generality. There's mathematical botany as well as... And you have that interesting or something very odd kind of observation. Who would have thought? He knows why that's so. Identifying zero and one on the lines to get a cubic curve, so you can imagine some other number theoretical error can be used to find this quotient scheme, and maybe that's what it is actually.

37:30 If the Fermat equation is actually true, what happens until it is true, it still is, but it doesn't just go away, does it? No, if the Fermat equation was true, it would be a non-singular cubic, but not modular. Yeah, I mean, the fact that it's a cubic curve is independent of whether it's a non-singular or not. Well, then there isn't any of them, are there? I mean, that depends on three numbers, A, B, and C. Yeah. Yeah, I mean, you've got a general cubit. Yeah, right. And its specialization to solutions of the money equation would not be modular. So they can say there is no such specialization, because there's no solution. The concept of algebraic methods and the habit of thinking of a given term is to encode a pair of numbers and register properties of them. When you saw, okay, or tripled them, you generally said, okay, well, if I encoded a couple of numbers, what would it be if you encoded them as, you know, yeah. You know, that would tell you why you get a cubic curve rather than something else. You just formally plug these numbers into them in general.

40:00 Thank you for watching this video. The emphasis on cubic curves is more than just Well, yeah, for a lot of purposes, it's not really cubic curves, it's genus-1 curves, which includes non-singular cubics, but also... No, I'm not going to say that you're walking or drinking now or something. Okay, let's go get some more water. That's very nice. Sorry, was that okay? I mean, it's not everybody's cup of tea, is it? No, no, no. I looked at my laptop. Yeah. Yeah, I mean, when you want to know rational points of genus zero occur, well, that's easy. They're projective lines. If they have any points at all, they have a rationalized projective line of points. You know, of course, there is a question of whether they have any at all over a given field, but that's a panel I think pretty thoroughly known, but it's just an exercise to completely insert other rationals.

42:30 So then it's natural enough to look at genus one curves. Genus one curves? Well, a lot happens. A lot of elegant stuff happens. The rational points form a group, but are not a finite group. Now that took some proving. I don't have the proof for that. The proof that it's a group is either a long calculation or using a rock. But should it be a finite group? And there really, for a lot of purposes, is genus zero, genus one, and genus more than one. Like they say about Australians, they can only count zero, one, and many. Yeah, yeah, but I'm sorry about that program. Whatever it is, it's hard to tell the truth. A lot of the kinds of things people want to know about curves, everything about genus one is like everything else about genus one. It's going to be, you know, in characteristic zero. You know, precisely not in the Bay conjecture. I was going to ask you how does all this line of work on this more concrete, specific topic relate to the, how does it relate to the, well, the stuff you've just been talking about, the treatment of, the fact that I didn't tell, and Angus wasn't overly explicit about this either, which is pretty clear. The first answer to the question, what does it teach you about the actual rationals once you know the things of the addicts, is nothing but some of this. Oh, it's a mark. Oh, it's still some left. To a first approximation, knowing solutions over finite fields tells you nothing about that. To a second approximation, everything that makes you a smarter person helps with everything you do in life. Then there are possible bridges, but these are going to be very subtle. Approaching by a while of oscillations.

45:00 Yeah, a while of oscillations would be the general in particular. But there are a lot of topics in life one has to approach. Hopefully not too while of oscillations would be the general in particular. But apparently the more particular, when you really get to the nitty-gritty of can you ever solve a rational equation by knowing something about the Theatic equation. The answer is, yeah, every once in a while you know a lot and you're lucky enough to know a lot of that. Well, I mean, there's this idea that you take the ultra-product of all the finite fields and you can still get the solution. Yeah, but the solution's there. But there's nothing to describe it. Yeah. It contains irrationality. Yeah. And you might say, well, I mean, in a way, that's what doing a field study was for taking a pietic solution and showing it wasn't that... So, Delaney got the Fields Medal, didn't get the Fields Medal for his work on... In the past, you could use a topo-syllabic term. Oh, no, hang on. I'm sorry. I'm getting mixed up with that. That's good. Sorry. Sorry. Cancel remarks. Well, who? Okay. But Delaney got involved. Well, you get these numbers that were integers. Well, they were algebaric over the integers. And if they are, they have the right absolute values to prove the Riemann hypothesis of finite function fields. Well, they are algebaric over the theatrics, and Delenius spent a few years proving them. In fact, they are algebaric over the rationals. And this has gotten a real name, and it's a long summer proof. It would follow that from a much... Much more general principles expressed in standard conjectures, if one knew the standard conjectures. Cartier is casting doubt on their truth. Angus, I think, just doesn't see that there was a breach any time soon.

47:30 And this is one point where I think it won't do to say, well, Broglie forms results. He says in the book of tonight that he would rather be Forty Hill preceding the Bay conjectures to the standard conjectures than had to prove by this Amazon-Ton method that he feels doesn't go to teach you anything because... Well, you can follow it if you want. He says that Savoy-Dauvigny has not done any great work since. I don't think that's a sentence you could easily say out loud in French. This is why I say, who is Savoy-Dauvigny? Savoy-Dauvigny. He actually says that it was a mistake of Dauvigny to go to France. I think there's a pretty general feeling that Zelenya has not done anything since the bake injectors on the level of what one hoped for given the bake injectors. I'd like to, if there's any at all, I'd like to get a feel for this, because obviously the technical details are not quite as good as I'd like them to be, but I don't think people are keeping things secret. I think of myself as just not having the equipment to follow you. But on the other hand, I know you're a fantastically good expulsor, and perhaps you can give me the essence of things. What is it in terms of the kind of unit of communication that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, You know, I think there's a difference between natural proofs and ad hoc proofs on a low level. This is sort of like this on a much higher level, you know. Yes, I've got, I have got the aspect just from listening to what you and people here have been saying about this, that I'm going to get across.

50:00 If you were writing a really responsible piece of... Exposition for a mathematically educated general public would be intended for a general audience. Well, I will some day, but I'd like to know more about it myself. Yeah, okay. You do just need to know a lot more before you know how to set about it, explain the issues. I think so. Very well. Yeah. That's the right thing to say. Well, I think a lot of the project is to understand the standards of the lectures. I can certainly see why it's tremendously important. If I come up with a proof of the model. I won't. Because it's all goodbye. But even failing that, it would be good to understand the newspapers, which are all motivational, because that's all there is for them at present is motivation. You know, I mean, actually understanding the concept of the static conjecture, that's quite what it is, what was to be, but it is not that. Well, yeah, that is. Oh, yeah. Now, what Angus says it comes down to is showing that somehow, I think maybe as long as they're all simple zeros, that you need trace. The trace of, well, specifically the provenience, is the number of fixed points, that your cohomology, your cohomology product, your theory of cohomology products, is in fact an intersection theory, because you can count points by intersection, the trace, unfortunately, is the cohomology, if you knew the products of cohomology measured intersectionally. Then what you compute by the trace of the venus would be a fixed point. Well, a fixed point total would be a fixed point here. It's not expressed anything like that, it's... Yeah, you think that might turn out to be the way that you would express it. Well, it says it's behind it.

52:30 We could be quite sure of the term. What intersection theory is, somehow to me the whole thing is very unsatisfactory in the beginning, because one assumes that that's the right tool to use, so the conjectures are formulated in terms of this, oh well, highly complicated machinery, which may or may not. In other words, the machinery itself is an answer to an unstated problem, an unstated universal problem, whether the machinery is or is not, in other words, on a much higher level. When you work on this present stage of knowledge, you're just sort of following instructions that you don't understand. You can develop such an incredibly sophisticated understanding of the instructions themselves internally, but... But you can't state what it's for, where it came from, but it won't become a universe, you won't get a clearly articulated universal property, a solution to it, unless you like the universal, faceable delta function. No, see that's already, that's already an abelian, the context, you're talking about abelian groups. Who knows if that's the right way to talk about these geometrical problems anyway? That's a traditional way to assume. You just start working on that without knowing why, usually. Well, Planck-Rey already doesn't know exactly why he's pursuing these cycles in homology. He sees it in surfaces that have a very plausible geometric view. He sees that he gets theorems on them. It's not, you know, not perfectly happy with the state of theory. And then Lefschetz, by the time he comes up with Lefschetz fixed point theorem, it was a problem from the beginning.

55:00 You knew that in some very nice cases, so let's just take a point there and point to exactly what he seemed to tell you, but only in some very nice cases, you knew it was telling you something about everything, except you don't know exactly what, and already for us, that's a problem. People have been working and working on that. I mean, obviously, I'm a complete outsider, but the thing which strikes me partly from what I've been listening to is that There's an aspect of development of mathematics which has not been made sufficiently explicit, which is that cross-fertilizing heuristic perspectives on stuff that you do understand, or think you understand pretty well, lead on to... To bold structural conjectures about you-know-not-what at the time you're making them, unless you're a genius of robotics, and it reinforces Samy's point that the foundations to the extent they ever come into view at all. ...crystallize very gradually as you work from the inside out, but as I say, it's that kind of cross-virtualizing heuristic perspective on different structures and then very gradually seeing how they interconnect that is giving you the clues, certainly not as people like John are practicing, working within a framework which if you press and press and press and press hard enough can be shown all to rest on long, you know, long lines of... Conceptual Dependence, which could be traced back a long time ago to begin with. Okay, I mean, I'm sorry, that's very halfway, but it just impresses whether we know this. It's not like any kind of philatelical is going on here. It's the first. Even if you look at the differential structure on topological surfaces, I mean, take them as differential surfaces, and now you know what's meant by the degree of a fixed point. Well, even there, I guess it has to be, you know, as long as it's differentiable, yeah, a fixed point of a differentiable map, you can clearly define the index of that fixed point by the behavior of the derivative around it, you know, and the Laschet's fixed point theorem counts them with that multiplicity, so you say, great, but the Laschet's fixed point theorem didn't use the differentiable structure, yeah.

57:30 So what was it telling you about non-differentiable automorphisms, where the theorem holds for them, but you no longer know reliably what it's telling you about them? That seems to be a common theme, you know, you've arrived at a point where you can no longer reliably tell what it is that results are telling you about, although you obviously have something to know. And if you really want an intersection theory for even just complex algebraic varieties of every dimension, I'm told it's very weird and not a good idea. For one thing, what if two varieties intersect at singular points of themselves? Apparently that makes it real hard to know how to calculate the multiplicity. Thank you for watching this video. Can I try it, buddy? Yeah, absolutely. You can regard it. Regard it? Regard it. You want to see what it's like? You can look at it, yeah. It looks fascinating. Quite good. Well, and it's with nuts, so it's a good thing, because we're going to be talking about a plus or minus law. Those aren't technically nuts, so we won't, that's all right, but. We used to have roasted chestnuts when I was a kid. I know, but they don't look like a walnut. They're a different, they're a different thing.

1:00:00 I know they are, but you, I'm sorry, you said to me that you had roasted chestnuts. They're not technically nuts. That's not that concrete. Yeah. Of course chestnuts are not chestnuts. Look at the behavior of the chestnuts. I've tried to eat, I've tried to make them. I think these people just ate very coarse, very plain-tasting food, isn't that right? But also just the farmers, I mean, basically everybody around there, except the bourgeois, which there weren't many of there. Except the whole point is that the cuisine of the bourgeois process told them the things. The peasants always have eaten better in terms of the ingredients in the food that they've had to cook. I mean, even, you know, French cuisine is really peasant cuisine taken from the Swabian sauce, in a sense, to a first-of-its-kind. So, Georgia and France, I mean, there was this huge group, and where else? I remember them only too well. I remember them only too well. They were apparently just pawns in his plans. I was going to say, you'll never see any of them again. That doesn't surprise me. I saw a few of them in Como 19. As long ago as that. There was one guy there, I remember, who was taller than the others, and who, not to go out of touch, but did look actually like Stalin and the other Jews, who stood out as resisting Stalin, because he had a much more independent mind. He didn't just rule the rest of the world. Well, he had some very good research ideas. I even corresponded with some of those when I got to see him. He didn't pursue these, apparently. He wanted to develop, well, cohomology, in fact, for...

1:02:30 Complexes of abelian monolayers, not groups. To do that you have to, a differential is not a single map, it's a pair of maps. Like what you would get if you took, let's say, some crucial group and added up the odd phases. There are a lot of examples where there are examples where that's not a group to begin with. Well, apparently one of these, what she knew about, was trying to pursue with it. You know, there's a general construction of cohomology theories. If you first define some function as a chain complex, then you take cohomology as a chain complex. There are some cohomology theories that can't be defined that way. The monster, they cannot be defined that way. Because in fact you can show that if the homology theory can be defined that way, and it's sort of trivially related to the standard singular homology, in the sense that the so-called coefficient, that is the value of the theory at this one point space, is in general a sequence of groups, the dimensional and whatever it's all about. One of the axioms of homology theory says that... You can relax that, and indeed, the so-called generalized homology theory, again, there's a place where I dropped generalized, but we learned from other people not too long ago, but they have an arbitrary coefficient sequence.

1:05:00 And you take ordinary singular homologies. There's a way of taking the convolution of these two to sound over all the pairs of atoms. And that's your theory. In other words, any theory that factors through chain complexes is merely the... True, it has a different coefficient sequence, but that's all. It's just convolving its own coefficients with the standard homology. So this means that there's a whole bunch of homology theories, particularly Bordism theory, the Bordism theory, which is not factored into the chain of context, but that was a known result. But then it was pointed out by Stanley Kupfman, who was at the University actually. Everything I said there was about abelian group complexes, that Borgism theory really does arise from the chain complex of abelian groups, of abelian monoids. Hence what he's working on is really backable to Borgism theory or to abelian monoids, interesting homology theory. Again, excuse me, there's this whole bit of derived categories and the way that intersection theory is trained, all that presumes that you're going to go through chain complexes. What has this simple, well, I mean, it's just a result of general topologies and operational topologies. It shows that, I mean, they use this bordism theory all the time. It's an important companion, it's an important companion to the singular theory, to the basic theory, but uh, yeah.

1:07:30 Somebody should write an expository article on the ubiquity of fixed points. Oh, right. Yeah. Jeremy Gray has an article about it. Yeah. Including, most notably, obviously, the theory of sequence. Well, of course, your own book, I know, you know. I haven't seen this article. I had the impression it was about the origin of my blog book. Well, yeah. No, it's not about... No, no, somebody should write, as I say, a general presentation. I'll just say, you know, I'll do some different stuff here, connecting with other people. What if there is a different approach to mathematics, perhaps particularly in the sense of doing such a different work? I mean, in the sexual organization of the subject, and it would, I think, be a very valuable project in philosophy of math. Well, yeah, philosophy of paper, I mean, people of math, mathematicians, are constantly surveying different kinds of fixed-point theorems. Sometimes, well, technically, fixed-point theorems for different varieties of convex space, this kind of thing. Fixed-point theorems and lattice theory, fixed-point theorems and truth theory, but then other, but of course those, you can survey those together, fixed-point theorems and lattice theory, or both of them, you can do that. Yeah, I mean, obviously, I'm not suggesting a completely encyclopedic treat for fixed-point theorems and algebras, because that's not the perfect way to do it. It's just a fantastic, fertile, connected idea. And just seeing how one can really see how one is set there in terms of this idea, it's exciting for me. I'd like to talk more about that. Yeah. Well, Daniel Scott was real interested in that.

1:10:00 Yeah. His models for untitling or for light untitling. Anyway, a bunch of his models were gotten as fixed points. Yeah, yeah. Well, and then you start looking for these classically impossible fixed points. And you've got the topos. Yeah. I'm going to get my crumble out for you, Louge. All right. I thought it was some custard or something. That was mine. Well, I don't know what kind of little herding dogs they've got in Georgia, but I'll bet they've got a lot of herding dogs in Georgia. Can't believe the amount of people, and they farm a lot of sheep. They're going to have sheep dogs, that's for sure. I can't believe they have those border collies. Well, they're getting more and more border collies. See, what these guys were bred for, these guys were bred for the transhumance, moving these sheep three or four hundred miles up into the mountains and back. Border collies were bred to get them out into the field and back out. Well, the truth is that French farmers now don't take them three hundred miles up into the mountains. A few people do just to keep it going. But, you know, what these guys were bred to do differently than border collies was to get them out. So, what are the sorts of areas in Georgia that you're talking about? Oh, no, in agility. These guys, they're cleaning up in agility. They're doing something. Well, isn't that probably good for the team in Georgia? Oui, pour moi, le crumble est au coulis. Le fromage de perdie. It's possible with the crumble, with a little bit of cream or milk cream, I don't know. With the crumble, it's possible. A little bit of milk? Yes, milk cream, it's possible. Excellent, thank you. Well, these are agility competitions. I don't know if you've seen them on TV. Dogs are running up and down ramps and jumping through tunnels and going through tunnels.

1:12:30 These little dogs, they adore it. They adore it. They're fast. They love obstacles. They love taking command. Because they're working dogs. They were bred for thousands of years. Take a command and give it up and pass it on to the future. They love knowing what to do. An inquisition. They brought in 40 people, questioned them for hundreds of hours each, wrote it all down, and it's still available. We have hundreds of hours of descriptions of daily life by these people in the Pyrenees in the 12th century. A shepherd would typically be a younger son who didn't inherit his family's business or farm. He would get himself a dog, and he would get hired by someone who had a flock of sheep. This clock might be 50-1200 sheep. But then they would get together, two or three or four of these shepherds would get together in one big group for the transhumans to have someone to talk to, and it's a little bit safer. These dogs would know their flock, and after lambing season, the dog would go out twice a day and bring in every ewe and milk from his flock to his shepherd. So these were not mountain lions. So they actually got their work heathen sense. There is a medieval scholastic literature on where the dogs can execute because of the syllogism. There's a bit about this in Belknap and Anderson. I've never really... Now, none is more sagacious than... Well, no, no, no, the claim is that the dogs... No, there's a literature about this and my only knowledge of it is that there's some citations of it and references to sources, scholarly sources. I know they're bad at some figures of syllogism, because any dog would conclude, given evidence that some boxes have new food in them, a dog will conclude maybe some others do. Yes. Well, that seems to me to be a very perfectly sound inference. It concludes that all boxes do.

1:15:00 They're not good at drawing conclusions from or to a negation, I know that. Well, the claim is that the proof that they can execute at least one native river facility is that when a dog, a dog that's trained to swim a trail, comes to a point in the trail where it balks, and there were two broken ones, and he will sniff one direction. If a dog is sent is not on that direction, he will immediately follow the other with us. A good dragon dog, yes. Yes. Ergo, at least some medieval writers claim, therefore he follows the human facilities that may or may not be there for him. So therefore the dog is... and it was quite a major point because, you know, if dogs are rational animals... Then, they have a claim on, which of course really screws up Christian theology because it's laid down in themselves. But this was quite a major issue in the 14th century. Vegetables have vegetable songs. Yes, but not Orthodox Christian theology. Well, they have anime, aren't you now? What they don't have is rational souls. No, no, in spite of the iconography. Yes, in the drawings, you do that so that people will understand. Yeah, you do that so that people will understand. According to, well, Milton, Milton is clear that this talk about arms, legs, falling, this is only so you'll understand. It would certainly damage the theory that calculation would reveal the rational soul. They thought that animals could do those calculations. Yes, which is interesting, why they got stuck with, so it just gave them a stronger... It would be a well-trained dog.

1:17:30 Certainly they did think about that. They did think about whether their dog could execute physics at the Syracuse. They could do that. It's not understood by the literature. The majority of people say yes, they could. But again, most dogs would conclude from... One of the people who just got here has a best guess. They will conclude that every person who just got here has a best guess. Which is an excellent first approximation of real-world reasoning. When I would teach the problem of induction, I would... Whatever exists, it comes from the university. But how could it exist here? When I would teach a problem of induction, I would talk about those high-temperature superconductors. When they first tested a sample with ceramic at 60 degrees absolute and it superconducted, how long did it take them to form the hypothesis that maybe it will always superconduct at this temperature? I claim they did that on one sample. Now they knew it was a problem, they need to check further. I claim they drew the conclusion on one sample. What Hopper says, there is no logic in induction. There is a process of hypothesis formation. There is no logic in induction. Yeah, well, I think that's one of the things that Popper says that is actually very sad. There aren't many, but that's one. But I would like to understand, well, to explain the errors. I'd like to understand more in the context of what kind of reconceived set theory.

1:20:00 We can see how clearly they also interact, except they're in the far whites. It seems to be pedagogically this is the kind of thing that you need to get across to people, to open their minds to this big conceptual reorientation that's involved in recognizing quite a different nature of mathematics. I think that was the first thing we had in the systematic study. According to the exam, they start shorter. In fact, rational functions, you know, specifically if you apply a new method to a polynomial, you get a rational function. In fact, you always get a rational function of 3-1. That's the difference. In other words, they're asymptotically linear, and have all kinds of poles and rules, in a lot of sense. But anyway, we're ready to start with a rational function of quantum physics, and then you can get a rational function, you know, an iteration of a rational function, you know. The journey they described is a very, very nice result for sure about this. I had never heard of these before until I read his article. I can't remember what they are now, but they're incredibly general in a way. They are an alternative. Either things converge very nicely or they grow up entirely. In fact, growing up entirely... Here's an example of chaos in the mathematics world. Well, he was actually one of the people I was emailing for the season. And I said, okay, it's because of what we're doing this week. So, if I email him back tomorrow, I'll say, Bill O'Gear asks me to remind you, asks you to remind him what your examples are, so he'll show you some of his points of experience. It makes sense to me, people would look then.

1:22:30 You're going to use Newton's algorithm, and you know that sometimes it works, a lot of times it works, and sometimes it doesn't work. You'd like a handler that's going to work, and you say, oh, there are these regions where it contracts. And thus you get, this is the fixed point theory analysis. But if it's going to contract in depth, anything's going to contract. So let's take a space where the thing contracts all over. Now and then we'll have a fixed point. But the Newton theorem is also important because it's the main tool of the proof of the crystal functions, you know, or you think it's not true and have to break it, because you, hmm. In fact, I always wondered if maybe you could construe this in some kind of generalized section. It certainly has an objective existence of its own, but in what you do, you have this math p and you want to show that there's a point in the co-domain, basically you want a section in some neighborhood of p, passes through, you have p of x equals y, so you want a section in a neighborhood of y which passes through x. So the way that it's done is exactly Newton's method. The point is, yeah, I mean the assumption is, the assumption is that actually that the derivative, just apply this to the tangent space, the derivative of the infinitesimal, that it does spin. And that's a way of saying it's onto a finite dimension. It just retains its validity in an infinite dimension. You have this linear splitting, which of course depends on which point in the neighborhood of Y you look at. So you have these pointwise, first-order Linton-Thesimon splittings, depending smoothly.

1:25:00 The formula for Newton's method says you divide by the derivative, but really you don't have to divide by it, you just need a one-sided, you know, a one-sided inverse, the size of a section, so you're multiplying, instead of saying f divided by f prime, you're multiplying f times this given section, you know, of f prime. And then, well, you iterate that, and you get the... If you take any of the such number of neighborhoods, that's small enough, that's small enough, yeah, sure in all cases, yeah, yeah, yeah. Well, I don't know if you'd like it on a larger scale. Yeah, I mean, Newton's method is also important, as you're saying, in algebra, I guess. You know, a nil-posed element has to stop after a finite number of steps. So, for example, there's this theorem about which is called... Thank you for your time, and I look forward to working with you again soon. Yes, yes, so the proof of that, I mean, is here. If there are any proofs, it's very elementary in some sense, and it's a fundamental element, but many of them are essential because you have to, given that some of it is sort of nilpotent, sorry, it is nilpotent up to an nilpotent direction. You then take it and iterate it and you get an actual, and that iteration is Newton's method.

1:27:30 Yeah, yeah, yeah, yeah, yeah, yeah. I've often thought that the conceptual re-application of history, of history of mathematics, on these slides was one of the most effective ways of presenting, and of course there are other methods too. Yes, not stalling the history, yes. I was going to say, you reconstruction, perhaps. Reconstruction of history. Reconstruction of history on these lines and using the insights provided by category theory. Yeah. It's one of the most effective life methods. Insights provided by all developments in mathematics. Well, sure. It's conveniently summed up by category theory. It's convenient, yes, but it's of course much more precise and explicit. Well, time for what I was trying to say, but yes. This is just a tremendously effective way of getting across the basics. To me it says something. Nearly every mathematical problem is really looking for sections. Yes, yes. Somebody should attempt a general history of mathematics bringing this out. It would of course be a huge project. So that says that killing infinitesimals doesn't change topological components. Right. But further, it doesn't change the category of the top covers. He's very excited about that. Doesn't change the category of the top covers. You can keep taking top covers in space, pull them back to the reduced, other reduced subscan. It's a painful process. It's a full and faithful function. Right. Faithful, I guess, is not hard. Full and faithful is also not hard, but really cool. It really says something. Yeah. It says basically if you had a covert neglecting of intesimals, there was only one way to make an entire opening of an intestinal system. Which pictorially, to me, is kind of obvious. So to me, this theorem gratifies that. It's good. You can think that way. Uh-huh. Uh-huh. Oh, yes. And of course it's just, so then it says, if you're looking at etal covers, to know all about etal covers, you have to know all about the etal covers of the irreducible schemes.

1:30:00 Well, yeah, maybe a little where, in a reducible scheme where the components cross, there might be some . But certainly it's equivalent to looking at reduced schemes. You can ignore ill-potence in the base when you're studying natal covers of a scheme. So if you want to know what's local in a scheme, you don't have to worry about infinitesimal fringe on it. If you've got it smooth on the part killing that, it won't extend very quickly. All right. So if you think it's not going to ramify over the infinitesimal fringe, you might think it might fail to extend. Well, if anyone wants to try the lovely chip cheese, I don't know any chip cheese, but I'm mindful. Satay. Do you enjoy this place? Sure. Yeah, I like it. There are not many places in Paris where you can eat well and you can very well find some sort of money. But, um, no, of course, you can absolutely see your point about bringing out the centralities of sections. Just to provide a history, a general history of writing mathematics designed to bring this idea to the surface. Well, that's just an example of a program. You could write this as string theory, and how everything gets represented if you want to, but also a lot of things really have to do with instruction stages.

1:32:30 I think Leo didn't quite believe me when I said that. You didn't quite believe me when I said the idea that it really occurs everywhere and shouldn't be made public. What did you say? Functions. Just functions. Yeah, yeah, yeah, yeah, yeah, yeah, yeah. Those are what's involved in an awful lot of segments. If you make them informal, you don't realize that's what's involved. I know John Bell was very resistant to him. We talked about his lunches today. John Bell was very well aware of it. What was the thing that you were discussing at your entrance today? You were saying, I can teach it that way, I can possibly teach it that way, but I'm asking the students. Oh, maybe. Well, no, I mean, advanced studies really ruin the ones that have been brought to graduate school at the time I did in Harvard. You know, the only place you hear about function space is in proportional analysis, and moreover, that is treated in an extremely complicated way now.