Afternoon Discussions, incl. FW Lawvere, Leo Corry, Angus MacIntyre, John L Bell, Colin McLarty (contd.)

0:00 Okay, it's now the afternoon of the 14th of June, and we welcome Leo Corrie to this series of discussions, and we had thought that this afternoon we might move from the discussion of history of algebraic topology and algebraic geometry, I should say, And particularly the Great and Deep School and their legacy, development and legacy of the Great and Deep's work, to look rather from the back, look particularly at the developments in the 19th century which led up to the fossilization of the, what I call loosely, the mainstream logical tradition, muscle and piano, vis-a-vis the alternative that stemmed from Schroeder and Bernstein and also particularly The fossilization of a certain view of foundations of what mathematics, what foundations of mathematics of the 19th century which saw the legitimation of and what, from the point of view supplied by an understanding of the basic notions of category theory, happened and might have been the alternative history if a different turn had been taken. I'm just suggesting that's a...

2:30 You did say that you didn't want a cumulism of this meeting. That's right, that's right. He wants to say that outrageously by summing up and to start out by... Okay, I'll say it. Go ahead, go ahead, go ahead. It seems to me quite evident reasons why it went that way. The demands of analysis. I mean, in my opinion, of course. If you see the development of it from the foundation, what they call the problem of the crisis, the difficulties in the foundation, the calculus, infinitesimals, and all this stuff, you can see that was the idea of legitimizing the notion of the continuum, making it some kind of object that had some kind of mathematical... Well, I'd say coherence, although that's opposed to cohesion. I know that's different. It is different, and this was the primary concern, I think, of the mathematicians at that time. We didn't even walk through. I think that it really was the demands of analysis. These people were not algebraists at all. Yeah, but what are the demands? What they thought were the demands of analysis. No, what they thought were the demands of analysis was this. Why did they think that? Because of the difficulty. I think that one has to take a view that tries to look at the entire complex. From the point of view of what happens later, we tend to see everything quite uniformly, You know, even if we look at Fields, Doran Dedekin, each of them has a different view and a different starting point and was looking for something.

5:00 Yes, there is a problem with analysis, but there are other things around. So when you look at your introduction, we should separate. The thing is logic, because even logic has a certain development at that time. It's not just that the idea of set developed, but also the very idea of what is logic, and what's the relation between logic and mathematics, and that's one point we have is the matter of the continuum, or I would say more generally, one thing, context of geometry, and it's not the same thing. I mean, they converge in the It's interesting to see sometimes how they move separately. For example, the question of continuity in geometry. People like Cayley and Klein see the interrelation between projective geometry and Euclidean and non-Euclidean geometry. And the fact that via the work of Cayley and Klein, people start to understand that there is a hierarchy. One interesting point, for example, this is a technical point that has to do with the idea of the continuum, is whether, since they are trying to subsum or to show that the Euclidean and non-Euclidean geometries can be subsumed under projective geometry, then the question arises to define a metric, not a metric in the sense of topology, but something that allows you to define coordinates in projective geometry.

7:30 We need to start from a projective concept. There is a lot of research on this point. Questions about what theorem depends on what theorem. If you look then what happens in the end is that Hilbert summarizes. This is the achievement of the Grundleiter geometry that he puts. And all the time there is a question about continuity. Do you need continuity assumptions or don't you? In a sense, it seems to run independently, this question, from the question of continuity in analysis. It's not the case. I mean, it's not completely independent, but you could see it as an independent question. And the people who are discussing it is not the same people who are discussing continuity in analysis. There may be one important point of contact between them, it's dedicating, because for example, I think it's in Continuity and Irrational Numbers, Stettischkeit und Observenzahlen, I can't even tell you the precise place and what is the concept of continuity. In fact, here we could even say this is a third context, because here we have the difference, he wants to understand the difference between the rational numbers and the real numbers. In the passage, of course, we have the idea of density, but then there is a jump and you jump into continuity. What's going on over here? And he makes a very clear analysis of everything and he suggests very clearly that this idea of continuity will pop over. ... has relevance to what's going on also in geometry and I think that even, at least partially, Hilbert, when he comes to deal with these questions in the Grundlagender Geometrie, which is a place that again connects very clearly the continuum of the real numbers in geometry, he is inspired or takes some information, I think inspiration is the word, from Deleuze.

10:00 So there is no linear story going on here. This is what I want to say. There are several threads going in parallel. Going back to the beginning of the 19th century, what intrigues me is the role of Bolsano. Bolsano and various particular historical... Cauchy met with Bolsano in Prague, and what did they discuss? Nobody knows, this kind of thing. Cauchy was a royalist and he was accompanying the king into exile and so, sheltered by the Habsburgs and Bolzano, but more specifically, I mean, I have read some things with Bolzano, and not very many, but the introductions to these, they say, well, basically the point is that Bolzano hated the age of reason, and he found the fact that the great, you know, French geometries... In the early 20th century, a geologist could occasionally make a mistake as proof that geometric intuition was totally invalid as a basis for reasoning, and therefore it was the first really to introduce this totally formal or... That's what he thought arithmetic, discrete... Discrete, yeah, discretizing the continuum, by way of saying that, well, this geometry is all... Right, right. It's a crisis in foundation, according to him. Now, on the other hand, people say that his influence was not very great. For example, there are many points of similarity between Bolzano and Frege, but these people, people, I mean, I've had emails with the experts on Bolzano, and say there's no connection, no influence whatsoever with Bolzano, which I continue to doubt in spite of this.

12:30 I know that Dedekind was... My friend Walter Felscher made a special trip to the library in Braunschweig to compare the successive issues, editions, successive editions of Dedekind's well-known book. Yeah, this is in Göttingen. I think you're talking about Dedekind because they are kept in the... No, just the books. Ah, okay. Anyway, I was told Brown's story, but the point is, he didn't tell me directly, a student told me. So you could see in the change from one edition to the other, because in the second edition, certain passages had been changed, and it was clear, apparently, that this change was due to the study of Bolzano. So certainly he did have influence later on, of course gradually. On the other hand, most of his stuff was not published until the middle of the century. I think that this basic idea of undercutting faith in geometry is definitely one of the driving forces. But I think geometric intuition played a different role in geometry and was never really abandoned by geometers as such in the 19th century. Then they did an analysis which was concerned, after all, with rather different kinds. The use of the term continuity, for example, in geometry, as you say, has different meanings, certainly, I mean, there are some connections, but it does nevertheless have a different meaning, I guess, when spelled out in analysis.

15:00 And in particular, there was this notion of continuity of form. Right? You find a continuity of figure, you know, the idea, especially in Projective Geometry, whereby some kind of continuous change, you go, you transform conic sections into one another. Right? Which is a principle that, well, I suppose it goes back a long way. Certainly Kepler had it. And these sort of ideas of continuity, if you like, in a structural level. In terms of figures, we're playing, you know, never were abandoned in geometry. I mean, they're the basis of protective geometry that continued to play, I think, this very important role in geometry. It's still due to this day, at least in one form or another. Whereas in analysis, I think there was really the connection between continuity and number, you know, the idea of... Right, of how one actually measures, in a way that geometry wasn't so directly concerned with, at least at the level of projective geometry, and that, I think, made it more geometric intuition, of course, Bill would say, well, you could still rely on geometry, but at the level of trying to, I think, debate and introduce numbers, mistakes were made. I mean, there were some very strange... Yes, exactly. And I think that was the... Geometry went on more or less, well, in the 19th century. And then right in the 20th century, it went on more or less independently of what was going on in the foundations of analysis. Well, no, not really. There are lots of non-Euclidean geometry and not... Okay, okay. There was a will to get geometric intuition out of geometry, and Pfeiffer's phrase is over. The Italian differential geometers were monotonic. I think also that is public. I don't know so much about Poincaré. I think even if you look at him particularly, you will see several things. One thing is when he speaks about the question of the foundation of geometry. And another is when he has to solve some problem. Even, you know, if you think about dynamic systems, let's say, he's using some kind of geometric intuition there. I mean, he's solving equations topologically, to put it that way. But from Cantor's notion of limits. This is where Planck very most proudly uses Cantor, is to study...

17:30 This limits the limits of orbits and recurrent points. I think Kim Conqueror, I mean I'm not really an expert, but he strikes me as really very much an eclectic in this way. He uses all kinds of methods. He's got to look at his attitude towards the continuum. He's not a straight constructivist as Brouwer or Watkins later on, but on the other hand, he doesn't believe the continuum can really be constructed as a power set. He doesn't really think that has a definite meaning. I mean, he varied his, you know, he does seem to be, and it's actually for that reason, I think, when Janet Felina wrote his book on, she admits it's actually very hard to kind of pin down exactly what Poincare did. Well, if you're going to try to find that he's a constructivist along the lines of the, what does he call them in that? If you're going to try to find that in his work, you're going to have a needle, and that's why he... He even says there, the contorians and the pragmatists, or whatever you call them, the contorians and these other people. I think he is pragmatist. He even says these people cannot understand each other. He goes on to understand them both. So he's telling you that he isn't either one of these things. But yeah, he's real proud of it. His analysis feeds directly into his models of non-Euclidean geometry. And he sees this as a victory over some kinds of geometry. Now, geometric intuition of continuity, yeah, that, he thinks there's one geometric intuition of continuity, but not of spatial form. I mean, that's his whole point. But, if you allow me, let me go back first of all to Bolzano. I have never read Bolzano directly, so I may be wrong, but I would ask the following question. Bolzano speaks about the infinite. I wonder to what extent he... For example, Dedekindik, this question with the question of continuity. Because look what happens, this is somewhere in the mid-19th century when Cantor is studying the convergence of Dirichet's series, right?

20:00 That's the point where he notices that something is not clear because there is the set of discontinuities of the converging function, etc. Something is not clear and this is what he says to clarify. So, I think that even if you look further, we know that Cauchy started with the epsilon and the delta, okay, which is a kind of arithmetization of, as opposed to a geometry, a geometrical analysis, but in the end, he used the geometrical intuition as a basis. And this is what Dedekin says very clearly. I think this is, I mean, at least... From what I have read, this is where I see the point that Deleking says to the foreman, look, I'm trying to teach analysis, like everyone, you know, every one of them was writing a new course, then at least, or something like that. So the question is, how do I prove, and he said mostly, let's assume most people start from the theory that says that there is a monotone sequence of rational even, and he says, show me one place where someone has... That's right. And he says there is no such place. People, in the end, rely on geometrical... He writes this in 1850-something, but his solution to the problem comes relatively late, let's say 1872 or something like that, so he understands perhaps what others didn't understand, that even if you want to move to a more arithmetic way of dealing with this matter, you always in the end rely on... Because we don't really know what is the continuum. So my guess would be that Bolzano doesn't really address this problem. He asks about the infinite. He doesn't really ask, I think, we should check that.

22:30 That's also the infinite, not the continuum. No, no, that's true. There was no question about continuity. And Debekin said here it is the problem. And why? Because the rational are not continuous. Well, he defined what is to be continuous. Not for the first time, but in a certain way, because also at the same time we have Cantor defining it with the sequences, etc. But Debekin is the person who is strongest of all in asking about the arithmetic. I want to speak about the arithmetic of analysis in the 19th century. It was not, you know, it was not that people were trying from the beginning to reach what we know now, that we have a clear definition of what the numbers are, and then our epsilon and delta is based on that. The epsilon and delta were around there, but not so clear, you know, as we can teach it today. From what I know, Deleking has this kind of obsession. About the numbers, you know, and trying to make a foundation of mathematics on the concept of set, actually. I think Kantor has even more of an obsession with it. But in what direction? Well, I mean in the idea that really the notion of continuous is really something that is not intuitive at all, according to Kantor. He says... We really don't have any. It's a notion of number that comes before our conception, our intuition of space. He says, really, as you know, it isn't in the gridlock. And he says that over and over again. And he also has disagreements, of course, to some extent, with Dedican and correspondence, where Dedican objects to the fact that he introduces horrifying... There are six continuities in the show that the spaces of different dimensions are actually by the same commonality, and Kantor has to be dedicated to trying to bring Kantor back to the problem of continuity. Of course, Kantor recognizes it. But even more than Dedekind, I think Cantor was quite determined to, he calls the continuum some kind of mysterium, you know, that has all been the source of some kind of almost religious reverence and so on, in a much stronger way than Dedekind does, at least that's my impression from my view.

25:00 Indeed, he starts with the ming, extracts the cardinal solemn, and throws the ming away for the rest of his life in practice because... I mean, at one point he's saying, oh, we can distinguish between matter and fields, because matter is countable infinity, and fields are purely a purely cardinality distinction, which is, I mean, it must have been pretty absurd even then, if one looked at it. Yes, I think he was much more obsessive, actually, in the... Well, that's my environment. He's got these orders of real numbers. I've never understood this. These orders of real numbers. There's the real numbers that are the limits of Cauchy sequences of rationales. Then there are the limits of Cauchy sequences of first order reals. Then there are the limits of Cauchy sequences of... And then he sort of says that actually every step after the first wasn't needed. But I don't know if I really understood it. But he takes them all. The proof is very simple, that if you take cuts of rationales, you add points. And if you take cuts of reals, you don't add any points. Therefore, this is not continuous and this is yes continuous. You don't go further. You stay in the same place. We already said that in counterpoint. And then he has these orders of will. He does say that. I've never understood why he has them. He sort of says he didn't need them, but I've never understood exactly what's said. From our point of view, it's a kind of presentation of this chapter rather than instruction itself. The fact that by diagonalizing the double sequences... I think there's one other point in connection with this question of the use of intuition. You know, of course, Cantor had directed the broadsides against Veronese, you know, have theories of infinitesimals and the geometries of the, and I remember reading, I'm not, again, an expert in it, but what I have read, Veronese objects to the idea, he says,

27:30 The productions of Cantor and Dedican are very valuable and very interesting, but how would we ever know that they had anything to do with the continuing of John? He says, you know, this is a point he makes over and over again. And Cantor, of course, he's really not, maybe Cantor would have objected to that as, you know. Actually, he objected mostly to Veronese's theory of infinitesimals, but I think Cagriot's whole point seems to have been that, really, we don't have anything, whatever, this has got, the intuition of continuous has really nothing to do with the mathematics as he's actually practicing it, whereas Veronese said, well, how would you even know, why would we even think that this construction is the, you know, is useful in that, or corresponds to anything that we've arrived at in the history of mathematics? That was his point. And he, as an example, he built the Non-Archimedean Geometry. Exactly, exactly. So again, also, you know, the same people, it's not just a matter even of division, like I said with Poincaré. Take Hilbert. I mean, he's not denying the use of intuition in geometry by all means. No, no, no, he doesn't. No, correct. When you are trying to... To prove a theory in geometry, try to use your vision or your visual capacity. This is one question. And another question is the question of foundations. The foundation, you want to be able to define the term in a very coherent way, etc. So, there is such a big problem, you know. Let's say calculus was doing very fine from the 17th century to the... people were doing a lot of... Of important things. So it's not that there is a problem that is stopping the development of the analysis. The main thing is the mainstream of analysis, you know, the question of variational calculus and differential equations and there is another question, which is foundation. And then you start to see that there is also non-Euclidean geometry and all kinds of things that make some people, some people, not very much of them, ask, is there any...

30:00 Well, can we still trust geometry and the guiding vision of what is mathematics? But it's not something that it's... It's really a main concern of mathematicians. It's a concern of certain mathematicians, and I would even say of certain mathematicians when they are interested in certain questions. Whereas they can go on, you know, working on other things. You don't agree with that? Not exactly. I think anybody who has more than half a head in practice in the final analysis takes a dialectical view. In other words, you don't rely on geometric intuition. To the last point, from geometric intuition, you make explicit some axioms. Then you try out these axioms. And again, geometric intuition suggests theorems. You try to prove them. So in that way, your knowledge of both aspects deepens. In fact, Ray says, no one would have a formal foundation. These are important discoveries. They said that Hilbert did know and praised Veronese in his earlier work. Not very strongly, by the way. But nevertheless, certainly he didn't condemn it in the way that... No, no, no. I'm just saying that he was keeping to himself part of the glory. Yes, that's probably true. And he saw the merit of it. There are also two questions on intuition here, whatever the exact balance between the theorems is, for just making new discoveries in geometry by trying to find intuition-free formal foundations. But I think Cantor was trying to just do that in one way. You see, he became so convinced, right, of Nisman, as you put it, you know, he gets his cardinal, he gets numbers.

32:30 He wasn't of the famous far origin, of course, but I mean that would be important, but I think that then he got, of course it was an amazing achievement in Merlion, but I think that approach became very rigid. It became, part of its power, of course, was the fact that it was very focused and it was revolutionary. Kantor's set theory was certainly revolutionary, but in some ways it was very rigid because Kantor says the only way, essentially, you're going to be able to extract new knowledge is by means of this general idea of number that he constructed, which led to the idea of an ultimate set. And this is more or less what he says in the Grunewald. It's all got to fit in this framework of number that he describes. And I think he then got to the point where it didn't allow for these. You know, for the further development of what Bill, you know, would call the dialectic, because I think he got powerful as a programmer. He became very narrow. It seems to me that Dedekind has a kind of an answer to Veronese's question, and what order it is, I don't know. Dedekind says, what is the continuum? As I read Dedekind, he said, he's willing to agree with Veronese, but we don't have an absolute intuition on that. But he says, here's a model that preserves one of the key things, one of the things we teach every first-year calculus class. This model, well, that the square root of six exists, you know, that the continuity in the ways that we use it in an introductory calculus class is justified by this kind of construction. He does seem to have a fundamental intuition that the real continuum has all the rationals on it, and it should have this continuity property. But he doesn't claim that this is the explication of the continuum. It's an explication that suits the needs of calculus. Contra precisely doesn't. I don't know passages in Contra. Where he says even that, he just lays down. Yeah, he does, he does. Because he doesn't admit that intuition, you know, even the spatial notion really plays any role. He says that that's something derived and something else. And what's the justification for this conception? You know why he does that? At least partly because he's very anti-county. He hates, he doesn't like that. He doesn't like that whole tradition that actually comes from some kind of introspection. He lays it out in the gridlock and he's got this fairly clear, he's pretty well presented, I mean he states it very clearly, but he does not like the idea, he wasn't the only one, there was a retreat from Kant going on among mathematicians and physicists during the 19th century, but Campbell was quite explicit about that.

35:00 Let me just correct one thing I said, it's not just that square root 6 exists, it's square root 2 times square root 3 times square root 6. This is a good example because it appears in a letter to Lipschitz, and this letter comes because Lipschitz, who was relatively open to Dedeckin's idea, as opposed to most, he asked him, why are you doing this? I mean, what's the point? We know what is this matter of continuity, and you come and give us an explanation which is more confused and more obscure of what we already know. It's a very funny passage because we mathematicians work in a very sloppy way. If there was a teacher of linguistics or language or linguistics who taught this way, we would all say, look at what he's doing. He asked, there's only one place where someone shows that square two times square three equals to... He said, what people do in order to justify this is the following. They write square... He makes the difference in the following that he does it for the natural number and explains it in terms of sets of change. He does it for the irrational number. He does it also for the algebraic number because he introduces the ideals as sets of numbers, etc.

37:30 So his problem is not limited to the definition of the continuum in analysis. To the idea of number in general, and particularly, he has to, in fact, in every one of these systems, he has one central question. In the numbers, in the natural numbers, his question is, what is induction? What is the justification of induction? And he does it, he shows or creates kind of an axiom in terms of sets. What Peano does, he has a kind of set theoretical version of it. In the case of the continuum, we ask what is the continuum and he answers in terms. In the case of the algebraic number, he asks what are the laws of unique factorization in a general algebra. And also, using the ideas, he gives an answer which is in terms actually of sets of numbers. I think that here we are looking at a... More comprehensive program, you know, that he, if you oppose it, I think, even to Hilbert. Hilbert also had some, of course, foundational concerns, but on the one hand, he had the concerns, and on the other hand, he's very practical about solving all kinds of problems in the, you know, variational calculus, perhaps the dialectic here, maybe he used to connect these two, but Derekin is single-minded. It's time to solve the foundational problems and there is a clear problem of continuity. What is the role of continuity in geometry? There are a lot of things going around. Some people are not happy with this, some people are not happy with that, some people want to do that. Of course, looking backwards puts some order into everything, but that's not the way things happen.

40:00 Things were in their development, not continuity in the 19th century. One has to look at all aspects together because... I don't see that a single aspect can, you know, can. And also, for example, what you say about Balsamori would say that I would, I would be nice if we had here the text to look at if he mentions continuity in some place. I know he mentions infinity. But I don't know if Balsamori gives a definition of infinity. Yeah, he gets a definition of continuity and a differentiability, which is more or less the definition that Cauchy, as far as I recall, in Paradoxes of the Infinite, although that wasn't published for, well, a long time after it was written. I think it is actually. It's a good question, I think. I think as far as I remember, because I was looking at it. Yeah? Well, I don't know. I'm pretty sure there is a definition, although he's concerned really more with, in that work, as far as I remember, with, as the title says. I'm not sure whether, I can't remember whether he actually gave it that title. I don't know, because you know, it was assembled and it was quite a bit later. It probably did, but it's true that that's really the thing that it's more noted for. You know, when he did other things, he gave a definition, I don't think it's there, but, well, he gave a definition of a nowhere continuous, nowhere differentiable continuous function, as you know, long before, long before Weierstrass did. I mean, he had a pretty clear, you know, conception of what, of what came to be the sort of epsilon-delta definition of continuity and differentiability, and I think, and I think you'll find it in, yeah, I mean... In paradoxes of the infinite, even though it's true that he is more concerned, I think, in that work to bring up the whole problem of infinity. And also, he also talks about infinitesimals. He talks about inverses of infinite numbers as infinitesimals, and he's really quite flexible on that matter.

42:30 But then I would ask you, for example, he makes a difference between the rational and the real. Because he may use the word continuous. He may have been thinking of this. That's a very good point. I wanted to know. I looked into that. Does he work it? He does work it. He's completely continuous. You know, that's a very good point. And I looked into that. At least, I'm not, you know... I'm an amateur. I was intrigued by that, that density was a notion. It goes back to the middle ages, like William of Ockham had, where density was what meant continuity for a long time. And the serious question behind angels dancing on the head of a bird. Yes, yes, and it was all... As opposed to the jokes that you can make about it. Yes, indeed. What are you saying about Bolzano proving it? So, I mean, does he explicitly have a notion of completeness and a notion of continuity for functions, or has he proven a mediability for specific things like polynomials or what is it? Well, he proves it for continuous functions. That's the upper bound principle. But you could sort of get away without having an explicit definition. I mean, do you have one definition of completeness and then you have, how do you define continuous? Well, the point is that he didn't make it... But you need a... See, his epsilon and delta can be rationals, but if he's going to conclude from a function being epsilon-delta continuous, that if it's positive here and negative there across zero, then that's going to be delta-delta continuous. It is, essentially. But actually, as far as I know, the first explicit, even Weierstroth, as far as I know, doesn't make... Explicitly saying that, but you have to define what a continuous, what a continuation is. The first one who explicitly does this, as far as I know, is Dedicate. Yes, Dedicate makes it explicit. He makes it explicit. He never said that it's just density, but I, this is what I really do. As far as I know, he's the first.

45:00 As far as I know, too. I mean, Cantor implicitly spoke about completeness. Because if you define the real numbers by series of... Sequences of rational, so you are assuming that there is a difference between the two, but Delecun was really worried about these matters. So, look, on the other hand, you know, Bolzano was an outsider. This is an interesting point because sometimes these outsiders have ideas and have insights into things. They, at the same time, they have many mistakes, but they make them. I am sure Bolzano is an interesting person to look at. I'm not too sure about... I think the story you mentioned with Cauchy is an article by the Ivor Grattan Guinness on the effects on sciences. He was the first to... Garrett Shubrin, I believe. Because it was at least as... I may have you heard it wrong. Yeah, yeah. In any case, Grattan Guinness did it in a very speculative way. And I think it doesn't really matter in the end because, okay, it's not who... No, no, that's right. He read this... That's right, that's right. He posed it as a method. Did one steal from the other the idea? I'm not caring about that. It's just that they must have had similar ideas and they must have left with different ideas or similar ideas. Part of the whole, even though these were two geniuses, at the same time it's a reflection of the collective. It would be interesting to know what they talked about, not to try to. I think there is another point that Ferreiros raises in this book, which is the difference between a theory of sets and the use of the idea of sets in mathematics. If you look at Dedekind and even Cantor, they didn't have, they started, etc., to try to define continuity.

47:30 The ideas may be, you know, the ideas start with Riemann. A connection of things or whatever before even starting to think that you have here a theory of the infinite or whatever. And in this sense, this is also some kind of opposition or parallel to the idea of continuum. The idea of looking at sets and studying the infinity, like Bolzano before anyone else in this thing, as a concept in itself, this is something that also leads you to the idea of continuum. In fact, Riemann, if you look at what Riemann did in the... With the idea of, in the sense of Riemann, it's try to see, you know, try to base, you see, you have the idea of geometry, and then you're basing the idea of geometry in something which is more general, the manifold. And to try to put the question here, in the manifold, rather than putting it in the geometry itself. But, yes, but Riemann, of course... One of Riemann's principal distinctions is between discrete and continuous manifolds, and really, no, he makes a very important point. He sort of anticipates. A discrete manifold really does end up being a set. Kantor then decided, really, Kantor, of course, is really new, and he uses the term a point manifold, and he uses this term, and he got much more concerned with simply the idea of point manifolds. I mean, that was the basis on which he then wanted to erect the theory of continuous manifolds, which I don't think Riemann, Riemann had simply drawn, you know, had drawn the distinction and then, of course, pointed out that if you really want to understand the intrinsic geometries of continuous manifolds, you've got to go to physics, whereas, of course, in the case of a discrete manifold, you have exactly something intrinsic, namely its number. But he says what he says. I mean, it's exactly what Cantor has in the case of a second, that's its cardinal level. So Cantor's big leap, in a way, was from the final arithmoid into the idea that arithmos could be actually infinite.

50:00 Because those... Discrete manifolds that Riemann talks about, I mean, he doesn't make it too precise, but you can well imagine that they're actually finite, maybe blending off into the counter, but basically finite discrete sets, pretend Cantor says, well, look, the whole continuum itself is really a discrete set, throw away all the cohesion. Absolutely, but I think it's very striking, though, that Riemann does say that... The explicit plan by Riemann that really a discrete manifold, I mean, of course, he may be thinking of the finite, but actually whether it's finite or infinite carries its own internal measure. It's called its cardinal number. I mean, the whole point of a continuous manifold is that it doesn't. Where does it come from? Well, Riemann says at the end, we've got to go to physics for that, at least. This is his final claim. But that requires that the discrete be somehow finite, because if it's... Even on the dense conception, the one inch doesn't have a different cardinal number than five inches. Right, right. Of course, that's a problem that Riemann never, you know, kind of got, I think, considered, but... So there are two kinds of discrete, as you're saying, and it's sort of in my paper, my early paper. I mean, Riemann's discrete is discreetly ordered. Kantor says, oh, it's continuously ordered, but you can see it as discrete in the sense that of well-determined elements every day. Every element is either equal to or unequal to every other. Going back to this thing of the discrete, even if it's true that Riemann has this difference, at the moment that you move from the geometry to the manifold, So you are saying something about the geometry, that the continuity of the geometry depends on what happens on the manifolds. It doesn't mean that I already know what is a continuous or a discrete manifold, but you move the question to some other place. And why? It's because it isn't... He knows exactly how to give a better foundation for geometry. He knows that now you have several kinds of geometry and he says, how can I address this question? He says, okay, move into a different idea, a more general one. And in this sense, this is the line that Derekin followed. Because he said, it's like saying, I will explain the continuous.

52:30 The continuum, sorry, of the real number in terms of something that is very similar to Riemann's manifolds. It's like generalizing the Riemann manifolds and showing that some of them are discrete, for example, the rational number, and some of them are continuous. I wonder, for example, if Riemann, when he wrote that, he was thinking that the rational numbers are a continuous or a discrete variable. I don't know, because if he was following the lines of the time, it's dense and so well, at least distinguishable from continuous. It was very, very vague. It's vague, it's vague. So, Derekin takes this vague idea and fine-tunes it very precisely, I think. Yeah. Also, there was a question in Riemann of, well, you know, there's a chordatization, he talks about You know, multiply, you know, infold, manifold, and implicitly, well, he's using some kind of idea of a parameterization or there are some quantities that are associated with the points and, or the, you know, right, and he doesn't, of course, go into the question of the nature, if you like, of these, right, of these coordinates, right, what they are as a, you know, intrinsically. Of course, that was what Dedekind and Kantor later did. It's an interesting thing, the connection, because since the term manifold was used, I mean, right up, oh, for a very long time, right up, I don't know, I forget the day in which Kanner really does begin to use the term set, because he, you know, in the 80s, he was still using infinite linear point manifolds. He's talking... No, there are many words. There are. System and... System was Denikin's term. Denikin also. But I mean, that's a... Menge comes quite late. Menge is late. I didn't mean, it's not just a terminological, because it's some kind of development from this much more geometric conception of manifolds. Manifold, yes, something with many coordinates or able to vary in many ways. A manifold variation, and that's evidently what Riemann made up. And then the idea is somehow you detach yourself in some way or considerations from that, and then you end up with the idea that it's just...

55:00 Yeah, I mean, right? Yeah, that's clear what that means. Manifold is not a language foundation for mathematics, but is it part of mathematics that reflects the rest of mathematics, or is it something? This is very serious. We know some foundationists who take the view that somebody who applies logic to geometry is a traitor to the cause and should be excluded from it. No, I think that in this sense, anything goes. In the sense that, you know, it's a matter of how a specific mathematician wants to work. And I think the point is different, and I want to make that... The fact that there are certain open questions, or more than that, uncertainties, in the foundation does not stop any mathematician, or most mathematicians, of just going on and, you know, if you take someone like Brauer, okay, he was following, first of all, methodological or philosophical principles, but I think that in this sense, the best example, or a good example is... ...that he was, on the one hand, working on questions, you know, proving the consistency of arithmetic, as if anyone doubted it. No one was doubting that arithmetic is consistent. They were looking for a proof of the consistency, right? But at the same time, you know, he's working on physics, and if you look at the articles... The first time I read this, I was quite amazed, because... I sent such an article to the Mathematische Analyse. It's full of analogies, speculations, for example, we need here a tensor that comes close to what Newton does in the linear and the only possible candidate is the Riemann tensor.

57:30 Two very nice questions. The first one says the following. He says, the building of science is not built like a dwelling house, in which you first have to build foundations and then start to build, you know, very big cracks in the walls because the foundations are not strong enough to support it. Then you go down and try to afford it. And he says, this is the right way. This is not just a mistake. Just one thing he said, and in other places he speaks about two kinds of mathematicians. Those in the forefront and make their way in the... and so on. And they come to arrange things and make them... I mean, people know, everyone knows that this... ...of geometry or of arithmetics doesn't say that he was concerned in the sense of... Here we have a problem and we cannot move forward until we solve this problem. So this is the kind of separation that I am doing, that you can separate even in the same person, certainly in the whole discipline. So one is going this way and the other one is going that way. But certainly, look, there is another, if you ask me about foundations, that most foundational questions are solved or attempt to be solved within mathematics, that was not always the case. In fact, there is also a quotation by Hilbert about anything that was common knowledge, like saying, for example, we have to reduce everything to the concept of natural number.

1:00:00 And what about the natural number? That's the business of philosophers. We mathematicians don't deal with that. This is something that changed later on, and we know that everything that came around was an attempt not to leave it to the philosophers, but to take it in, and of course the same goes with words. There was a kind of division of labor that emerged, you know, because as I suppose has happened, you know, in mathematics generally that you have geometers, algebras, and so on, although the greatest mathematicians or even the best did solve the connections among these things. The foundations in this sense became another branch of mathematics. It wasn't an actual process, it occurred. It may not be a very healthy one, perhaps, if it gets completely cut off from the rest of mathematics, but then of course you have a similar, if they're not mathematicians, who quite well, you know, like... I think you got the wrong end of the stick, actually, but like Morris Klein, for example, who had this notion, mathematics, the loss of certainty, and those members of the Courant School in New York were very worried about the fact that actually mathematics maybe doesn't become completely abstracted. There's been this real split, it's a sort of big division of labor now between mathematicians and physicists. There was a point, I mean, I think it was exaggerated, partly because Klein and Koch didn't. You have to say that set theory did affect some kind of unification. It did provide what is called a foundation.

1:02:30 Of course, in practice, most mathematicians ignore it. I mean, they pay lip service to what is a group and then rapidly got on with the development of their own subject. It was a kind of obligatory introduction to set theory. You know, at the beginning, the books are pretty well every subject, and then within ten pages, it's all been forgotten. It's hardly played out, not in all areas. It plays more roles in some areas than others. But what I meant was that the idea of, you know, providing some kind of unified view of mathematics, including, as you say, then got sort of separated out in something like the same way as other branches of mathematics have diverged to professional, you know, ramification, if you like. I think that's changing. I mean, although I'm not sure many mathematicians sort of... Well, they're not interested in foundations as practiced separately from the rest of mathematics, let's put it that way. That doesn't mean they may not be interested in foundations in a broader sense. That's a different question. But as practiced, if you like, by, you know, let's say set-in-set theory, well, you know, you've had these Dauntless 400s of battles on this sort of question. We used to be hounded by the so-called foundation, you know, the foundational, you know, and Simpson and Co. I don't know what you mean. Yeah, okay. What I mean is that foundations did get this, you know, we're at the second world of foundations. People with foundations thought they were not getting copper recognition and so on, you know, as with Friedman and Simpson. Yeah. I mean, really. Nobody here? Sorry, what else? I think there is an interesting point in that poem, that theme. I think many mathematicians look for the interconnections between various branches, and the question is whether some of the foundational theories remain outside this wave of... And I know many mathematicians who say that theory has nothing to offer. For example, you are asking questions about number theory. The set of functions or whatever, you can get from topology, you can get from here, you can... What do you get from set theory? It doesn't allow one to formulate new concepts. I mean, it hasn't for a very long time.

1:05:00 He did it first. ...towards the analysts and so on, and towards the wild geometry and topology, but I think that's really the major point. I mean, I've never forgotten the time I had Quill and he said, I mean, totally, he destroyed everything. Even those who may need some set theoretic logic, there's still someone who needs some fine-tuned set theoretic logic. He's not using really advanced set theoretic logic, he's using gamma, the field's not just gamma, it's in there, there is some sort of mixture. And what about the work of Khrushchev? Yeah, but see, Khrushchev has done a tiny, tiny amount of work on what one might call set theoretic logic, the very tail end of the Schellach program. Most of what Krzysztofowski is doing is really completely, I mean, Krzysztofowski talks, tries to talk most of the time nowadays, in editorial terms or in terms of, currently he's trying to define general integration theory and logic, it's not, it doesn't involve real numbers at all, so much as anything that Cartier was talking about today, I mean, model theory at least, of which he is a questionably leading person, largely detached from said theory now. I mean, Scharach made enormous contributions to our subject. He did them in secondary terms, and the subsequent people had to kind of do the mining, get the stuff out, and then see that actually there was some geometry there, explicitly in terms of geometry, both something fairly close to algebraic geometry, but also close to all the notions of geometry, I mean, microids and dependence relations, cornetization from the 19th century has played an essential role in geometry. I mean, we have vast generalizations. The whole point being, we eventually forget...

1:07:30 There are groups of coordinates, sometimes fields of coordinates, and so on, and this is our best way to understand these things. So I would say that not everyone in the subject agrees with me on this. I've written on, essentially, the detachment from within this part of mathematics. We're interested in the structure of definitions. I mean, I suppose we will eventually do things properly in a funtorial way we will come to. Or close to the category theory, even model theory. I mean, Tarshish was so strong that it has been very difficult to deny. I would say that that's essentially all. For example, one can expect in certain situations that an open problem in a certain discipline may be solved by taking some development in said theory or having logic. That hasn't happened in the last, I don't know... Using monetary ideas, which one can quite readily trace back to Scheller, but I mean, they had to be joined, as Pruszkowski has succeeded in solving problems, not utterly mainline problems, but very hard problems in which he talked to people about in, in, in, in, in, in, in, in, in, in, in, in, in, in, in, in, in, in, in, in, in, in, in, It's suspicious that this technology could solve it. They didn't understand the proof. But it's not a set theoretic, which these people would never ever use.

1:10:00 Between one and the other? You see, any limit between, say, contemporary model theory and geometry, there are quite a number of cases. We mentioned duality this morning, the concept of duality space in the theory of complex. This has been done over, significantly improved. There is a very fluid frontier now. No, and what about the other, between logic and set theory? I mean set theory, we'll see papers in the in the JSL sometimes called something like an application of set theory. This is a theme worth putting in your title that you're using set theory here. Because of the way that history has developed this phrase set theory has about at least seven different meanings. If you say set theory opposed to something else. And so people often even mix up these completely distinct oppositions. So, okay, so what are they? I mean, what is, yeah, I mean, how, how it's being used. So, for example, there are some places where, say, in analysis where you say you're using set theory, it means that you have to explicitly invoke the notion of measurable cardinal as opposed to non-measurable cardinal, you see. So, in other words, these... Particular, more or less technical things that you set their work on, which occasionally come up in other mathematics, but you're actually independent of the formalization. So that's one thing that's set theory. Some discussions, this is what it means.

1:12:30 Very often, it means membership-based formalization as opposed to categorical formalization, in the sense that the sets themselves are structuralist, are they structuralist or not? Well, that's another important opposition. Anyway, go on. There are people who, if they think you're speaking in the plural, will say, aha, you're using sets. The way all these are quite distinct. Well, there's also one other way, I suppose, which, you know, is the use of axiomatic set theory, I mean, where you... Yeah. Well, really, there are cases where you actually use, I mean, you actually do use features of the formal system in some way or other, which is different from those that you mentioned so far. Although those are rarer usages because most mathematicians don't even know what the actions of set theory are and why they should do it. Okay, another important opposition is simply whether you explicitly use function spaces or the power set. Typically set theoretical operations play a key role, then it's set theoretical. If they don't, it's not. But I mean, the kind of thing you were referring to, an application of set theory, one can be almost sure that it is, first of all, able to... The algebra is absolutely thoroughly in the import of Schell's work and the Heide improvements were of that nature. Sarah had given a perfectly straightforward proof for the... The Algebras don't have a means of carrying, typical Algebras did not have a means of carrying out trans-federal recursions and getting through the limit ordinals, or showing that you couldn't.

1:15:00 So Schellach typically came along and said, what can ZFC, because here I can do it, I can give you this particular model, but you're not doomed to failure of destructibility because you have such a very, very strong form of... There's a lot of control over the way the universe is generated. There's combinatorial principles far stronger than the continuum of things that will get you through to the end. I mean, that is an important lesson for algebraists. On the other hand, as far as I can see, this is exactly the same game. It's a trans-finite set generalization of combinations. I mean, that's what goes on, right? Man used to account for all the parameters of set theory. That's right. And no one suddenly has this technique. And Tarski said that. It's not, well, I'm not connected with topology, but it's a very good thing. To me, it isn't topology at all. But what was called analytic topology, it had to be done at some point. Well, it was a kind of dead end. It was a... Yeah, but there was no geometry whatsoever. It's all, the topology is entirely based on set theory. Yeah, okay. Well, I mean, stone's theory, stone's, stone's, stone's, I mean, boolean algebra and, well, we can't know all that, but I mean, boolean algebra is rather more sanitary, because there is, you attach it geometrically.

1:17:30 Very interesting topological space is inextricably linked to something that we might actually call geometry, but it's, it's a good idea to have that in mind. I mean, it's just, I mean, well, on the one hand, You know that Gibraltar topology? This one would be Gibraltar topology. These people were using differential equations, they were using complicated metrics, they were using K-theory and Marxist-architecture. In places in the world, Madison, Wisconsin, the separation axioms where there may be a cardinal involved or some kinds of compactness or new kinds of topology, but all of them variedly. The general definition of topological space indeed is a reflection of metrizable, locally compact, and so forth and so on, but then you take this abstract definition, that creates a huge category. Now you want to explore the fringes of this category. Well, the main way to get at it is through cardinality. Cardinality of the size of the open set lattice. I mean, they're very deep questions. I mean, sort of master, inshallah, who can direct. It was pretty well a dead end by that, by the full of the late people. They were, but they were as courage, as you say. They all showed to be independent. There was no progress you could make on the basis of just the theoretical methods that were being used by them. I don't believe there were many of these questions. No, no. I mean, something like the Sousslin problem was a pretty natural thing, but even there that was a little bit of a problem. Almost all the others were. It was a nice problem. I mean, the Sousslin is a bit more connected, but to Dedekind and so on. I mean, the very definition of the metacardinal is not really there in it.

1:20:00 But abelian group theory would also have been. Because, I mean, all the basic things have been discovered. The beautiful facts are essentially always about finitely generated abelian groups or quant-ve-agent duality or something like that, which are... There is nothing inherently to do with the unfoundable, and I mean, anybody who has the trans-finite induction, or maybe you can, if you're Sherlock in some particular way. And how will the independence of the continuum hypothesis help you? Well, it's not the continuum hypothesis, it's the technology. I mean, sometimes the models for the continuum hypothesis will do it very straight away, but not always. I mean, the method of corner, the time scale is a bit different. You have some set-theoretic question, and you think you couldn't prove it from ZF, and you're up against the wall. All you can say is you can't prove it. You show it and you do it by introducing variable sets. Yeah, yeah, yeah. No, no, I mean by puts. No, essentially by introducing, you know, a lot of different set-theoretical universes, which Cohen, and that was a sort of mystery of Cohen for ZF because he had this, but he does, he has two, you know, it's a whole idea of going from...

1:22:30 You couldn't just sort of start an accountable model and then pull one of the missing elements of the power set of a maker, adjoin it, and then close off because you were not going to get it. I mean, it was the process of closing off. If you go in some sort of generically, you would succeed. I mean, there was some category intuition there and so on. And then, of course, this we now understand in a much broader... But it did shift. The interesting thing is that it was a real shift of problem. I mean, you put it very well. Now we know. But then that... Well, that contracted too very quickly. That is a situation where set theory provides recursions in many fields of mathematics. Well, the thing is, to say many fields, one has to be careful. I mean, it has essentially no implications whatsoever for geometry. It's supposed to be somewhere else. And you add the... you'll prove... Specific. You can write them down, but they're not... Yeah, yeah. And it's morally certain that one cannot use these methods to get down to questions about curves and so on, because if you could, this would really have, this would tend to overturn major contexts and process, and in these questions of topology, I mean, it kept the field.

1:25:00 These fields might themselves have always had topologies, but others would have none, and certainly not a job. If you take the Cartesian product of the unit interval with a normal space, a normal is a well-known problem. So that's sort of on the borderline. That sounds like it might be a mathematical problem. It turns out not to be, in some sense. It's a problem. It's a problem. It's a problem. It's a problem. It's a problem. It's a problem. It's a problem. It's a problem. It's a problem. But with relative to set theory, in a sense, it overthrew the idea that there is an absolute theory, so we learned gradually to, I mean, I don't know how many set theorists have really done this, but in practice they do, that there's really a range of almost equally being set theories. And then one searches for counter examples. We've got counter-examples, we've got calculus, calculus leaky tens, you know, and all that stuff. We've got to a point where you couldn't find these counter-examples just in one universe, you see. And Cohen came along and offered a whole load of new ones where you could actually find, locate these counter-examples. Because you couldn't see it if you were just working with a, you know, with one model of those concepts you said. I mean, there's one other kind, which are essentially combinatorial problems, but they're real numbers.

1:27:30 ...passing into some special universe of set theory, which nobody... ...structure of the theory. Let's say if one thinks of the Poincaré conjecture, does it help solving it, some of these developments? No, no, no, no. I mean, I was slightly surprised a few years back to hear Frank Quinn, he's a well-known, low-dimensional... ...the ones who wrote this article with Jesse... Yes, yes, yes. I mean, I drew him back at Yale, a controversial, in that sense... ...the con-surgery... ...set theoretic topology...

1:30:00 ...practice identified set theoretic topology... If you can show they're independent of CF, then you're doing set-theoretic topology, and then you'll say, okay, if V equals L, we get one answer, if the continuum hypothesis is false but Martin's axiom is true, we get the other answer, the canonical one. So that was their definition of set-theoretic topology and their goal was always, like they brought up this one theorem and said, you know, there was this project in the homotopy of manifolds that seemed to depend on this at one time, and if only it had turned out that it really did depend on this, it would have been great. Well, it didn't. But anyway, there's still this... Translating, or trying to insert something like a homotopic group into this in a totally artificial way, you see, like this is... Now, if I can take this a little bit backwards again, in the context of Hilbert, and say about what I said before, the separation between foundations and the... But there is also unions, which is... And it has been somehow implicit in all this is what Hilbert thought about the role of the axiomatic method, which is not a formal thing but a practical thing. In the sense, for example here, you have a theory, but things are not completely clear, so try to clarify the interdependence. And the interesting thing is that for Hilbert this could have a use in Penrose. This is the point of the axiomatization of physics, that you build a theory and then if you have a new empirical discovery which doesn't fit the theory... Then instead of throwing the entire theory, you only have to make a little change, change one axiom here. So the idea of this kind of foundation, an axiomatic foundation, is to allow you in the future to correct your theory so that it fits the new situation. Here you're speaking about an empirical science, but in a sense it fits to what you say here about your motives or whatever.

1:32:30 You know very well the logical structure of what you are saying, and since change, you know exactly how it will influence this or that corner of the theory, and this is a strong connection between a foundation and the upper layer, you know. And it does empire. It's kind of remarkable. I mean, no one has ever made... All I'm saying is that that hasn't been... No, I haven't mentioned... I don't know. I used to wonder about that. There are some remarkable features, but... They're still exotic. I mean, there's still a core in which mathematical practice sort of goes on as a melodrama, at least in principle, and that core act has not really changed for a long time, and nobody, nobody even, Woodin is trying to show, he's not going to succeed. None of these people are going to change it. No, I mean, I think experience, a good deal of experience has shown that we can't expect set theory anymore to have any, I mean, set theory in that tradition at least. This is the curious thing about Panmatti's project. He wants to say, how do we establish new axioms for set theory? Well, not on philosophical grounds.