Pm session: FW Lawvere, C McLarty, L Corry, A MacIntyre, JL Bell

0:00 And they need to record. That is now recording. Okay. I am now going to stop. And the story is, at least Mrs. Truesdell told me, that Jurewicz was trying to impress some lady. So therefore didn't wear his glasses. Oh no. And yet attempted this very dangerous climb. Oh no. He fell and his body was ruined and in a couple of days he died. And so they, they, they, they... He was quite young actually, 54. Still obviously had great work in front of him. So they were still assembled in Mexico City and so Eilenberg says it was a dark day. He was there of course too, Eilenberg. Eilenberg was sitting in Mexico City and got the news, they all got the news. What an utter waste. I had no idea that that was what had happened today. It's very near the supposed impact point of the comet. Ah, yeah, that destroyed the dinosaurs, allegedly destroyed the dinosaurs. It's one of the steeper pyramids. So that's in the Yucatan, isn't it? Yeah. Near Merida. It's astonishing the number of great mathematicians who have died in accidents. One thinks of Herbrand, who died in the mountaineering. Of course there's Galois, who died in a stupid duel. It's astonishing the number who are accomplished climbers. That's true as well. Love to climb, yes, Herbrand and... That's a kind of challenge. That's like doing great maths, of course. Frank Adams, which was actually... Which the other Excel player. ...was supposedly so good that he could take any room, say this room, and he could climb around in the middle of the wall without ever touching the floor. There's a special cult of this in Cambridge, they have a special club, you probably know about it, it's a special cult. They actually go around at night climbing around some of the college buildings, and they're often very high, you know, 50 feet above the sidewalk, they go around climbing on the Senate House roof and kind of jumping across from one building to the other, completely crazy, suicidal, I don't know why more of them don't get killed, but it is said that it tends to attract a disproportionate number of mathematicians.

2:30 But then unfortunately, Lighthill too, because Lighthill, Adams, who was he? That was Peter Johnston's supervisor. And Lighthill, yes, yes, yes. And Lighthill, who was Dirac's successor in the Lucasian chair, who did a lot of the work on distributions, in a theory of distributions, he died in a swimming accident. He was very, there's a fit, and he made a point of swimming. He had a holiday home on one of the Channel Islands, a very small Channel Island, I think it was either Alderney or Sarp, but he had a holiday home, and he always made a point of swimming twice around the island, you know, every day, and he was about six, you know, he was well into his sixties, he was very fit, but he went round and he got a crab and, you know, he was swept out to sea in the currents and drowned. And so it's astonishing the number of outstanding mathematicians who have met accidental deaths. I'm sure it's... Do you see this one or just the algebra one? Only the algebra one. I haven't seen this at all. Oh, is this the one with the paper that you were talking about? Yeah, this is the full-fledged version. Oh, this is the full-fledged version. I haven't seen this at all. I'd love to have a look at this. Unfortunately, the price is really outrageous. It's $170. That's what Kluwer take. I mean, this is... Yeah, Kluwer are the biggest price garages in the whole business, I know. I mean, it really upsets me because, you know, people can really get angry about that and not buy it. It was nice because this, of course, I was able to afford and do have on my shelf. And still it's expensive. I got this on Amazon for about $50 for a paperback, but these days that's not. The Burkhauser are much better than Kluwer. I'd love to have a reader. They put those things on after you've read them. Yeah, yeah, they just sent it to me. They didn't ask.

5:00 Well, ask the author, do I have the right to abuse your copyright by photocopying, obviously not the whole book, but a fair chunk of that. Good for you. In fact, I can even send you the... I would love to. I would love to study that. I really would. That would be my payment for the... Oh, no, no. You pay me. No, no. Other way round. No, this is... It's great having you here. Let's, let's, let's start, shall we? Shall we? Is Anderson... Oh! Yes, he is. Yep. Oh, thanks. If I chance to see my view of Grassland and the story of Mathematica, it would be worth being able to consult with us as we go along. Well, I sit here, but I don't know you. Yeah, I know that's what I mean. You want to take Pierre Cartier. I think I'm sitting here, but not importantly. Pierre Cartier was probably over there. He was there. In that case, you'll be expected to talk, or something. Tell us some more tales. Yeah, yeah, yeah. Well, we'll keep you warm. I'll be seeing you in the... We're looking forward to more than just tales, so it's fascinating to know who they are. 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 the history of algebraic topology and algebraic geometry, I should say, and the legacy, to look rather further back in the history.

7:30 I looked particularly at the developments in the 19th century which led up to the fossilization of the, what might call loosely, the mainstream logical, vis-a-vis the alternative. Also particularly perhaps to look at the response which backed the fossilization of a certain view of foundations of what mathematics, what foundations of mathematics ought to consist in. And the role that that played in response to the crisis of geometrical intuition in the 19th century, which saw the point of view supplied by scientific history. I'm just suggesting that's a... You certainly stacked the case on one side. Ah yes, it's not a... You did say that you didn't want a cubism at this meeting. That's right, that's right. I'm sure he wants to say that as an outrageously biased summing up and to stock up my... Okay, I'll say it. I'll agree. Get it, John. Go ahead. I just said it. There were, it seems to me, quite evident reasons why it went that way. Because it was the demands of analysis, not algebra. I mean, it might have been, of course. If you see the development of the foundations, what they call the problem of the crisis, the difficulties of the foundations, 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…

10:00 Well, I'd say coherence, although that's opposed to cohesion. I know it's different. It is different, and this was the primary concern, I think, of the mathematicians of that time. We're reading right through. I think that it really was the demands of analysis. These people were not algebraists at all. Yeah, but what are the demands of analysis? Well, what they fought with the demands of analysis was this. Why did they think that? Because of the difficulty. I think since we have someone present who knows a lot more about the history than we do, we might ask for his opinion. Sure. I mean, you and I just have this... No, we're mere amateurs. We have this body. We're mere amateurs. That's true. Isn't it? I think that one has to take a view that tries to look at the entire complexity of the problem, because, of course, from the point of view of what happens later, we tend to see everything quite uniformly, and it was far from that, you know, even if we look at a relatively narrow field. I mean not field of mathematics, but people. Weierstrass and Kronecker, let's say, and Cantor and Dedekind, each of them has a different view and a different starting point and was looking for something different. Yes, there is a problem with analysis, but there are other things around. So if I look at your introduction, we should separate. One thing is logic, because even logic has a certain development at that time. This point has been made very interestingly by Jose Ferreiros in his book on Cantor and the theory of, at the beginning of the theory of set, that it's not just that the idea of set developed, but also the very idea of what is logic and what's the interrelation between logic and mathematics and the that's one point we have to be very careful about. The second is the matter of the continuum, or I would say more generally, continuity, because continuity arises or is dealt with within the context of one thing, but also within the context of geometry.

12:30 And it is not the same thing. I mean, they converge in the end, but it's interesting to see sometimes how they move separately. In geometry, let's say, starts from, if we take, for example, all the works of people like Cayley and Klein in the 80s, 40s, and 50s, where they tried to see the interrelation between projected 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. And one interesting point, for example, this is a technical point, but it 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 are, or can be subsumed under projective geometry with certain additions, then the question arises, how do you define a metric? Not a metric in the sense of topology but something that allows you to define coordinates in projective geometry and in order to do that you need to start from a projective concept, let's say the cross ratio, and not from a measure like in Euclidean geometry. Can this be done or not? So there are a lot of research, there is a lot of research on this point and then there are all kind of questions about what theorem depends on what theorem. The sarks on Pascal, Pascal on Papus, or three-dimension, two-dimension, and if you look then what happens in the end is that Hilbert summarizes, this is in fact the achievement of the Grundlagen der Geometrie that he puts order into this question and all the time there is a question about continuity. Do you need continuity assumptions or don't you? So, 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.

15:00 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 Debekind, because, for example, I think it's in Continuity and Irrational Numbers, Stettisch Keidung der Absinth-Alle-Zahle. I can even later find in my book the precise place where he explains. That, of course, he's doing this in order to understand 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. What happens in the passage, because 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 or has relevance to what's going on also in geometry. And I think that even, at least partially, Hilbert is aware of this, when he comes to deal with these questions in the Grundlagender Geometrie, which is a place that, again, connects very clearly the continuity of the real numbers and the continuity considerations in geometry. He is inspired or I know at least he takes some information, I think inspiration is a word, from Deleking because he is very aware of it. So I think that 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 Bolzano in all this. Bolzano, well I mean there are various particular historical questions which apparently are unanswerable according to the fact that Cauchy met with Bolzano in Prague and what did they discuss nobody knows this kind of thing but because Cauchy was a royalist and he was accompanying the king into his exile and so forth sheltered by the Habsburgs and then here appears

17:30 But more specifically, I mean, I have read some things of Bolzano, not very many, but the introductions to these things, they say, well, basically the point is that Bolzano hated the age of reason, and he found the fact that the great French geometers and mechanicians 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... Well, it's really arithmetic, discrete... Discreet, yeah, discretizing the continuum. Right. But 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, the people, I mean, I've had emails, but people are trying to be experts on Bolzano. They say there is no connection, no influence whatsoever with Bolzano. Which I continue to doubt in spite of this documentation, but in any case, I know that Dedekind was influenced by Bolzano because my friend Walter Felscher made a special trip to the library in Braunschweig to compare successive issues. There are a number of 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 Göttingen. The various editions, do you mean the manuscripts? No, just the books. Anyway, I was told Brown's story, but the point is, he didn't tell me directly, his student told me this.

20:00 So you can see it 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, studying Bolzano. So certainly he did have influence later on, and of course gradually he became known. On the other hand, most of his stuff was not published in the middle of the century. But I think that this... 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 than they did in analysis, which was concerned after all with rather different kinds of... And for example, one thing that's striking is that 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. 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, at a structural level. In terms of figures, we're playing, you know, never were abandoned in geometry. I mean, they're the basis of projective 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... The idea of continuity and number, you know, the idea of how one actually measures, the way that geometry wasn't so directly considered at least at the level of projective geometry, and that I think made a more geometric intuition. Of course, Bill would say, well, you could still rely on geometry, but at the level of trying to... I think Dimension introduced numbers, but mistakes were made. I mean, there were some very strange... Coordinatizing. Yes, exactly. Coordinatizing. And I think that was part of the source. Geometry went on more or less...

22:30 Well, in the 19th century, in the 20th century, it went on more or less independently of what was going on in the foundations of analysis. Not really. There was a lot of non-Euclidean geometry and not much non-Archimedean geometry. There was a will to get geometric intuition out of geometry and white, red, gray, and silver. No, I mean, it was different. Italian differential geometers were not... I mean, they... I think that also that was a complex... I mean, I don't know how much... 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. He's using some kind of geometric intuition there. I mean, he's solving equations topologically, to put it that way. But from Contra's notion of limits. This is where Poincaré most proudly uses Cantor, is to study limits of orbits and recurrent points. I think Poincaré, 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. Look at his attitude towards the continuum. He's not a strict constructivist, as Brouwer might get later on, but on the other hand he doesn't believe the continuum can really be constructed as a power set as well. 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 this book, she admits it's actually very hard to kind of pin down exactly what Poincare did, what his genuine view was, because he was a genuine kind of eclectic, and he did have very, it's hard to extract a kind of, how can I put it, a sort of core position that he held. 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 1912 paper, If you're going to try to find that in his work, you're going to have a real tough time. He was going to be. He isn't there. And that's why he even says there, the Kantorians and the pragmatists, or whatever he calls them, the Kantorians and these other people.

25:00 I think it is pragmatists. 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 geometric intuition. 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. No, no, no. 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 mixes, as for example Dedeckin did, this question with the question of continuity. Because look what happens. The problem arises somewhere in the mid-19th century when Cantor is studying The convergence of Dirichlet series, right? That's the point where he notices that something is not clear, because there is the set of discontinuities of the converging function, etc., and something is not clear, and this is what he says to clarify. So, I think that even if you look, for example, we know that... Cauchy started with the epsilon and the delta, okay, which is a kind of arithmetization of analysis as opposed to a geometrical analysis, but all of them, in the end, use 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 Dedekind says to the foreman, look, I'm trying to teach analysis, like everyone, you know, every one of them was writing a new course, the Analyst or something like that. So the question is, how do I prove the fundamental theory of analysis? So you can start from the intermediate value or from this or from that. And he said mostly, let's assume most people start from the theorem that says that a monotone sequence of real numbers, or rational even, has a limit.

27:30 And he says, show me one place where someone has proved this thing. That's right. And he says there is no such place. People, in the end, rely on geometrical intuition. And we are speaking about relatively... I mean, he writes this in the 1850s or something, but his solution to the problem comes relatively late, let's say 1872 or something. 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... The geometric intuition. And why, he says, because we don't really know what is the continuum. So my guess would be, first of all, that Bolzano doesn't really address this problem. He asks about the infinite. He doesn't really ask, I think, we should check that, that's a paradox of the infinite, not of continuity, but for anyone else, including Weierstrass and other people like that. There was no question about continuity. And Dedeckin said here it is the problem. And why? Because the rational are not continuous. Well, he defined what is to be continuous. Let's say not for the first time, but in a certain way, because also at the same time we have Cantor defining it with the sentences, etc. But Dedeckin is the person who is strongest of all in asking about the arithmetic. The people speak about the arithmetization 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 clearly, you know, as we can teach it today, and from what I know, Dedekind has this kind of obsession about the numbers, you know, and trying to... To make a foundation of mathematics on the concept of set, actually. I think Kantor has even more of an obsession with this. 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...

30:00 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, he says it in the gridlock, and he says it 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, dizzying... Discontinuities in your correspondence, showing that the spaces of different dimensions are actually by the same commonality. And Kantor has to be, you know, Dedekind tries to bring Kantor back to the problem of continuity. Of course Kantor recognizes it. But even more than Dedekind, I think Kantor 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... and so on, in a much stronger way than Denikin does. At least that's my impression from my reading. Indeed, he starts with the menge, extracts the cardinal solid, and throws the menge away for the rest of his life in practice. At one point he's saying, oh, we can distinguish between matter and fields, because matter is countable infinity. It must have been pretty absurd even then if one looked at it. Yes, I think he was much more obsessive, actually, and... Oh, yeah. Well, it's my... I don't know. It's my... Yeah, well, it's... Okay. And also his hatred of infinitesimals, the color of the... What are these real numbers? I've never understood this. Which? What are 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 limits of Cauchy sequences of... Yeah. And then he sort of says that actually every step after the first wasn't needed, but I have not really understood because, well, but he takes them all. The thing, at least in the Stetisch Keining Rationalzahlen, 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 have any points. Therefore, this is not continuous, and this is, yes, continuous. So, after that, whatever you do with cuts, you don't go further. You stay in the same place. Maybe we need to say it again. Contra. But contra doesn't have any points. And then you have these orders of reals. He does say that.

32:30 I've never understood why he has them. He sort of says he didn't need them, but I've never understood exactly what says. From our point of view, it's a kind of presentation of the structure rather than the structure itself. So the fact that by diagonalizing the double sequences, you could reduce it back. That's a relation. But he doesn't reduce it back. He just says he didn't have enough. I mean, there are similar problems with the intuitionistic reals. Intuitionistic reals are not really reals in a way, because they're never actually passed to the quotient. That's right. The presentation. Could I just say that I think there's one other point in connection with this question of the use of intuition. You know, of course, Cantor had directed these broadsides against Veronese. You know, we have theories of infinitesimals and the geometers who were... And I remember reading, I'm not, again, any expert on learning, but what I have read, very basically objects to the idea. He says, well, yes, the productions of Cantor and Dedekind are very valuable and very interesting, but how would we ever know that they had anything to do with the continuum of geometry? He says, you know, this is a point he makes over and over again. And Cantor, of course, he's... Cantor, 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 Cantor's whole point seems to have been that, really, we don't have any, whatever, this has got the intuition of continuous, has really nothing to do with the mathematics as he's actually practising it. Whereas Verdi said, well, how would you even know, why would we even think? That this construction is useful or corresponds to anything that we've arrived at in the history of mathematics, but that was his point. And as an example he builds the non-Archimedean geography. Exactly, exactly. So again I think that 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, quite right. When you are trying to prove a theorem in geometry, try to use your vision or your visual capacity, this is one question. And another question is the question of foundations. What is the foundation? You want to be able to define the term in a very coherent way, etc.

35:00 So the problem is not that even in the calculus that 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 important things. So it's not that there is a problem that is stopping the development of the analysis. One thing is the mainstream of analysis and our question of variational calculus and differential equations and there is another question which is foundations and then you start to see that there is also non-included geometry and all kind of things that make some people, some people, not very much of them ask Is there any question? Can we still trust geometry as the guiding line, 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 mean, I think anybody who has more than half a head in practice in the final analysis takes a dialectical view, right? In other words, you don't rely on geometric intuition. To the last point, from geometric intuition you make explicit some axioms. And 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. And it's questioning intuition leads to the non-Euclidean geometries, which are clearly known to be valuable by Hilbert's time. What Planck-Rey most praises Hilbert for questioning intuition and on is leading to the non-Archimedean geometries. And Planck-Rey says, no one would have arrived at this if they had said it. Well, okay, so Planck-Wright's wrong. So Planck-Wright says and praises Hilbert. It says that Hilbert makes discoveries here that no one would have made if they had relied on intuition instead of looking for a rigorous formal foundation. These are important discoveries. We have to face that, that Hilbert did know and praised Veronese's earlier work. Not very strongly, by the way. But nevertheless, certainly he didn't condemn it in the way that…

37:30 No, no, no. I'm just saying that he was keeping to himself part of the glory. Yes, I think that's probably true. And you saw the merit of it. Yeah, yeah, yeah, of course, of course, of course. So, yeah, I mean, to whatever extent Poincaré thinks, well, okay, Veronese, there are also two questions of intuition here, whatever the exact balance between the theorems is. At any rate, Poincaré praises Hilbert for making new discoveries in geometry by trying to find intuition-free formal foundations. Now then you get an intuition for these new geometries. But I think Kantor was trying to just do that in one way. You see, he became so convinced, at least as you put it, you know, he gets these cardinal, he gets numbers. He was a number theorist by origin, of course, but I mean that would be important. But I think that then he got, of course it was an amazing achievement, but I think that approach became very rigid. Part of its power, of course, was the fact that it was very focused, and it was revolutionary. Cantor's set theory was certainly revolutionary, but in some ways it was very rigid because Cantor says the only way, essentially, you're going to be able to extract new knowledge is by means of this. This general idea of number that he constructed, I mean, which led to the idea of an ultimatum set. And this is more or less what he says in the Grunewald. It's all kind of 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 the program is, it became very narrow in a certain way. It seems to me that Dedekind has a kind of an answer to Veronese's question, and what order this is, I don't know. Dedekind says, what is the continuum? It seems, as I read Dedekind, he says, he's willing to agree with Veronese that we don't have an absolute intuition of it. 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, the continuity in the ways that we use it. In an introductory calculus class, it's 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 serves the needs of calculus. But Contra precisely doesn't. I don't know passages in Contra where he says even that. He just lays down...

40:00 Yeah, he does. He does. Because he doesn't admit that intuition, even 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-Kantian. He hates it, he doesn't like it. He doesn't like that whole tradition that actually comes from some kind of introspection. You're right. He has his other prints. He lays it out in the gridlock and he's got this fairly clear, it's very well presented. I mean, he states it very clearly. But he does not like the idea, he wasn't the only one. I mean, there was a retreat from Kant going on among mathematicians and scientists during the 19th century. But Kant was quite explicit about that. 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. So let me tell you something about this. This is a good example because it appears in a letter to Lipschitz and why this letter comes because Lipschitz was relatively open to the Edequine idea as opposed to most colleagues at the time. 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 than what we already know. And Edecky says, no, you are wrong, because look, it's a very funny passage, because he says... We mathematicians work in a very sloppy way. If there was a teacher of language or linguistics who taught this way, we would all say, look at what he's doing. He asked, show me one place where someone shows that square two times square three equals... He said, what people do in order to justify this is the following. They write square two. Square root of two, square, times square root of three, square equals, but they said this is what we need to prove because we don't know that this property of the rationales extends to the algebraic properties. And I think this points to a very important difference between Dedekind and Cantor. Practically, let's call it, solve the problem. Whereas Dedekind has a kind of very... a problem of principle and I see the difference in the following that Dedekind does this kind of analysis for three kinds of number systems. He does it for the natural numbers and explains them in terms of sets.

42:30 He does it for the irrational numbers in terms of cuts and he does it also for the algebraic numbers because he introduces the ideals as you know sets of numbers etc so his problem is not limited to the definition of the continuum in analysis but To the idea of number in general. What is a number in general? And particularly in this case, 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 sense. So what Peano does as an action, he has a kind of theoretical version of it. In the case of the continuum, we ask what is the continuum and he answers in terms of cuts. And in the case of the algebraic number, he asks... What are the laws of unique factorization in a general algebraic field for the algebraic integrals? And also using the ideals, he gives an answer which is in terms actually of sets of numbers, so... I think that here we are looking at a more comprehensive program, you know, that if you oppose it, I think, even to Hilbert. Hilbert also had some, of course, foundational concerns, but on the one hand, he has the concerns, and on the other hand, he's very practical about solving all kinds of problems in the, you know, in the variational calculus. Perhaps the dialectic here, maybe he used to connect these two. But Dedeckin is single-minded. He's trying to solve, as it were, only the foundational problems and to think. So, I think that by looking at these kind of people in Canter, of course, or what was said in Poincaré...

45:00 They were not really like solving the same problem, let's call it. It's not that there is a clear problem. What is the problem of continuity? What is the role of continuity in geometry? There are all kind of things going around. Some people are not happy with this, some people are not happy with that, some people want to do that. And of course looking backward we can... There were some conceptual problems, but everyone was looking at his own thing. So, therefore, I think that in looking at the question of continuity in the 19th century, One has to look at all aspects together because I don't think that a single aspect can, you know, and also, for example, what you said about Bolzano, it 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 he gives a definition of continuity and 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. 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, well, I don't know, a few months ago, and I'm pretty sure there is a definition. Okay, it could be interesting to see what... 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 translated, 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, but he did other things, he gave it, I don't think it's there, but... Well, he gave a definition of a nowhere-continuous, nowhere-differentiable continuous function, you know, long before, long before Weierstrass did. I mean, he had a pretty clear conception of what, or what came to be the sort of epsilon-delta definition of continuity and, and, and differentiability. I think, and I think you'll find it in, I mean, in Paradoxes of the Infinite, even though it's true that he is more concerned, I think, in that work to...

47:30 bring up the whole problem of infinity. And also, he also, in this connection, as far as I recall, he also talks about infinite decimals, he talks about inverses of infinite numbers as infinite decimals, and he's really quite flexible on that matter. But then I would ask him, 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. That's a very good point. And I looked into that, at least I'm not, you know, I'm an amateur, but I was very intrigued by that. You know, far as density was a notion. It goes back into the Middle Ages. It's really in the long run. Yeah, yeah, yeah. Density is not wrong. Density was, you know, continuity for a long time. And the serious question behind angels dancing on the head of a pin. Yes, yes. And it was all... As opposed to the joke that you can make about it. Yes, indeed. What do you say about Bolzano proving it to be a value theorem? Does he explicitly have a notion of completeness and a notion of continuity for functions, or does he prove an immediate origin for specific things like polynomials or what so ever? He proves it for continuous functions, and 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? How do you define function? Well, what function does he use? Point-wise functions, yeah. The identity function? Yeah. Yeah, yeah, yeah. I tend to doubt it. Well, how does he define continuity? Yeah. Epsilon, delta. Epsilon, delta, what epsilon and delta are? The point is that he didn't make... Are they rational? Exactly. He clearly doesn't think that he knows that the square root of 2 is right there. See, his epsilon and delta can be rationals, but if he's going to conclude from a function being epsilon, delta, continuous... If it's positive here and negative there, it cross zero. It does, essentially, yeah. But actually, as far as I know, the first explicit, even Meyer's cross, as far as I know, doesn't make...

50:00 The first one who explicitly does this, as far as I know, has done it. Yes, he makes it explicit. He makes it explicit. People have said that it's just me, but this is what I really... As far as I know, he's the first. And 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, but then it was really worried about this matter. So, look, on the other hand, you know, Bolzano was an outsider. And this is an interesting point because sometimes these outsiders have ideas and have insights into things. They at the same time may have many mistakes, but they... I am sure that Borsan is an interesting person to look at, no doubt. I am not too sure about... I think the story you mentioned with Cauchy is an article by Ivor Grattan Guinness in the Archive for Historic Effects and Sciences. He was the first to mention that. I don't know if this is... I think Gert Shubring had it later. Really? Ah, of course he did. I may have the order wrong, but 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, if he read this or... That's right, that's right. He posed it as a matter of did one steal from the other, the idea. I'm not caring about that. It's just that, you know, they... They must have had similar ideas, they must have left with different ideas or similar ideas, you know, so it's part of the whole, even though these were two geniuses, at the same time it's a reflection of the collective views and practices of mathematicians, so it would be interesting to know what they talked about, not to try to subdivide the credits ad infinitum. I think there is another point that Ferreiros raises in his book, which is the difference between a theory of sets and the use of the idea of sets in mathematics, because, you know, if you look at Dedekind and even Cantor, they didn't have, they started to wonder about the properties of sets, etc., and to try to define continuity, but the ideas may be, you know, the ideas start with Riemann...

52:30 And people like that. Riemann perhaps was the one who used in the strongest way the idea, without trying to analyze it, using it as, you know, it's just a collection 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. Because the idea of looking at cells and studying the infinity, again perhaps Bolzano came 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 with the idea of manifolds, manifolds in the sense of Riemann, right, is try to see, you know, try to base your... So you see, you have the idea of geometry, and then you are 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 for him really does end up being, you know, a set. Cando then decided really, Cando of course is known to you, 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 manifold. 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. This is what he says. It's exactly what Cantor has in the case of a second. It has its cardinal numbers.

55:00 There are a number of metaphors that can be used to describe arithmetic and the idea that arithmetic could be actually interesting because those discrete metaphors that Riemann talks about, I mean he doesn't make it too precise, but they could well imagine that they are actually finite, maybe blending off into the calendar, but basically finite discrete sets, but then... The whole continuum itself is really a discrete set of things. Throw away all the cohesion. Absolutely. But I think it's very striking, though, that Riemann does say... I mean, it's that explicit point by Riemann that really a discrete manifold... I mean, of course, he may be good at thinking of the finite, but actually whether it's finite or infinite kind of... It 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 the distance to be somehow finite, because even on a dense conception, one inch doesn't have a different cardinal number than five inches. Right, right. Of course, that's a problem that Riemann never kind of got. There are a few kinds of discrete, as you're saying, and it's sort of in my paper, my early paper on sets of sets of points of spaces. Riemann's discrete is discretely 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. But look, let me, going back to this thing of the discrete, even if it's true that Freeman had 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 manifold. 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 he didn't know exactly how to give a better foundation for geometry. He knows that now you have several kinds of geometries. And he said, how can I address this question? He said, okay, move into a different idea, a more general one. And in this sense... This is the line that Dedekind follows, because he said, it's like saying, I want, I will explain the continuous, the continuous, the continuum, sorry, in the, in the, of the real numbers in terms of something that it's very similar to, to, to Riemann's manifolds.

57:30 It's, it's like generalizing the idea of Riemann manifolds and showing that some of, some of them are discrete, for example, the rational numbers, 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 number. I don't know. I don't know. Because if he was following the lines of the time, it's dense and so at least distinguishable from continuous. It was rather vague. It's vague. It's vague. Dedekind takes this vague idea and fine-tunes it very, very, very precisely, I think. Yeah. Also, there was a question in Riemann of, well, you know, there's a coordinate, coordinateization. He talks about, you know, multiply, you know, n-fold, 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 he doesn't, of course, go into the question of the nature, if you like, of these coordinates, what they are intrinsically. Of course, that was what Dedekind and Cantor later did. It's an interesting thing, 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 date at which Kander really does begin to use the term, because he, you know, in Atiyah, he was still using infinite linear point manifolds. He's talking... No, there are many words. System and... System was dedicated to it. It was dedicated also, so... But I mean, that's a domain... Mania comes quite late. Yeah, mania's late. It's not just a terminological mistake because it's some kind of development from this much more geometric conception of manifolds. Manifold means something with many coordinates or able to vary in many ways. A manifold variation. And that's evidently what Riemann had in mind. And then the idea is somehow you detach. And then you end up with the idea of pure mania. Pure fuck.

1:00:00 Yeah. I mean, right? Yeah. That's clear what that means. Manifold is... But I think that it's different for us now. But I think to a German in 1830, pure fuck and manifold. That's not what I'm saying. That's what I'm saying. Manifold, yeah. There's an aura of meaning around the word manifold that we don't have around pure fuck. Yeah. Just two. How do you call that, a wake-up signal or something? An another signal is a wake-up signal. So you distinguish foundation from mathematics, but is it a part of mathematics that reflects the rest of mathematics, or is it something distinct? This is very serious. We know some foundationalists who take the view that somebody who applies logic to geometry is a traitor to the cause and should be executed. No, I think that in this sense anything goes. You know, it's a matter of how a specific mathematician wants to work, and I think the point is different, that 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 Brouwer, 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 Hilbert. 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, he's, you know, he's working on physics, and if you look at the articles on physics, the first time I read this, I was quite amazed, because I am sure that... I sent such an article to the Mathematische Analyse, they wouldn't accept it, I mean, it's full of analogies and, you know, yeah, speculations or even, you know, for example, we need here a tensor that comes close to what Newton does in the limit, and the only possible candidate is the Riemann tensor. I mean, you know, so...

1:02:30 He was playing both games and he says it very clearly, you know, he has a very nice quotation, we have it there in the book. He says... Some mathematics... he has two very nice quotations. The first one says the following. He says the building of science or science is not built like a dwelling house in which you first have to build... Very good foundations and then start to build, you know, floor after floor. He said no, science first of all builds very big rooms with very big windows and corridors and only when you see that there are cracks in the walls because the foundations are not strong enough to support it, then you go down and try to fortify. And he said this is the right way. This is not just a mistake. This is... This is one thing he says. In another place, he says, He speaks about two kinds of mathematicians. One that goes in the forefront and, you know, make their way in the woods and probably makes lots of mistakes and contradictions and so on, and then come in the back, so people that start to arrange things and make them, I mean, people know, it's not, everyone knows that these things exist, so, but... Therefore, the fact that he was working on foundations 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. Someone is going this way and the other one is going that way. But certainly, look, if you ask me about foundations, the way we see it now, 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, but I think that was common knowledge, like saying, for example, a chronicler said, we have to reduce everything to the concept of natural number. 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 you know we know that the Hilbert program and everything that came around was an attempt not to leave it to the philosophers but to take it in and and of course the same goes with works like yours and others that say no this is a question of mathematics but it has not always been the case I mean this is a relatively recent phenomenon we can call it so so I think yeah in this case this separation can be done to that extent.

1:05:00 Well, I think also the case of, there was a kind of division of labor that emerged, you know, because as I suppose it's happened, you know, in mathematics generally that you have geometers, you know, algebraics and so on, although the greatest mathematicians or even the best ones did solve the connections among these things, and working foundations in this sense it became some... Another branch of mathematics in some sense. I mean, it wasn't an unnatural process. I mean, it occurred... Of course, it may not be a very healthy one, perhaps, in some sense, if it gets completely cut off from the rest of mathematics. But then, of course, you have a similar, if not... The mathematicians, 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 that mathematics is the loss of certainty. And those members of the Courant School in New York were very worried about the fact that actually mathematics really doesn't become completely abstract. 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 think it was exaggerated, partly because Klein and Co. didn't really understand. But nevertheless, these are the sort of tendencies that does arise. And then you get developments as in category theory, for instance. Well, for instance, there isn't any other one. Why is this called a unification? It did provide what is called a foundation. I mean, it didn't give... exactly. Of course, in practice, most mathematicians ignored it. Well, you know, 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.

1:07:30 You know, at the beginning of books are pretty well every subject, and then within, I mean, within ten pages it had all been forgotten, I mean, it was, it hardly played, not in all areas, it played 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 foundations, as you say, It is a more recent one because they 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, most mathematicians, you say, are not really... 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 law, you know, you've had these doubtless four-hundreds of battles on this sort of question. Well, I know, because you used to be hounded by the so-called foundation, you know, the foundational, you know, in Simpson and Co. People at foundations thought they were not getting proper recognition and so on, you know, as with Friedman and Simpson. Yeah. I mean, really, it's... No, but here, I think there is an interesting point in the following sense that I think many mathematicians look for the interconnections between various branches. Yeah, it's natural enough. And the question is whether some of the foundational theories... I know many mathematicians who say set theory has nothing to offer. For example, you are asking questions about number theory, the set of function 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 very long time. It did it first. It did it first. It did it first towards the analysts and towards the wild geometry and topology. I think that's really the major point. I've forgotten the time I had Quillen at the Blackboard in Oxford. He was doing an argument and he said, and he's not about to know,

1:10:00 there's no way I can do this argument. It would have been totally, it would have destroyed everything in there, to put it into set theory. For granted that the majority of mathematicians feel this, even those who may need some set theory. There's still some who need some set theory, fine-tuned set theory to get around this even, I suppose, even there. And so on is fairly clear. He's not using really advanced theory. Gowers, the field is my least team. Gowers ain't there. There is some sort of mixture. And what about the work of Khrushchevsky that we were mentioning? Yeah, but see, Khrushchevsky, I mean, Khrushchevsky has done a tiny, tiny amount of work on what one might call set theoretic logic, the very tail end of the Sherlock program. But most of what Khrushchevsky is doing is really completely, I mean, Khrushchevsky talks, tries to talk most of the time nowadays. Almost in factorial terms or in terms of currently he's trying to define general integration theory of logic. It doesn't involve real numbers at all so much. It's the kind of thing that Cartier was talking about today. I mean, moral theory at least, of which he is unfortunately the leading person, largely detached from said theory now. I mean, Schellach 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. I mean, we talk explicitly in terms of geometry, both something fairly close to algebraic geometry, but also close to older notions of geometry, I mean, The whole point being eventually to get, not because, not for any foundational reason at all, but just because there are hidden in very abstract structures in mathematics, 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, I mean, not everybody in the subject agrees with me on this. I've written up quite a bit on it. But my feeling is that we are essentially in a process of detachment from set theory, even within this part of mathematics.

1:12:30 I mean, we're interested in the structure of... Definitions, I mean, and I suppose we will eventually do things properly in a functorial way. I hope we will come close to the category theory. I mean, even model theory, we're almost ready to kick away the notion of a set theory model. I mean, Tarsier's influence was so strong that it has been very difficult to do this, but I would say that that's... That's essentially all there. Of course, one could be wrong, but that's my interpretation of it. For example, one can expect in certain situations that an open problem in a certain discipline may be solved by taking... Some development in set theory or even in logic? Yeah. I mean, that hasn't happened in the last, I don't know... Well, I mean, Froschowski has produced using model theory ideas, which one can quite readily trace back to Schellach. But, I mean, they had to be geometrized before they could be used. Froschowski has succeeded in solving... Problems, not utterly mainline problems, but very hard problems in which top people are not. In algebra? They did not really believe. They didn't understand the proof, but they could not believe that this could achieve. And we're having a meeting in Cambridge at the moment. We've had a six-month meeting. We've had a number of leading people from Daifat and Germany, French and German people, who have taken on board this logical language, but it's not a set theory language. These people would never, ever use it. Well, I mean, I don't really see any limit, any boundary between, say, contemporary model theory and algebraic or analytic geometry. There are quite a number of cases. We mentioned Duody this morning. I mean, Duody had a very elaborate conception of Duody space in the theory of complex analytic. This has been done over.

1:15:00 Significantly improved by people than one. There's a very fluid frontier now. No, and what about the other, between logic and set theory? Well, I mean, set theory... There's a phenomenon I've been interested in that you'll know more about. You'll see papers in the JSL sometimes called something like an application of set theory to the theory of groups. This is a theme worth putting in your title, that you're using set theory here. Right, okay. So I wanted to say that because of the way the 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 make them mix up these completely distinct oppositions. So, okay, so what are they? I mean... First store their logic. What? First store their logic. Or predicate... Well, I mean, that's supposed to be the dominant concept within mathematical logic, and the one that say that big people have something wooden and so on and so on. These are models of Sarmiello-Frankel theory, where the ordinals, you know, right down the middle, I mean, the ordinals which are... I don't mean for that sort of thing. I mean, how it's being used. So, for example, there are some places where, say, in analysis, where you say you're using the 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... There are particular more or less technical things that set theorists work on which occasionally come up in other mathematics but which are actually independent of the formalization. So that's one thing that's set theory. Some discussions, this is what it means. 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. What else? I wouldn't defend what Leo said. I have about seven of these, that's three. Anyway, go on. There are people who, if they think you're speaking in the plural, will say, aha, you're using sex.

1:17:30 That's right, any collection. Anything that sounds like a collection. Which is one idea. In a 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. Really, there are cases where you actually do use features of the formal system in some way or other, which is different from those that you've mentioned so far. Although those are there are usages because most mathematicians don't even know what the actions of set theory are and why should they? Another important proposition 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, but if they don't, it's not. I mean, the kind of thing you were referring to, an application of set theory to algebra, one can be almost sure that it is... First of all, it will inevitably be about an uncountable case of a case which the algebraists understand absolutely thoroughly in the important countable case. I mean, all the things that came out of Schell's work on the Heide problem were of that nature. Serre had given a perfectly straightforward proof for the Heide thing in a countable case, which was really the main case that was of interest. But the algebraists don't have a means of carrying, the typical algebraists did not have a means of carrying out long-transfident recursions and getting through the limit ordinals, or showing that you couldn't. Attempt to go from the countable to the uncountable shall have typically come along in ZFC. You're doomed to fall under if you want to work in ZFC because here I can do it. I can give you this particular model, but it simply doesn't work. On the other hand, you're not doomed to fail if you say work with the axiom of constructability because you have such a very, very strong form of control over the way the universe is generated. There's combinatorial principles far stronger than the continuum of thought that will get you through to the end. I mean, that is an important lesson for Algebras. On the other hand, as far as I can see, almost all of this, the kind of result which is proved is far, far away from the central interest of most mathematicians. I mean, it's very hard, but in essence, in a certain sense, it's...

1:20:00 And so topology is the same kind of thing, with some transfinite generalization of compactness, for which some conjecture is now actually hard, which hopefully means independence of ZF. That's a very interesting thing in the history of mathematics. I almost should have said that topology would have been dead by now. Even if Paul Cohen did not come around with the technique of forcing, they hadn't proved anything many years ago. They made all the answers, every problem they were trying to solve was turned out to be independent of set theory. And in context you then know that, that's now a positive. Cohen said he had this technique, and Tarski said that. The moment he heard of it, he said, oh they have a method now, they will do everything. Why do you connect that with topology? I don't connect it with topology if that's a very good thing. To me it isn't topology at all. But it's what was called analytic topology. It had to be done at some foundation, at some moment. Or set the original topology. Set the original topology. It was a kind of dead end. It was a... Yeah. There was no geometry whatsoever. Yeah. It's only... The topology is entirely based on set theory. Yeah. Yeah. Okay. Okay. Okay. Like with stone... Stone work with... Well, I mean, stones... I mean, stones... No, I mean, with Boolean algebra and... Well, yeah. Oh, no. That was a different thing. But you can even... You can also sort of go into degeneracy with Boolean algebra. It's easy enough. The only knowledge was rather more central. How distributive are they? Because you detach the geometrical intuition and you make it... I don't want to say that every interesting topological space is inextricably linked to something that we might naturally call geometry. But it's a good idea to have that in mind. I mean, these people who are doing that kind of topology... I used to see it very clearly in Oxford. I mean there were two groups of people claiming to be topologists there. They could never exchange the same thing. Can you tell me the groups? Well, the distinguished group of Atiyah and so on, who were really doing, I mean Atiyah, Penrose, etc. We're finding real spatial topologies. Hitchin, I mean, we're finding connections to topology. Non-algebraic? These people were using all mathematical tools. They were using differential equations. They were using complicated metrics.

1:22:30 They were using K-theory. K-theory and lots of algebraic topology. On the other hand, there was a small group, as there are small groups in various other places, like in Toronto and in a few places in the world, Madison, Wisconsin, They became interested in, you know, generalizations of the separation axioms, where there may be a cardinal involved or some kinds of compactness or new kinds of depositors, but almost invariably linked to... A 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 and 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, there are very deep questions. Unfortunately, we have a sort of master in Schiller who can direct the world around it. I don't, I personally don't believe it. It was pretty well a dead end by that, by the thought of it. They were, but they couldn't wait as courageously 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 these guys. I don't believe there were many of these questions that we needed. No, no, I agree. There are exceptions. I mean, something like the Sousslin problem was a pretty natural thing. But even there, that was a little bit. Almost all the others were so- It was a nice problem. I mean, the Sousslin is a bit more connected to Dedekind and so on. The definition of the reals. That is fine. I mean, the cardinal is not really there. But abelian group theory would also have been essentially dead. All the basic things that have been discovered, the beautiful fights, are essentially always about finitely generating the building groups or upon the eigen duality or something like that, which have got nothing to do with, nothing inherently to do with the uncountable. But there were these questions about uncountable... There are three groups. How do you get at a basis if there is one, or how do you show that some prima facie weaker condition in fact employs as a basis there? Anybody who ever thought about them knows that this is a problem in set theory. You just cannot run the trans-finite induction, or maybe you can if you share that in some particular model of set theory.

1:25:00 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 for you straight away, but not always. I mean, the method of Cohen was so fantastically general. I mean, there's probably never been a method in mathematics which so quickly came on the scene and, I mean, it led to a solution. ...thousands of problems, I guess. Maybe differential calculus. Yeah, differential calculus, yes, yes, yes. But it was a bit... That took much longer to evolve, as I said. I mean, the cool thing, one day we knew nothing, and then the next day they could just... Well, there were more mathematicians around than in the differential calculus. Same with differential calculus, of course. Maybe the time scale is a bit different. But it used to be you'd have some set-theoretic question, and you'd think you couldn't prove it from ZF, and you'd run up against the wall. All you could say is you can't prove it. With Cohen, you could show it didn't follow. You could show it, and you'd do it by introducing variable sets. Yeah, yeah. No, it's actually by introducing a number of different universe. That is another point. But he does, it's a whole idea of going from one, because he does it with standard models, but fundamentally he's starting with one and introducing variations, in other words introducing some new ones which you couldn't have got it by any other way. Because people didn't know about, you know, models of set theory before. Of course they did. They'd been worked by Sheppardson and various other people. Yes, yes. But you couldn't find them easily for a given project. Exactly, exactly. Cohen introduced with this very... You couldn't just sort of start an accountable model and just pull one of the missing elements of the power set of a maker and join it. That's right, that's right. Because you were not going to get it. I mean, that's right, that's right. It was a process of You would succeed. I mean, clearly, there was something there, 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 that it doesn't follow, but it actually gave a whole... Well, for a while, that independence thing, but also there was an enormous... But then that, you know, that contracted too really quickly. So I think that's the equation where, sincerely...

1:27:30 It provides an entire point of view, technique, etc. that has repercussions 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, except possibly some... Or for the other theory. No, only... I mean, there are very... there are general nonsense things that one knows, like, for example... If you work in set theory, if you do number theory inside set theory and you add the immeasurable carbon, you'll prove the unsolved will of the Newton-Einstein equation. So that is a bit of a thing. You can write them down, but they're not the things that mathematicians have. But there's more than that. I mean, it's not merely that people haven't looked at these particular ones. It is that if we simply do the blind decoding of this, write them down, they are... You know, they'll be in dimension 10 or something in the geometrical entities you're dealing with. No one has ever had any pretensions of understanding these and current things. And it's more than certain that one can't use these methods to get down to questions about curves and so on. Because if you could, this would tend to overturn major contradictions in knowledge. So that is, I don't regard that as a genuine thing. I mean, there's no doubt that for... Availing group theory, some other parts of group theory, but always where it involves a trans-finite iteration of some process, and in these questions of what they called either general or analytic topology. I mean, it kept the field going, but these fields are not really mainline fields. I mean, there was really our analytic... The topologist in Oxford tried to sell himself always as a topologist, but the others would have none of it. You know, he wasn't all the topologists for the number one, and certainly not a geometer in any sense of the word. As I recall, just to give a typical problem, I don't remember if this one was actually shown undecidable or not, but... If you take the Cartesian product of the unit interval with a normal space, a normal is a well-known condition in general anthropology. So that's sort of on the borderline. That sounds like it might be mathematical. It's not a problem in the sense of the distribution of the prime numbers. It's a problem that has to do with building a theory.

1:30:00 Well, let's do what's the generality of this. You make this definition. How general is it really? In a sense, what it did, what it was willing to sit through, it... It overthrew set theory, in the sense that it overthrew the idea that there is an absolute short of it all. 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 categories that could be sort of almost equally... And then one searches for counter-examples, and then of course it got to a point where you couldn't find these counter-examples just in one universe. There's one other general methodology, but... The first set theory has been used in a very delicate way, I mean, taking seriously the idea of actually working inside a model of set theory, much smaller than they could be in whatever universe there is of set theory. I mean, again, Scheller, for example, has occasionally solved problems which are essentially combinatorial problems about the real numbers by going, by passing into some special universe of set theory which nobody... Which satisfies probabilities that no one expects or wants, even wants to be true in any perceived absolute universal sense, working in there and then using some metamathematical principle to drop back down to show if it's, if the thing is true in this special universe, then it's true in the real universe, then Kreis will give a beautiful use of this in connection with Gödel's model for the consistency continuum hypothesis. If you make a statement in arithmetic, which is demonstrated in arithmetic, and that includes very often statements in homotopy theory and so on, they can be coded that way, if you can prove them using the accident choice and the continuum hypothesis, and the other accidents I've said to you, then you can find another proof where there's no accident choice and no continuum hypothesis.

1:32:30 That's more for people's concepts than anything else if you don't like that sort of talk, because you want to really get more information. But I would say, I mean, all of them are very special kind of problems which have to do with the structure of the theory. Yes, yes. For example, if one thinks of the Poincaré conjecture, does it help solving it, some of these developments? Apparently, no. No, no, no. I mean, I was slightly surprised a few years back to hear that Frank Quinn, I don't know if you know him, he's a well-known low-dimensional topologist and somebody has worked on him. The one who wrote this article? Yes, yes, yes. And I knew him back at Yale, and before he was sort of controversial in that sense. And I mean, there was a while where they needed to do very complicated kind of surgery in low dimensions, and they at least believed that they were eventually using fairly complicated set theory things. There were limit arguments. There could have been, I suppose, even transmittal recursions, which were dovetailing arguments that we would use systematically to do recursion theory or something, but they didn't typically use. But again, I think it was a technology which was temporarily useful to them. It didn't lead to any real fundamental advance. Well, at the Schenectady meeting, they had the set theory session. The Schenectady meeting, a long time ago, they had set theory, number theory, category theory. I went to the section on set theoretic topology. And first of all, they, in practice, identified set-theoretic topology as topology getting to questions that are independent of Cf. 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 action is true, we get the other answer, the canonical one being the white-headed conjecture. 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... I mean, there are masters of that kind of thing, for, you know, translating or trying to insert something like a homotopy group into this in a totally artificial way. Now if I can take this a little bit backwards again in the context of Hilbert and say about what I said before of the separation between foundations and the... but there is also a union which is...

1:35:00 And it has been, somehow it's implicit in all of 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. We've all been waiting for it. And the interesting thing is that for Hilbert this could have a use in physics. 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 he's speaking about an empirical status, but in a sense it fits to what you say here about the homotopies or whatever. 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, if you want. I mean, set theory has actually not changed. I mean, actually, my set theory has been no change in it since, well, essentially since Ramello, Ramello-Frankel. Except, I mean, except maybe for the, I mean, there is this large cardinal program. But that's one sale. No, no, that's what we did as a collection of people. Yeah, okay. Some kind of coherent picture of the large cardinals. And it is extremely coherent. And it does impact on... Well, analysis in the sense that it has implications about the vague measurability and so on. I don't totally buy it. It's kind of remarkable. I agree. No, no.

1:37:30 Who has ever made it look like this kind of cheating with measurement cards? No, no. All I'm saying is that that hasn't been... No, I have measured... Well, I don't know. I used to work with that. Not to any real depth, you know, 25 years ago. Yes, and there are some remarkable features, but... They're still exotic. They're still the core of mathematical practice, sort of those odds of Melo-Franklich, at least in principle, and that core act has not really changed, not for a long time, and nobody even, Witten is trying to show, he's not going to succeed. None of these people are going to change it, far as I know. 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. That's not a legitimate contribution to any significance for any kind of unity in mathematics. I just don't think it has. Perhaps it has at some point turned into a language. And this is the curious thing about Penn Maddy's project. She wants to say, how do we establish new axioms for set theory? Well, not on philosophical grounds. Sure, in olden times, up through, say, Zermelo and Franklin, they gave philosophical explanations. But that's not how we do it since World War II. Here's how we do it since World War II. Oh, incidentally, we have never done it since World War II. Well, when the French, when Bourbaki says ensembliste, it always means just the contrast between structured and unstructured. But again, it has to be structured in a very informal way. Yeah, I'm saying in an informal way. It always means things are less so. You know, the categories that have that flavor of being unstructured do play a preferred role, but of course all the models of ZF are more or less dependent to that. Yeah, yeah. So that would be inherited from Emmy Netter's school, their use of set theoretic. Yeah. Set theoretic is, it's a subtle thing, it's certain ways of invoking set theoretic properties are set theoretic.

1:40:00 Right. So what I'm getting at, I think it's especially Dedekind but also Cantor who made a little more conscious this idea that you study the mathematical objects as interpretations of some kind of abstract structure into abstract sets, where the abstract sets are relatively structureless. And if you, you know, if you independently of these historical controversies and so forth, if you say, well, why should one do that, you see, it seems to me that one answer is basically the completeness theorem for first order logic. It means that you're interpreting your theory into this background. The background doesn't really have any contaminating effect. It's sort of neutral. I think that at the early stage it's more a practical thing again. It's the gradual understanding that starts with Dedekind and then goes a little bit with Hilbert and then strongly with Netter that if you start treating things in this way you discover certain Let's call it mathematical, the underlying processes that you were not aware. For example, it goes from dedekind, the question of factorization in fields of numbers, algebraic numbers, to polynomials. You know, that's when the click comes in, the netter puts them together. Suddenly, I mean, things were, people did know these things. It's not that it was completely unknown, but then she's able to see, look, there is a structure. If you have this property, which is a property of sense, you have a chain. If you have a change, then you have unique factorization. If you have an ascending change, you have this. So it's a great discovery. People see that you can look at those things that you haven't looked before and then you have a lot of theories. It's a tremendous instrument. What I'm inquiring about is why is it so useful and to what extent is it useful? So I'm saying, I'm making a general statement about that. You have the completeness theorem for the first order of biology, so that, you know, anything that's going to be true about the interpretations of the theory in this particular kind of background is going to be provable, so that there's, it doesn't introduce things beyond.

1:42:30 Intuitively, this is exactly the sort of thing that's not true about the cohesive world, the continuous continuum. It surely must be having some influence, you see. So that has various expressions. I mean, one is the fact that there is no completeness theorem for higher order logic in the same way that there is for first order logic. In other words, it seems that once you go beyond this, this, well, it happens always when you try to make universal elements which have no properties, well, that's precisely a property, you know, so that this point of trying to make things property-less. In fact, introduces a very strong, is a very strong property on, on, on, so that, so the higher order things do not interpret faithfully in the same, in the same way. So just, so in terms of the, the quest for some kind of completeness for higher order, that, which is seemingly very difficult. But the other point is that simply, again, simply from a practical point of view, One wants to interpret these theories not in a featureless background. If you take, for example, the theory of groups and interpret it in a smooth background, it becomes the theory of Lie groups automatically. Or you take the theory of vector spaces and you automatically get Bonac spaces or something like that. In categories that have a lot of cohesion, it produces a lot of stuff. And this is certainly in a practical way a very useful kind of point of view because on the one hand you get a lot of stuff for free and on the other hand you create a lot of questions that are... So this question of the discrete background has to be taken in a dialectical way, at least in these two aspects, the higher order part and the... I think it also, set theory did, and this is of course the philosophers of... There really was a question in the 19th century, and it arose also, I think, in connection with Dedekind. What were ideal numbers? What are they? There was always some kind of... Mathematicians may not have taken ontology as seriously as philosophers, but nevertheless, these questions do arise. The whole nature of ideal numbers, points at infinity.

1:45:00 All these, and look at the debate over imaginary and then look at the terminology. So I think ontology was an important background issue in mathematics, you know, and I think that in the case that there did actually, after all, come to answer certain kinds of ontological questions. For example, The use of ideal, you know, a set, if you like, of numbers or whatever is an ideal number. The whole term ideal comes, of course. And also, what do we mean by this whole thing of using equivalence classes, which was a very important use, early use, of set theory. And where did Dedekind's theory of change live? Dedekind just talks about these things. And Dedekind, I think it's fair to say that from a certain point of view, Dedekind really does confuse... Proving there are infinitely many things from proving there's a set of them. Right, right. And it was worth spelling that out. Yeah, so I do think the, you know, there was a fairly simple... Of course, certain things were lost with this discrete bed, but I think the ontological thing was an important drug. You say, what is a group? Well, it's a set, da-da-da-da. It's an answer to the question, what is? And I think that question, although maybe not a primary, it's not perhaps the major question in mathematics, but it always played an important role, I think, in the underlying development of mathematics. Except that we provided an answer, which is now, I think, being transcended. Yeah, but on the other hand, people, once you make, you know, a definition of group and define your various group theory notions, your quotients, et cetera, set theoretically, the most unhealthy thing to do, and many people did it, would be to continue to study general set theoretically. No, that's right. No, that was what, I mean, that's still going on. That was the downside. It was helped a little bit, again, by Cohen, can we all make sure that some of these problems are going to be done. I mean. What you've said is unquestionably correct, we wouldn't be where we are now, we probably wouldn't have any of what we're talking about now, had one not been able to get that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that, that,

1:47:30 It did, in some form of what do we mean by an abstract structure or form, it did give, it's true that you could question the answer, and Bauer did, and a number of people questioned set theory, but nevertheless it did give that kind of definite, apparent definite answer. Cantor, I think, carried it. Perhaps to extremes in his rejection of saying everything really has to fit into this and that's really all that mathematics is. But nevertheless, I think it did answer those sort of questions. I'm not sure that's what I'm misreading of Cantor. I like your local point that Cantor on analysis says this is how I'm going to describe analysis. That's all there is. I'm not even going to justify it. But on the other hand, to say that he's done all of mathematics is this, I think. All right, all right, I may, I exactly, but I do think, I, I, yeah, okay, all right. It's just that I was so stuck in reading the gridlock in recent years that, again, that I, you know, he's... And then other things that he really does have this vision, this great vision of discreteness and somehow, John, everything else has to kind of fit in. No, no intuition of cognitive. Anyway, maybe I exaggerate, but because I thought it's a recap from the other side. I was a set theorist, biology. And so, and of course, Cantor, wow, it is amazing. You're fascinated. It's a fantastic achievement. Well, I was looking at basically writing a book on the continuous and the infinitesimal, you see, from the other side, you see, and it's very interesting to see what Cantor, in other words, trying to understand why he came to have this rather, well, it's essentially a rather narrow view of the continuum as opposed to this really quite revelatory view of the infinite. Anyway, so perhaps I exaggerate, but anyway, it struck me. Anyway, reading that book, reading Cantor's article, incredibly. I had to point out to Jose Ferreros several points of a very fundamental nature. After studying Cantor and all this for a long, long time, he still hadn't picked these things out. It's very dense, I guess, very dense with content. And plus we come with a real strong preconception. There's a whole literature on Kantor explaining the role of the extensionality axiom in Kantor. Well, he never quite says it but he meant it here. He never quite says it but I can't have imagined it wasn't true.

1:50:00 Because we know that sex are just collections and extensions so he must have believed it. And then you read that Zermelo in 1935 knew perfectly well that Cantor could not state extensionality for his turn on Stalin and criticized him for having a theory that wouldn't let him do that. Yes, because the whole point is that, yes, it's Zermelo's conception that sex are just collections and extensions. You can't have a set of dots, some sort of a P of dot. There is no P of dot. There are no, they don't have any properties. At least not in that sense. One of the things that most strikes me about this Cantor thing is how hierarchical. Leo mentioned the Noether. Nobody can doubt that these are the right thing for huge areas of mathematics connected with geometry. Because they always involve dimension. And also we get them in this business of connecting the number theory deal case and the geometry deal case. This is a central theme leading up to the solution of the Debye conjectures. But Cantor, so it's a kind of induction, we start with the introduction of Debye and then we have this Maltheian induction which is... But Cantor's induction, which is so much more fantastic, in fact, is rarely used in mathematics at all. It's a big mystery to me. Of course, it was used in counter-example time and in attempts to get at the continuum of forces. But in geometry, for example, does anybody know an example where there's an essential use of any kind of trans-financial induction in geometry? There's the Tohoku paper. I don't exactly know what that bit occurred to me when you said it yesterday. That is fine. That is true. Yeah, that's perhaps the answer. A lot of that could be right, I'm sorry. Yeah, that's true. That is a more sophisticated notion of geometry than I had in mind when I asked the question, but you're right. But it is 50... Yeah, 57, right. That's a good answer. There were constructions using tractions. I thought I was... I'm no expert in abelian... You know, there's that... There's something more... There have been some... Well, like the earlier part of the paper, I mean... Well, also the other... This is completely unnecessary for a topological case, but still, tohoku could be used as a long transponder. I just want to put an emphatical point that transponder induction, well recursion really, was used rather a lot, I think, up to the time, I mean, with the use of the axiom of choice.

1:52:30 Well, you know, for example... It must have been used in the proof of the Hamel, you know, that there's a continual... Oh, yes, yes, yes, yes, yes. Then all that got swept away by the use of maximal principles. But this, yeah, but even more, I mean, this is the realm of... This is the realm of wild analysis, basically. Yeah, I guess that's true. But the example of Karmus is... Yeah, you're right, you're right. I can't think of any examples where, despite the fantastic power or beauty of the thing, I mean, it's, it just doesn't seem to... Sayer or Atiyah had ever, in their lives, used translated... But what about the construction of maximal ideas? Oh, wow! No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, On the other hand, he certainly is quite avid at work with, I guess, with Zorn's lemma consequences earlier on. It's the most fundamental or powerful seeming application of the ordinals. So it has been pointed out that you don't need it there because in fact the same kind of trick that Dedekind used in the infinite intersection from above to find the smallest set that's closed under the derivative, you don't have to do it that way. The uniform method of Dedekind actually applies to that case as well, without ordinals at all. But I mean, the Cantor-Bendixson theorem, I wouldn't put that quite in the realm of wild analysis. I mean, that's already set theory more properly fit. Yeah, it's definitely classified. That's just a theorem ultimately about, it's a non-trivial thing, about closed sets, the structure of closed sets, which is maybe a... A tame version of the continuum, or something like that. I mean, that's a borderline case, I would say.

1:55:00 I think another important, just to try to give some context to this thing, is that we spoke about Cantor, we spoke about Dedeckin, very nice, and then we have all these developments. The way from one another was not a necessary one, historically speaking. I mean... Certain things happen. For example, the role of Hilbert was fundamental in attracting the attention of mathematicians to the continuum hypothesis, let's say, and things like that. And at the same time, adopting Dedekind's point of view in number theories as opposed to chronicles. So what I want to say with this point is we look at these words very strongly and we of course find very interesting things, etc. Contemporary mathematicians not always did. And if, for example, Hilbert had opted in number theory to go the other way and choose the chronicle approach and not to put, for example, the continuum hypothesis in the list, I mean, at least chronologically things would have been different. I'm not sure at all that people would have paid so much attention at the beginning to these techniques, to this approach, and things may have developed differently. Perhaps in the end, I mean, when you have powerful tools, they tend to appear somewhere and to be used, but at least it was relatively quick to happen, from the cantor, let's say, papers, to the centrality that it was accorded, and it was because certain historical things that could have been otherwise. Very much so. I mean, at least from the fragmentary knowledge I have, it seems that there was a major discontinuity. Things could have gone many different ways, but because they're malo in particular... I swore a philosophical allegiance to Frege instead of to Cantor, therefore a whole line of development. If people had followed Cantor and Dedekind more directly, it might have been quite different, at least what we know as set theory, not mathematics, because this decision of Zermelo, why ever why he made it, he never even gave an argument. But it was not just on one point, it was about, you know, Frege versus Cantor, I am for Frege, you know, and so he followed this other method, but then you come to all the later people, as far as the philosophical starting point is concerned, they just refer back to Zermelo, von Neumann refers back to Zermelo, you know, so…

1:57:30 So this bifurcation, this seizing upon the iterated membership business and making that the central formalism. It could have gone quite a different way, and apparently it was just this quirk of this one man who... So I'll ask you something about that. Sermeno started as a mathematical physicist. His doctoral dissertation was with Planck, and he was involving questions of kinetic theory, you know, the criticism to... The paradoxes, the criticism to Boltzmann, and so on. And he came to Göttingen. That at a certain moment was a very powerful center and Hilbert was telling people what to do and he just started with his axiomatic program and he said to Cervelo, here we have this problem, please go on and take it. Of course Cervelo could have said no, I don't want, but something happened, you know, there is a lot of historical contingency, I would say at least in what has to do with chronology. Because maybe in the end you have no choice and you would have things, but things happen in a certain way, in a rather contingent way. This is the very nice thing of this story. But the order in which they happen does affect the perception of the seminar for a long time. But Zaneva was concerned with developing his... because he's trying to secure his proof of the well-ordering theory. Yeah, yeah. I mean, there wasn't a specific problem, I mean, it was exactly, he didn't even, in the first paper, the 1904 paper, he doesn't really have a system that worked out, so, you know, and it was really criticized. We know from Moore's book describing it, you know, there was a big criticism of it, they thought it was circular, you know, you, well, all your choices, what do you say? There is a book by, by, what's his name? Not by Moore. What I mean is that it's very interesting that in the later development, I mean, yes, he did develop this and he did it, of course, he came up with this set of domains and then, you know, structured by membership and the probability of membership.

2:00:00 Well, of course, it had to play a central role in his... Oddly enough, the iterability of membership plays no role whatsoever, really, in his proof of the actual choice, of well-ordering from the actual choice. I mean, you can look at the proof of... The beautiful proof of this is what's called the Zermelo's theorem of Zermeli, they call it in Bourbaki in chapter 3, and I've analyzed quite a lot about that proof. It's nothing to do. Of course, there's not really any set theory as such in Burbanki at all, and they don't use it. But in Zermelo, it's somewhere between a trick that he found worked when he needed axioms and a philosophical... Exactly. ...that he eventually attributes to something he learned from Brady. Exactly. It's not in the mathematics that it requires that. No, it isn't really, yeah. No, you're right. No, no. I think he wanted to get it beyond Krighiunt to remove it from criticism. And it's brilliant, there are reasons, but it was rather effective. I mean, it actually convinced, I suppose, more mathematicians than before. It also, incidentally, of course, had the effect of interest in that system. I mean, we later have Frankel, you know, and all these people who got interested in the actual... There's a lot of formalism, if you like, the framework, if you like, that Zermelo had introduced. But it's these sort of intricacies that really don't come from a specific problem, as so often in mathematics. I like the thing you're introducing. It's precisely if you believe that mathematics would necessarily have solved the well-ordering problem that you have to realize it wasn't necessarily going to be through Zermelo. Yes, that's right. And get it again in 1905. But you know the argument that... Because he doesn't do the remedial number in 1904, and in fact, you know, sure, Frager's construction of the natural numbers and Ehlers' proof and actually the construction of the von Neumann ordinals are all special cases of the same kind of general result. I mean, and you could formulate it to really, well, you do it, it's more easily formulated, you know, you could formulate it, of course, also in terms of, of, of maths from power, you know, from, from, from exponential objects to objects, but it only needs one, you know, it only needs one, one membership, one, one. Remember, between an object and a set, perfectly reasonable notion. It's iterated membership that's really questionable. You need such and such, but you don't need any homogeneity.

2:02:30 No, you don't. And indeed, Zermelo doesn't use it. It's not really a... And that's why Zermelo states it clearly, not in 1905, but in 1933 or 1935, when he writes his footnotes. That's right. That time he's... That's when he writes his footnotes. Also, very, I'm not sure, very happy with what had happened. Well, you know, he had, he had, with what happened in logic, he didn't understand Goethe's. Well, he felt that Feigl had betrayed him. Exactly. Exactly. Exactly. Yeah. So, but yeah, but the, this, this stress on, on iterated membership, it becomes explicit years later. Yeah. Yeah. Yeah, which, I mean, the notation is due to Peano, actually, but Peano and Frege conspired on it. Yeah, absolutely, there's ST, you know, the singleton, then that stuff, yeah. But this was, Frege explicitly wrote to Peano about, oh, right on, that's what we want to do. So in that sense, it's Frege's influence as well. Although Frege, sorry, no, I just say, although... Frank, of course, wasn't doing set theory. Frank actually has one level. He has objects and concepts. And in a sense, of course, the concepts get, they get stratified in some way, but basically he's only got two levels. And he doesn't use mentorship. He has these things called extensions, which he admits to. He doesn't want to go into what they are, and in fact it was later assimilated to sets, but actually he doesn't say that. I mean, they're different, Frege isn't doing set theory and such. He's working in extensional logic. That's right. It's a different, I mean, there are other reasons why one might find that objectionable, but it isn't because he's doing set theory. Well, one of the things that I recall Bill remarking in Lawrence is that the whole line of development of Frege... Both reflected and itself resulted in the persistent neglect of codomains. Perhaps you could elaborate a little bit on that point and get me as... I keep coming back to this idea that one of the main advances of category theory is simply to note that maths have definite domains and definite codomains. In order to be able to compose, the co-domain of one has to be exactly the same as the domain of the other.

2:05:00 The things that come even before associativity and all that. Well, first of all, the properties of maps depend very much on this. Whether a map is surjective or not obviously depends on fixing the codomain. And surjectivity is dual to injectivity. There are these contravariant functors that you apply even in elementary set theory to transform one into the other, so it's very unnatural to... To restrict oneself to subjective functions, on the other hand, to assume that the codomain of every map is the universe, which is essentially how, you know, if you look at the notion of function as formalized in ZF, that's essentially what it is. If there is a codomain, it's the whole. You've got a domain, and for every element of the domain you've got a value, but not a set. Well, where is the value? It's to live in the universe. You don't know. You have to add an axiom or an axiom replacement to ensure that it really does have a domain. To put a bound on it in general. So just this simple fact somehow, which I think could have been appreciated. Well, topology is already in the 19th century. A continuous map went from this space to that space because it didn't make sense to worry about its geometrical properties unless you knew where they were. Did it divide off a region? Well, I mean, a map from the circle into the plane divides off a region, not into three-space. You have to know it's into the plane. So topologists did that from the beginning. The first algebras to do it were the Netter school after they got interested in topology, the ones who were doing topology. Because they needed to talk about maps between homology groups, and they needed to know whether it was onto or not. For Netter, still, a group homomorphism is onto. But her school, all of a sudden, it is not always there's a group here, a group there, a homomorphism onto some group of this one. So how do you relate this to Frege? Well, Fred, because once you decide that everything in the universe is identifiable in its own terms, you apply a function to an object, you get another object. You don't have to ask what set is it in.

2:07:30 No, because there's one... Frege makes this unjustified identification of concepts as properties. So everything is really a property of this one universe, you see. But that comes a bit later, because what he does... No, that's true, but... He actually starts with the idea of a function, right, which is defined on the universe of all objects. Even the domain. Yeah, yeah. He has no typing, you know, he just... And then he says, well, a concept is such a function. Right? Which only has two values. It takes the values 0 and 1. And of course, he then assimilates that to property. Right. That's right. That's right. That's important. And that's the order he—but of course, because he's got this kind of unbounded notion in the first place, where the domain of the thing is already huge, right? It's the universe of objects. Right. Well, what's the code in it? Well, the universe of objects. For Graham or for set theorists through most of the 20th century, you can talk about what's the square root of two. It's the square root of two. You don't have to say, did you mean the algebraic number square root of two, the real number square root of two, the complex number square root of two? It's just the square root of two. Whereas computer programmers now, these are different functions. They need to know which square root of two it is. And in fact now in set theory books, a function as defined in set theory books today, has a set that it's defined on, takes values in another set. It is normal now. There's a lot of information out there that says that a function does have a domain, but it was not normal up to 1950 or so. That's right. Because, for example, if you want to talk about whether a function is onto or not, you've got to have said onto what? Of course. I mean, it doesn't appear before 1950. It doesn't become standard in set theory books. It doesn't become part of the basic... ...conception of what a function is. Yeah. Before that you'd say, here's this function, each one... It's a reflection of Frege's point of view into... And the change comes because of the stress given by category theory to that or for other reasons? Category theory summed up and concentrated the features of topology and algebra... Where that notion wasn't actually...

2:10:00 It's an explicit one. It's absolutely essential because simple functors, like, take the set of components of a space... This does not preserve the fact of being a monomorphism. A monomorphism may become a surjection or vice versa, you see. And these are very basic functions to this conception. In other words, if you're in a context where the functions did really preserve monomorphism, then you can sort of get away with this view that there's no photoman. But when you're dealing with, immediately when you're dealing with the qualitative features of cohesion and non-adjective policy, without even introducing groups or anything, just a set of components, it doesn't preserve the model theory. I mean, Tarski somehow insisted on this directly in Fregeian tradition. I mean, functions, he didn't like our functions. There's symbols in there at all, there's relations, there's a one-sorted structure, which just suppressed almost all the most delicate things you wanted to find in the structure of definitions. There's somehow a misguided idea that this is a simplification. You simplify things because you omit a dimension of a code. But that, you see, on the other hand, the type theory did the opposite. Type theory did the opposite thing, because in type theory, the types themselves are co-domains, but you don't specify clearly what the domains are. You have to look at the formula and figure out what the few variables are, and even then, you often want to consider functions which are nominally of certain variables, but don't really depend on some of them. Things that factor through projection maps and so forth. So there's a complete ambiguity in the domain. So many foundationless sort of defending set theory, they say, well, okay. We'll look at category theory. Well, that's really type theory. You already know what type theory is. They're equally wrong because you need both domains and co-domains. It's the most elementary thing. It requires a lot of... Well, current type theories do have to make that distinction. Of course. Because they're actually being applied. It's important. You can't get it off the ground without... I saw this with the impact of computer science, but it has to work in computer science, so this has now come into focus, that this neglect of domains has become more apparent.

2:12:30 Also, if I can pick up on what Bill was just saying, one of the ways in which he has certainly taught me to think about the axiom of choice, which is clearly one of the conditions which... The idea of abstract sets satisfies, which reflects the constancy that's being enforced on the objects in that, is precisely in terms of the, it's expressing the idea that there are no obstructions to the existence of inverses of maps into the domain, which again, of course, applies to this particular one very special. ...instance of the categorical cause of the constancy of the objects. One has to think a little bit harder in order to recognize the related code, and perhaps it's a little less obvious to spot the geometric meaning of the... Yeah, I mean, with these abstracts, there are no properties in a way. There are only properties in additional given data, in a way, and rather than something that's just there intrinsically, so... If you take a set, maybe you know it's countable, but is it the integers, the naturals? You haven't decided yet, until you've decided what structures you're going to put on it. You're going to see it with a successor function. Then it must be the naturals. But it only became the naturals because you saw it this way. You've got a countable set, another countable set mapping into it but not onto. What's that? Well, it's only if you say, okay, on my base I'm going to put this successor and this is the evens. So even becomes this insertion. It's not that anything down here was even. It's the insertion of another set into it is the evens. In other words, a subset is not a set. It's more than a set. It's an insertion. I think of the action of choice. If you think of the action of choice as one... Look, the arguments against the action of choice you'll see as well... I think there were two of them. One was the question of uncountably many choices. There was that issue that what does it mean and so on. It came up from the criticisms of Burrell and some other people. But the other one really was...

2:15:00 I think the idea that the choice is really definable. I mean, a set was something that had to be thought of not just as a bunch of a combinatorial thing, where indeed combinatorial arguments in some intuitive way tell you in some sense that a choice set exists. You simply shrink, you have a bunch. A bunch of disjoint sets, say, you know, you think of them as, you know, bags of dots, whatever you like, you don't know that, and then you have a choice set is one that intersects, right, each set, no, no, no, no, no, no, no, a choice set, a general choice set, just either set, so you know they exist, you simply take the union. Now you shrink. You know, you have this idea that they're just like you can throw out. Eventually, you sort of intuitively, in this combinatorial conception, you'll get a set which only has one element in common with each one. Now, Bernays actually, I think somewhere, he calls, and he has, he's an argumentative guy, he distinguishes this combinatorial notion of set, which he clearly means something like Like bags of dots, you know, I think that they're just, yes, I think that's what I understood what he means by the other notions, you know, extensions of properties and so on. Now for the combinatorial notion, in other words, I think the notion of an abstract or a cardinal sum, the axiom of choice is correct, it's intuitively very natural, you don't have to specify. It's all about how to define and make the choice. That's not the point. It's something built into the kind of combinatorial structure, and I think that's why I take it that that's the reason, what good reason for taking the axiom of choice as being correct, you know, for the category of, for the category of constant sets conceived in this way. Only gives us this combinatorial information we can't get our hands on and it can't answer any of our questions. It's not that it's not true, it can't answer our questions. Now this is also a mistake. I didn't know Parker, wait, the way is this? I didn't know that Parker... In this 1912, he's got like four... I didn't know he said anything about the absolute choice. Yeah, pragmatists say that Zerbino's well-ordering theorem is...

2:17:30 Oh yes, that's right, right, right. Because it gives you a well-ordering. It shows you there is a well-ordering. But it doesn't really tell you anything about it. You have it. You can't have it. It's a constructivist... Well, no, because he doesn't... Even his pragmatists don't say the well-ordering doesn't exist. They just say they can't use it. And he doesn't unambiguously endorse the pragmatist. In fact, as I say, he says they can't understand the Cantorians, and he can. So evidently he's not just one of them. Now orienting people differently on this question, on this kind of broad foundation which starts that everything could be reduced to a set of three, that one could see, geometric and analysable in terms of additional structure imposed on some underlying set, was precisely the Banach-Tarski paradox, which is obviously… Well, I mean, it just meant that you analyze it. It's true that the Banach-Tarski paradox is correct for a couple of them. It's just not a paradox if you conceive of it in the... In this pure combinatorial sense, it's not really a paradox at all, it's when you represent continuous structures that you get this paradoxical aspect, which is a very startling thing, evidently. Then, since we should expect our mathematics to be able to make explicit concepts needed to capture the notion of intuitively, the concept of the continuous, or the many forms of intuitively, but certainly... There were a lot of notions of set and set theory floating around which got conflated and what appears quite natural, so to speak, in one manifestation, if you like, of set theory appears quite paradoxical in another. But yeah, it did, of course, have a startling result. It wasn't the first of them, of course. It was Hausdorff, two-thirds of the sphere being equivalent to one-third and so on. Yeah, Hausdorff really did the main step.

2:20:00 Yeah, the main step was Hausdorff. Sipinski did something along those lines as well. But I guess it's true that since set theory was going to be used as the official, well, it had become the kind of, well, foundation, framework. Or language within which advances in analysis, particularly in this case, and measure theory and so on, were being made. I mean, all the work being done in Polish school, for example, at the time, I mean, Badak Tarski would consider this problem because, you know, as Polish mathematicians, many of the advances that were being made in the use of set theory and analysis were being made in Poland in the 1920s. I'd like to be able, or someone to be able, to say something more about how this, all this idea of using logic and theory, etc., in the Polish school arises, because I have the feeling that it's not the same continuity like, you know, in Germany, from Ganderling. No, no, no. Something strange happens there. The first issue of Fundamenta. Yeah, when is that, 1913, something like that. Yeah. I think it's 1920. It's after the First World War. On the front of the piece, you will see a Polish officer in full uniform. And then there is a manifesto explaining that this is for Polish nationalism. Yeah, they definitely have their own ideas and their own... Well, I think as far as logic is concerned, there was a strong influence of scholastic... If you notice, all the Polish logicians, they could quote all those, all the medieval philosophies. Neotomism. Yes, and it's quite a strong tradition, which, certainly as far as logic was concerned, and you could see that in the work of Lech Niewski, you know, the more specifically philosophical. The school was in fact founded by a humanist who wasn't really one of the practitioners and he brought in this scholastic. It may have been him, but I'm not sure. The only thing the Pfeffermus book is good for, we'll get all of this stuff.

2:22:30 The Pfeffermus book does have an adequate account of all of this stuff before it gets... I think also Grattan Guinness has a chapter on that. But it's like a chronology and I really, also with the Hungarian mathematicians, but that's a different story because it doesn't have so much logic and set theory, but all these topological, set theoretical ideas, it's a strange historical phenomenon I think. I think it was because there were philosophers who were not really very mathematical at all, who had been instrumental in educating people like Tarski, and of course there were domologists. It was a strange way to do it, it's true. And there was a strategy that Brouwer seized on and that I suspect the Poles picked up from him, although Brouwer got in on it sooner, of saying this topology, this is hot stuff, this is new, it's not being done in the world centers of mathematics. We can be great in this, in Amsterdam, or in Le Boffre, right, I mean the so-called general department at Le Boffre also played a key role, not a math department, but a general department. And who was there in Le Boffre? Rotated. Rotated. When Piłsudski wanted to step back and not be the absolute dictator, except in reality he wanted to put up a figurehead, then there'd be somebody like that, and then he would change his mind, that person would go through the same general department, so this kind of people were completely mixed up and sometimes identical with the ones who were actually doing the mathematics. This is described in... A book by Kwiatkowski, I believe. Kwiatkowski, I believe. It's called Fifty Years of Polish Mathematics. Mastowski wrote a book. No, Mastowski wrote Fifty Years of Foundational Statistics. It wasn't specifically Polish. Thirty years, it wasn't specifically Polish. It probably was Kwiatkowski. Mastowski's books are not very useful at the time. No such thing has been written since. Many of his logicians were working in the field of science.

2:25:00 This predilection of Charles Tasker's for a single-sorted universe, do you think that is early exposure to the... No, I think this is probably... I mean, you didn't want it to be sort of definite. And also this kind of Occam's razor stuff and all the rest of it. You know, they didn't want too many... There was no reason at all to go into any other... No, no, I wouldn't. I plead guilty. Dover's going to reprint Bell and Slobson. We're not going to get a chance to change that, Bill. Oh, do that. We don't want to reprint Bell and Slobson. Bell and Slobson was, I mean, okay, maybe you understand what I'm saying, but Bell and Slobson was a profoundly un-Tarsian work in life. I hope you take that as a compliment. In this respect, I do. Because almost everybody, even now, who reads it, regards it as a delightful book. We went down that road initially, and then of course we reintroduced them, because they really need it. When you're doing algebraic theories, naturally. That's the problem. Even now, many of my contemporaries that sit at the Newton are insistent on using relational language. I mean, Zilber, who has a very deep understanding of algebra and geometry, It's funny because, you know, it's like, I think Church's lectures also, but you see, somehow these are basic principles which are not justified at all. That is, there's no argument given why you should accept, you know, single-sword relations and not functions and so forth. No, this is just the way we do it, and you learn that on the first day. Well, and this is how it looks, and this is how the thing, and it's incredibly infectious. I remember from one year at Berkeley, From one year at Berkeley, I had all these incredible habits, and in my early papers you'll see, you know, like, the variables are x to u, x1, x2, I mean, this is utterly crazy, but they always did it that way, and so I thought, well, I want to make use of this, you know, this logical framework. Non-standard analysis in Robertson's original formulation is very much like that.

2:27:30 Robinson was like a slobby version of Tosca. I know that. I have a story about him that's actually in Daubens, not very good, but he does reproduce this story. I remember when I, this I'll mention it, I heard Robinson lecture in 1965 in Oxford when I was my first year as a graduate student there. He came for a term, he was visiting, I can't remember what it was called, and he gave these lectures on non-standard analysis of Maché Mako, who had been his student, you know, that's how I got to know Maché. Anyway, a little bit of that, but it's sort of funny. He gave these lectures, and he was a very charming person, of course, while he was very nice, and had a party at the end, really very, very nice. I have the impression, in these lectures, that he had a rather muddled... I remember thinking, I can't remember what the result was, although the result, the actual presentation was fascinating because you were seeing the notes of the first time and marvelous things he did, but his method of proving things puzzled me because there was one point at which he had written some proposition, say P, and then to prove P he would seem to assume not P. And then you'd think he'd be arguing for a contradiction. Right, so we're going to not P. What he'd do is then somehow prove P quite independently. And then that contradicted not P, and that gives you the double deviation of P. Hence, P! This is what we're supposed to be talking about, and then I'm going to tell you that. I sort of tried to trace it out, and I think this is a momentous method of proving P. Somebody else taught me exactly the same thing. It's very strange. I don't know who told me this. I mean, I'm your audience. That's what he ends up with. But this is in Dauben's, I didn't read Dauben, but he asked for it and I don't think, but this little thing is in his book. Somebody else noticed that. Yeah, somebody else told me. It's very strange. It gets into writing in Brouwer at one point when he's proving the degree theorem. Degree isn't very another hot topic. He gets these four numbers, and he says, I want to show these two are equal. Well, I can show these two are equal, and these two, and I can show these are equal.

2:30:00 Suppose these two were equal. I remember once shortly after I went to Yale, no it was after Robinson's death, I went to New York and I met Wilhelm Magnus, who was a great geometrical group theorist and a very charming old man, and he almost made me fall off my chair, he said to me, Robinson is a brilliant man. I mean, nobody I ever ever heard said such a thing. I mean, there was always, there was always nuggets of anything he said. His presentations were very, very odd indeed. Anyway, of course, all this, you know, Moshe Markover had been Robinson's student, of course, in Jerusalem, and of course he was really impelled to try to clean up, you know, the presentation of non-standard, and of course he wrote that thing with Hirschfeld, you know, non-standard analysis without tears, and, you know, and because he said, there's got to be a better way. Is it better? Oh yes, oh yes. Oh yeah, no, because of course it's become, if you look at... Well, it's great to say, Moshe has a very nice account of a kind of distilled version of his approach to it, to his and Hirschfeld's approach. In that book I wrote with, he wrote with me, Bela Machover, you know, Mach and Bela over and over again. Well, the other one is Slav and Nelson. I'll leave you to draw your own conclusions. But, yes, it's a very nice presentation. It's very pretty. He substitutes, of course, the types. You just do essentially what he called... Because there may be those structures, you do build this type structure by simply iterating the power set up. Anyway, it's much prettier than that. But you see, here I can raise a question related to what we spoke about. You have here this new approach to a foundational problem. People are doing, did a little bit but it of course did not became mainstream in any way and some people, I don't know if still are people who are researching something about it, so what happens is, does it mean that we have to forget it and this was just a dead end or in a certain situation it could have been different and perhaps it will be different, I don't know, 10 or 20 years from now.

2:32:30 The situation is similar, structurally speaking, to what happened with Cantor. The outcome is different so far. But what will happen? How can we judge it? There was no Hilbert to push this view. There were historians. For example, Dogen always speaks about the new revolution in mathematics. Well, there are people like Kiesler. Kiesler. What did he do with that? Well, he was very vigorously pushing non-standard analysis. He wrote a thick textbook. I asked myself whether in physics there is any... Well, there have been attempts. There have been attempts. I simply cannot judge the value. I mean, at least done by people with very serious credentials and parts of his statistical... Albeverio's work in statistical physics is quantum field theory. He certainly has attempted to write some of this in the non-standard forms and that's looked okay. I can't really judge it. On the other hand, there are others like Feinstadt who give accounts of this and it seems to me utterly shallow. So, I don't know. In the development from Keesler, there were a number of people... and others who made serious attempts to use the non-standard formulas in the combination of PDEs on Syria and some problems in the theory of PDEs. But it's very difficult, I mean it's difficult for the logicians to understand it and it's certainly true that the practitioners say in the UK that PDEs... We came with you and now your thoughts are not totally hostile to what's been done by these people, but they also find it very difficult to judge. So there might be some development? Yes, I think so. I mean, in these high-level stochastic things and so on, enough has been tried. We haven't probably got nothing sensational that's been done, but I think it's more than just honest work, that something might happen. Not by Robinson, but by others, it was way oversold, including by Gödel.

2:35:00 Well, he says that he thinks that it's a surprise that he wasn't discovered earlier and that somehow it's the analysis of the field and so on, and then we, I can't remember. One of the real solutions to, for the most part, he says that it's extraordinary that we've made all this natural engineering and yet we still have unsolved these arithmetic problems which can be stated in such an elementary form. Maybe non-standard analysis helps the key to their solutions. We conclude that at the end of the chapter on non-standard analysis, I think it's the section which Moshe wrote, of course, I think it's called a cautionary memoir, a cautionary tale. Yeah, yeah, no, I mean, that was, I mean, I saw it. Because we say afterwards, well, you know. Unless my memory is wrong, I saw it. Because there's no unique model, you know, it's not really, it's a sort of relative. But I think, unless my memory serves me wrong, I saw the The original of that letter of Gödel to Robinson as Robinson received it. I mean, Robinson, I saw Robinson every day when he was sick. When, sorry, what year are we speaking now? 1974. No, Robinson died in 1974, and he was ill from the end of 73 onwards, and he was, much of the time, he was in the health center, just across the road from the math department. So I went in every day and spoke animation. I think it was Gödel letters on Facebook. But Gödel's had more, Gödel's of course had... I think he reassured Robinson that he was more than just a batch of proteins or something. There was something comforting about that. High grade and deep! A blob of protoplasm. I don't know what happened to that later. I presume there was a Robinson archive somewhere about it. There's something very sad here because Goebbels was obsessed with... But you see, there is another point here, if we compare that, it's not just that by chance there was no Hilbert to push this, also perhaps it's not strong enough to do it, I mean the theory, but also there is no possible Hilbert to do it.

2:37:30 Because, you know, just compare the list of problems in 1900. Now came 2000 and there was no Hilbert to give a list because mathematics has become so diversified, disunified, let's say, that you have to put together some people, right? There is no single person who can have a vision, a general view of the field and say, we should do this, this is unimportant. You have to put together, I don't know how many people were in the client commission who came up with the list of seven problems. It's very, I mean, the disunity of mathematics at this moment makes such things very difficult. Or the diversity, I don't know if disunity is the name. As a social fact, that's undeniable. The profession is far huger than it was then. I'm not happy with people who say that math is conceptually less unified now. I mean, it's difficult to say. I mean, it is broad, and you can't expect somebody in the center of a potential Hilbert to understand. They will see into every little corner of algebra and analysis. But I mean, there are people. Konsevich, for example, among the younger people. Hermione. Manin certainly... Manin came up with a list, I think. But Manin is certainly one of the rare individuals who has done, as I think I understand, almost, still almost everything that's going on in mathematics. I mean, of course he writes the book on mathematical logic, which I like very much. I think it's an extraordinary book. He lends himself only to attacks by more scholastic people. George Boole has attacked him for it. Well, okay, his treatment of the continuum hypothesis was not as elegant, and clearly he was unsightly, thin ice at times, but... I mean, the thought that one of our community might go out and write on the kind of things that Manning has done is beyond me. I think it's a really remarkable work. He just concentrates on some of the most beautiful things. Absolutely. And there are also speculations in it which no one's ever followed up. He's got various diagrams about recursion theory which look really very much connected to serious issues in geometry and so on. He had pictures there about exactness and things like that. I mean, Manning is one of these stories. Hilbert was unquestionably the leader of the, what, three dozen German...

2:40:00 Well, and German mathematics was relative to mathematics in the rest of the world, you know. But I think there are two quite distinct issues here. We're talking about the disconnectiveness or disunity of mathematics in two quite, at least two, possibly three, quite different senses here. Some of mathematical activity in the sense in which it occurs, in some sense I suppose can be of exchange, and that quite clearly for the conceptual organisation of the subject and for the kind of integration of mathematical knowledge as a whole are such as to provide greater unity to mathematics in now than they were in 1900, and I would say that actually a situation now was more hopeful. More helpful. More helpful. Yes. In the 1900s. Well, principally because of the development of category theory. Well, no. If I compare physics, I look at physics. There are two, at least two main axes. Let's say quantum, quantum whatever you want to call it, and relativistic. People can talk through these things. Physicists working in different fields can always find and... More or less easy way to communicate. I don't see them that much in mathematics. I see that in a more difficult way among mathematicians.