Parts of discussions

0:00 And so I thought, well, I want to make use of this logical framework. Non-standard analysis in Robinson's original formulation is very much like that. Oh, it was so kind of tight. Admittedly, that was slightly influenced by the Russell. And so on and so forth, and so forth, and so on and so forth, and so on and so forth, and so on and so forth, He came for a term, he was visiting, I can't remember which college, and he gave these lectures on non-standard analysis of Maché Marko, who had been his student, you know, that's how I got to know Maché. Anyway, a little bit about him, but it's sort of funny. He gave these lectures, and he was a very charming person, of course, Robert, he was very nice, he had a party at the end, really very, very nice man. I have the impression in these lectures that he had a rather muddled... I remember thinking, he gave a pro, 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 he, to prove P, he would seem to assume not P. And then you'd think he'd be arguing for a contradiction. All right, so I'm going to prove it not P. What he'd do is then somehow prove P quite independently. And then that contradicts not P, and that gives you double the answer. Hence P. This was wrong, and I want to tell you that. I sort of tried to trace it out. And I think this was Robert's method of proving P. Somebody else taught me exactly the same thing. It's very strange. I don't know who told me this. That's what he ends up with. But this is really, this is in Daubin's, I didn't read Daubin, but he asked for it and I don't think, but this little thing is in his book. It's funny, somebody else noticed that. It's very strange. It gets into writing in Brouwer at one point when he's proving the degree theorem, that degree isn't very good or half-topic. He gets these four numbers.

2:30 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. Now, suppose these two were equal. That's similar. It's a similar... I remember once, shortly after Wendy Yale, no, it was after Robinson's death, I went to New York, to a geometrical group, he was a very charming old man, he almost made me fall off my chair, he said to me, Robinson is up there. I mean, there was always, I guess, anything. Anyway, of course, all this, you know, Washington 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 there? Oh yes, oh yes. Oh yeah, no, because of course it's become, if you look at... Well, as I was about to say, Moshe has a very nice account of it, of a kind of distilled version of his approach to it, to his and Hirschfeld's approach, in that book he wrote with me about Lubach over a room. Well, the other one is Slavik Nelson, but no, yes, yes, it's a very nice presentation. It's very pretty. He substitutes, of course, for the types, he just do, essentially, what he called... Because they're Maillot 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 see, 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 become mainstream in any way,

5:00 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. 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, Dobbin, who always speaks about the new revolution in mathematics, but he wasn't. There were people like Keesler. What did he do with them? Very vigorously. They were pushing. It's like textbooks. I asked myself, for example, whether in physics there is any... Well, there have been attempts. I simply cannot judge. Albeverio has worked on quantum field theory. He certainly has attempted to write some of this in the non-style books, and that's looked okay to me, but I can't really judge it on the other hand, there are others like Fenstad who give accounts of this and seem to be angry, so I don't know, it's serious, I mean, on serious unsolved problems. I'm not totally hostile to what's being done by these people, but they also find it very difficult to judge work. So there is, there might be some development. I think so. Yes, I think so. I mean, these high-level stochastic things and so on, enough has been tried. We haven't, probably nothing sensational has been done, but I think it's more than just honest work. Something might happen. The term of non-standard analysis at the very beginning, not by Robinson, but by others, it was oversold. Yeah. Including by Goethe.

7:30 Well, we make that remark in a lot of them. No, no, we, we, we, I made this, uh, because we, we... Well, he says that he thinks it's a surprise that he wasn't discovered earlier, and that somehow it's the analysis of the field, and so on, and I can't remember the exact answer, but he always says it's extraordinary that we've made all these natural instruments, and yet we still have unsolved these arithmetic problems which can be stated in such an elementary form. Maybe non-standard analysis holds the key to their solutions. We concluded at the end of the chapter on non-standard analysis, I think it's the section, which Vache wanted to put, I think it's called a cautionary memoir, a cautionary tale. Yeah, yeah, I know. I mean, I saw it. Because we say afterwards, well, you know. Unless my memory is oversold, I saw it. Because there's no unique model, you know, it's not unique. It's a sort of relative... Unless my memory serves me wrong, I saw the... The original of that letter of Gödel to Robinson as Robinson received it. I mean, I saw Robinson every day when he was sick. When, sorry, what year are we speaking of? 1974. No, Robinson died in 1974, but 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, and then he showed up. He was getting better too. But Gödel's had more than Gödel's, of course. I don't think I'm confident about that. I don't know what happened to that letter. I'm pretty sure it was in an archive somewhere where it is. No, there's something very sad here because we know that Goebbels was obsessed with... Yeah, that title itself was quite... Yeah, it was quite 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, because, you know, just compare.

10:00 The list of problems in 1900 now came to 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 Clay Commission who came up with this. 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. In the center of potential to understand little corner of Manning certainly Manning came up with a list I think and it was Manning is certainly one of the rare individuals who does I think understand still almost everything that's I mean of course he writes the book on mathematical logic which I like very much I think it's an extraordinary he needs to attack George Bull has attacked him for it okay his treatment of it was not as early he was on slightly thin ice at times but I mean You might write on the kind of things that Manni has done. I think it's a really remarkable work. He just concentrates on some of the most beautiful things. Absolutely. And there are also speculations 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 he had pictures there about them. I mean, Manning is one of these who is virtually a universal mathematician. But the institutional situation is different. For example, even the case, you know, you have so many important centers. Of course, by 1900 you also had, but not that many.

12:30 Yeah, Hilbert was unquestionably the leader of the, what, three dozen German mathematicians. Well, and German mathematics was relative to mathematics in the rest of the world, you know. No, but also the French, the French speech. The French are there in Germany now, I mean, it's just... Yeah, exactly. But I think there are two quite distinct issues. We're talking about the disconnect or disunity of mathematics, the enormous increase in the total sum of mathematical activity in the centres in which it occurs. There are more quantum mechanics in now than there were in 1900 and hopefully in the principle because of them. If I look at physics, at least two main axes, let's say 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 that that much in mathematics. I see that in a more difficult way among mathematicians. Which is Atiyah in? Is Atiyah in geometry, analysis, number theory, topology? You can't even say which field Atiyah is in. Of course, if you mean that the average physicist can communicate across this barrier, or as the average mathematician can, you're probably right, but there are still quite a large number of people, you know,

15:00 Conceptualized, abstract, and so on. People like Atiyah who thinks... No, no, I'm not talking about the exceptional. I'm talking about the other... That's true, I mean, that's true. But, yeah, there is, again, there is a dialectic here with the exceptional and the algorithm. Yeah, yeah. It's a very important issue because it raises this whole issue of the relation of individual and collective. No, it would be nice, for example, if category theory could have provided such an axis to... Most likely that that is the category theory. It has to be involved in any kind of... Yeah, but would the average mathematician agree with that? No, well, the thing is I can speak from better experience within logic. It seems to be close. I mean, given the many forcings and so on, you cannot get them to go just a very small distance and look at a... A vast generalization of the thing. This is quite bizarre, but then logicians have notoriously done it. Weird, I mean. Yeah, but they know algebra, perhaps they know the categorical language, the mainstream, let's say, algebraic, but I don't know, what about functional analysis? Is he able to make some, you know, to enter the dialogue? Functional analysis is a difficult case because functional analysis went through a period of near degeneration too. I mean, there are independence problems in... Functional analysis as well, but functional analysis gets revitalized, I mean, in what direction? Well, for example, in the direction of, no, no, I think Gauss is a bit, Gauss is a combinatorial kind of thing, but functional analysis is certainly involved in Kohn's work and so on, and this is at least, in terms of judging, it's obviously a very rich intellectual mix here. So this is an instructive case, because it's certainly true, and it's true in Britain at least, that the... By the 60s it was frankly washed out, but there were still a lot of people doing it, you know, but they were doing little, little things, they had no ground. People came back, Connes was coming out, or coming to the province 20 years ago.

17:30 At that point I think, historically speaking... Because if we look at the mainstream, let's say, German, French, British is a little bit different, but let's say American. Let's think about German, French, and American, and I can say also, let's say something like South American, strongly influenced by, at least, let's say, Bourbaki, in which you learn theories. You learn this theory, this theory, this theory. The Russians deal with problems. Here you have the problem, now bring whatever you can to solve this problem. I think this creates a completely different kind of education. Thank you for your attention. Even Gelfand, who's 92 or 93, Gelfand's mind is still here. Yes, and also very categorical. Yes, yes, absolutely. Yes, yes, categorical. So the point is, as you say, being so immersed in the problems, they take whatever they can. So most of them, by this time, have realized that sometimes it's useful to take categories. So it's part of their mind. I'm not making a big deal out of it. Yeah, exactly. I mean, the Russians should categorize their representations. I mean, they are, they just think categorically, that's clear. I mean, fundamental contributions to hardcore representations. And yet, no one would call them category theorists to be nonsense. But category theory has become, in their case, a part of common language. Absolutely. Well, people would not call Pierre Cartier a catechist. A lot of people would say, functorially, he thinks. Yeah, but that's different because he works, let's say, in fields. that have always been prompted to use Pythagorean for language, like homology or algebra.

20:00 I mean, one thing, we probably want to try and draw to a conclusion fairly soon. One thing I thought we might touch on tomorrow, part of, was to say a little bit about what, you know, which is the, to ask whether that doesn't in fact provide some of this underlying conceptual unity of all. Prior to that, whether that may in fact be the principal theme, the principal axis of our... Mathematics may well only emerge. Yeah, just one very quick question, but I don't know, of course, but wouldn't the attitude that you say, you know, to have this huge mathematical culture, to be able to work in many different fields, I mean, that could also lead to a rather instrumentalist, what I call, kind of toolkit of mathematics, that, oh, you don't have to think about the overall unified concepts as long as you've mastered, you know, how to use this particular piece of the toolkit on that problem. A focus on sheer versatility might be the other way around. But unification has, in just bare practical terms, been so powerful. Yeah, in spite of... The question is, what is the aim? I mean, if we look at, I mean, I know this better, what the Germans did in the beginning of the century, and Bourbaki later on, everything very systematic, very clear. Is this more important than that? I don't know. I mean, I don't want to have an opinion on that. I'm just looking at that. And if you can solve so many problems, that's also an advantage, of course. Look, I think the good comparison, or a possible comparison, is with the Italians in the late 19th century, beginning of the 20th. You know, you have all this geometrical, algebraic geometry that do all kinds of crazy things that later on come the Germans or Americans and say...

22:30 Well, you have this problem here, this is not defined, this is not defined. Okay, not defined, not defined, but they solved so many problems, so... They were also unreadable to most people. That's a problem. The proofs were good. And that's where Springer-Verlag is important, Hilbert standards for proof are important, you just read student diaries in 1900, 1920, 1930, mathematics became so much easier to learn as textbooks appeared. You didn't have to go to the reading room. In 1920 you still studied by going to the reading room because you couldn't own those books. Springer-Verlag puts out these series of textbooks in a very uniform notation. And then comes Bourbaki, who takes it a further step. We will say the terminology for all of mathematics. And that really helped. And it helped, and some people say it also had negative effects. Possibly it had both, because it limited, delimited at least, the kind of things that were allowed. For example, we spoke about geometry. Then it's the complete barring of, at least on the text. ... of picture and intuition and etc. So I think, I mean, you know, you have to take from wherever it comes. I think everything has its positive sides and the individual mathematician can only do that much. But if you look at the, you know, from the vantage point of the historian who can look at many things, this was good, this was good, this had advantages, these are limitations. I mean, I don't see any problem with that. I mean, the fact... I'm just pointing out, I see a difference because I look at the physicists, how they can communicate with each other, I think, better than mathematicians. That's my impression. Yeah, there are clearly people in physics who do not communicate with each other, it's a great difficulty communicating. No, no, I see more than that. Oh, it is so. Sometimes. And also bear in mind what I'm trying to commit whilst you've said, what was it, von Neumann, there's a famous remark, and I'd like to remind you in a related way, but sometimes you know, I don't know who it was, just for the sake of argument, let's say it was von Neumann speaking to Lyle, I don't know. Hermann, you know, physics is a... Huge, huge subject. I mean, physics is almost as big as, you know, any one branch of mathematics.

25:00 It's as big as group representation theory? Yeah. Is physics as big as group representation theory? Maybe it's a classical group, perhaps. Yeah, I think it may even have been the group. Okay. Where do you want to go this evening? Good question. Let me just... I'll maybe just hang here. Hang on, hang on. I think this is for me. Sure, sure. That's what I meant. You don't need to know the question. I'm trying to figure it out. No, I don't. I feel like... I think I was interested in something that I was speaking to, but it wasn't there. I think it was. I don't think it was. I don't think it was. I think it was. I forget who it was talking about. I'm sorry. And the thing is, it was. When I say any one branch of mathematics, there was a specific one he mentioned. I couldn't think of one. It probably was something. Physics is absolutely huge. Well, it's his biggest practical group representation. Well, yeah, in particular, he was talking to Hermann Weyl. He makes the point. Hermann Weyl believes that physics more or less was group representation theory of a few very special groups. Yes, of course. But I mean, that rather does make the point that it's not a very fair comparison. That's the other thing I was going to say, yeah, it's not a very fair comparison. How much is being communicated? When Penrose talks to, actually when Penrose is written for, it obviously must be communicated, but when, you know, if Penrose is talking to someone who's doing, you know, low temperature physics, I mean, I do, I mean, you know, do, do, yeah, I mean, I'll just say, do, do, let's, I mean, there's some field theorists to tell me about. Is quantum gravity comparable to classifying manifolds in all dimensions, or is it comparable to the 19th century theory of elliptic curves? It's not a terribly sensible question, but the 19th century theory of elliptic curves is communicated to graduate students a semester now, while the gravity has never been worked out at all yet.

27:30 If we decide those two subjects are the same size, we're going to say the mathematicians are way ahead. In terms of conceptual implications, on the other hand, if a side quantum theory is as big as the classification up to diffeomorphism of manifolds, then it's huge. I think the point is what you just made, the conceptual unification. You get different answers depending on what you were talking about. There's an unhappy tendency nowadays to think that it gets it from the best sources. They're wrong. Yeah, my impression is that it's less disunified than it was a hundred years ago. Actually, I'll have to say it gets it from the second to the best. There's even someone like Barry Major who considered the class Unity of Mathematics today created by Christine Rod and Tom Broganique. Well, let's talk about that tomorrow. OK, John wants to know where are we going to eat tonight? OK, well, you know, I'm open to suggestions. We have got to strike that Leo can't, so we can't do the pizzeria. And I wanted to save the... Yeah, they have salads, but he doesn't want to eat just a salad, does he? The problem is the other stuff they've got. I mean, we want somewhere... Yeah, I won't be able to give away, you know, he's got some choice, you know, anything he can get on the menu is a chicken. I'd be perfectly happy with going for the Indian again, but if you're not happy with that, we might try something else. You were underwhelmed with the Indian. Well, it's just that that's dang ugly. Yeah, yeah, this is true. Okay, tomorrow night I was going to do a... Just cheese and wine and meats and things like we did for lunch, but a bit more, you know, organized. And then we'll have the final... Yes, Leo! Sorry, we're just coming. Yeah. So... Yeah, where are we going to eat? I know. No, no, no. Because I was assuming that you would say that even if I didn't ask.

30:00 So, what do you need to know? First of all, look, I have a calling that I can phone and be charged in Israel. That's fine. Go ahead and use it. So, just pick and use? Yeah. In this case, it's a positive statement. We can form this dual in a second. As a matter of fact, it's even better than that because... These things are monads, you see, and so the exponential of an algebra is always an algebra, you know, you take a to the power x, well a is an algebra, a to the x is an algebra, that's true for monad algebras in all generality, and so in fact if you exponentiate two fixed points of the monad, you have that, okay, it's not a fixed point, but at least it's an algebra, it has an algebra structure, which is a half. So there's a definite impotence, you see, on that space that you just have to show is the identity. It's sort of very extremely algebraic. It's not just that there exists, you know, an inverse, but we already know what the inverse would be, we just have to show that it exists. And for the classic ultrafilter case, that's saying we know how to embed the set in its ultrafilter, right? We just have to know whether that's invertible. If f is a finite set, then f to the power n should also be finite. So we have to recalculate the base of the ultrafilter space of that and compare that and so forth. But at least we know that it has the retraction of the Dirac delta. Just because any algebra to any power... To show that that independent is the identity, we have to use the fact that the exponent n itself is also finite. My problem is, is that sufficient? So that's a complete explanation of a very down-to-earth algebraic problem in two-dimensional category theory, which is sort of a necessary lemma to all this, because it would show that whatever mechanics you started with, at least the fixed points, are still a model for all the mathematics you want to do.

32:30 You can always arrange it to any given object, but you're measuring these sets in this sense, you're measuring them in terms of very small sets, relatively speaking. You're measuring the finite sets in terms of two. You're measuring the positive property being a fixed point under this monad. In the world at large, there ought to be an object, a single object, which represents that. So this object U is the natural dualizing object and so forth. But notice, by the way, that I've committed heresy because I've said that a measurable card must exist because U itself, you see, it doesn't contain anything. By very definition it doesn't contain anything, but therefore, since it's all of the things that don't contain, it either is itself or something like a measurable card. It might actually be a little bit less than a measurable card, but I'm not sure. In other words, I don't put forward this axiom outside this room, but there is this very desirable thing that you can have some kind of positive property which you're collecting. You can go and try to build that, fine, but that should be a way of solving some problem. But this is a problem. There's some parallel between that proposal and, you know, the introduction of inaccessible cardinals in early mathematics. I mean, you know, more or less, because there really was closure. I mean, there was essentially closure conditions, if you like, that were trying to be realized. Yeah, yeah, yeah. I mean, no, no, no, exactly. But it was realizing the idea that you have a domain, which in this case is closed under, you know, explanation and so on. I mean, to that extent, similar. And this is... Surely, I mean, Hausdorff and, you know, they were, I mean, he was thinking of cardinals perhaps in, you know, in terms, we were introduced by the idea, I think, was thinking about maybe perhaps more arithmetically, you know, the idea that somehow... An inaccessible cardinal was one. The cardinals below it had the property that they're closed under all these operations, but nevertheless, it's still an idea of a domain which is closed under certain natural sorts of operations.

35:00 Well, it's closed under the things Cantor wanted to be able to do. Yeah, yeah. It's an extension of Cantor's constructions. Well, if you've got an inaccessible domain below that, he can do all the things he wants to do. Of course, the difference, I think there's a striking, one strike, as far as I understand it, one difference that strikes me anyway between you is that in some way the earlier, you know, to introduce large cardinals and to provide universes... For set theory, which are closed under the operator, you really, it is in a way done from the inside. I mean, you know, there is this idea somehow of closing what you have already and in that respect it's different from the idea that what you're looking for is some kind of fixed point right of an object that ought to be there somehow because it produces fixed points for it. So in other words, it's not really being done internally. I mean, what I mean is that the closure, the construction of the idea of measurable, not of measurable cardinals, of course, but of the so-called arithmetically defined large cardinals, inaccessibles, malo, malo, I was trying to put it that way, malo cardinals, in some way is something that is done in some way, iteratively, if you like, in some way, closing things up, whereas I don't think, that isn't true of the description you're providing, right, at all. I mean, it's not really a closure, it's something that... It's a solution, if you like, to a fixed point, or a solution, really, to an equation looked at in that way. And you can look at certain animals... Well, I don't know, I mean, it's... I was talking about objects which are a fixed point. Yes, but the... I'm not saying it's a universe. Yeah, but the universe... It's also a fixed point or something. Yeah, but the universe you actually get... the object should be there, right, to provide the kind of backdrop. Within which this, you know, these, which satisfies these conditions, right, that sets have, you know, that you have fixed points and so forth. I mean, it's not constructed, if you like, iteratively by starting with something and then closing it under, under, under, which is given by, it's given by global considerations, right, rather than local ones. In the, right, you have the category of categories. You have objects. That is some kind of given, although you don't know exactly how extensive it is, whereas that's not the way at all, you see, of the iterative construct, even in the case, I think, of monocardinals. It's really done, so to speak, from the inside.

37:30 Zermelo's picture in Grenzatlan is, we're going to look at universes of sets that we might have thought was all there were. We might have thought this was all there was, and yet we've got the idea of all the ordinals in it. And that's going to break us up to something bigger than we might have thought if all there was. You're never saying, let's think this is all there is. You're just characterizing something here. Well, I think this is proper. I can just say one retrospective about what we've been discussing. We've certainly seen just how deep and the many separate areas of general reality and the general notion of structure, how they affect our understanding of the end-stopping. That might just be one topic for perhaps an hour or so. No, that's an excellent topic. Thank you! You have to go slowly because we have to make it into a categorical one and then perhaps in the second half we could get on to what we would call because we must keep at least one leg of the discussion, at least a half of the discussion Well, that's the other thing. I think I'd like to keep at least half a day for the Scouting Challenge Projection, maybe another half day for discussing other aspects of Steve's work.

40:00 I think it's important to take account of what it is and the state of work on it and why it's so important to this anonymity program and why it connects so deeply with these ideas of potential topology. And you're then going to say a bit about how it connects with the bigger picture of how one should think of geometric and algebraic ideas as fitting together. I mean, obviously, there's much that could be discussed. And then that would naturally lead us back to a more general exposition of what we were very loosely calling the Lafayette-Chamuel theme. Oh, sorry, I forgot to turn it off. Oh, no, those have run out anyway. It's this one that needs... Those are finished. The tapes are gone. Yes, boss. Oh, no, no, no, don't say it like that, please. No, no, really, hard drive. I said I was going to speed up the line, didn't I, a couple of days ago, and the chief shop steward raised no objection at the time, so he was sleeping on the job. So, well, the work is boss now. That's it. There's so much more to discuss in the next lecture, absolutely. And I think I should... ...conjecture that... The key number of your...

42:30 I mean, it's just that in the case of this one seems to be a bit more... It's constrained by its whole... None of it can ever be matched. These numbers are areas where you find out that there's some hidden geometrical connection that works. They were infinite. It didn't change much because, you know, whatever is proved for regular primes... ...was an expert who failed in the... He didn't put it as a conjecture, and to a certain extent, people took it as proof of a pharmacy theory, because it would be very easy just to take out the non-regular case. So it turned out to be different. Yeah, no, I mean, it's tricky, but I mean, I did somewhat deliberately want to use this here. Right, I mean, Comer at that time perhaps didn't really have a right to an opinion. He had discovered things, but there was no... But better, he was saying he didn't offer his opinion. He offered it as a hope or something. Yeah, I mean, for example, the case of this Hodge conjecture, the topology, the classes, the formulations of this work include... That's right, yes, yes, that's right.

45:00 Then he was eventually a professor. It's one of these uncanny situations. You've got these penetrating theorems, which are true, whose original proofs were just clouds. But it's uncanny that something that's not on its face, light, is correct. People who discovered this had no utterable reason to believe it. There's a similar thing that you often see, for example, in Italian algebraic geometry, where as long as you're not dividing by zero, and commonly enough, there wasn't a zero in there. But this is something else. This is a proof. You can't add any conditions, you can't define the terms in it. It's just... No, but look, this is something simpler. Ruffini's proof of the impossibility of magnetic. No one understood it there. Do you think there were reasons for someone to believe that this would be the result? And nevertheless, he published it, or people knew it, and people started to accept the fact. You know, I didn't really understand. Then came another proof by Abel, which also has its problems, and then comes the general proof by Galois.

47:30 But insolubility of the quintic, the quintic hasn't been solved and there's only two possibilities, one is or it isn't. No, because the very idea of insolubility doesn't exist. I mean, now you look at... Okay, Lambert, Lambert conjectures this kind of thing. That some equations might not be solvable in radicals. In general. Yeah, yeah. So now the question is, is it true for the quiddick? The quiddick is the one they haven't solved. The question exists. He says, ah, no. But Poincaré, from that Poincaré duality, the question didn't exist. There are lots of alternative answers it could have had. No one is going to... You assume that you will not solve it unless you know something. You try to look for things, okay, if you are an established mathematician you can take risks that a young mathematician cannot take. I think it plays a role in what kind of things people are willing to undertake and it seems that there is today more legitimation to go into things that are unknown, into things that are based on conjecture. I think nature is saying, of course we've always had the deduction theorem, you can always prove it. But there's a bunch of thises that we're now going to publish additional proofs on that we weren't before. You know, we were talking about the Gurbakir. I don't think that anyone, not only a young person, I would say even an established mathematician would publish a proof of if that conjecture is correct. I mean, probably people did it, but not people that were under the influence of Gurbakir. In the office of some professor they were tending to each other, I know that if this is true, then another. But from there to publishing it. But you certainly didn't take that as a starting point for trying to crystallize architectural principles.

50:00 Exactly. The organization had an attitude. It has the opposite effect too. The stuff is based on conjectures. Then there arises a small community which believes that this is a fruitful way to move and so if you... you have to work in that community if you... I mean, nobody who's close to that kind of professor is going to suddenly take up the, you know, the exponential rigs of... Even if this might lead to some important results, since it's solid mathematics instead of this shaky kind, get into the... The community grows in some sense. Yes, of course, in one sense it's more adventurous, but in another sense it's less... The continuum hypothesis provides an example. The way that the set theorists thought about the continuum hypothesis provides exactly an example of why even today they just simply don't get the... No, it's in the community. You're affected with this community going this, but is it in Cambridge, is it in Jerusalem, or in some other place, you know? The hypothesis is false that this happens. Later we discovered the Riemann hypothesis is true. This proof probably still has some content. Yeah, it says that one ideal is engaged in another. No, it says something happens. Yeah, okay. If you are willing to look at it from the computer point of view... Might have to be... Some people say, ah, you wasted your time and... I mean, this might turn out to be really important to different corollaries. Right, and you might be able to extract, you know, once you knew that the premise was faultless, you know, to extract something that you hadn't noticed before in the proof. It shows that this situation does imply this situation, even though the zeros of the zeta function never give rise to this situation, nonetheless, this situation exists, and it implies that one. It just doesn't bear on what you thought you were doing. And one way to find this is by abstracting, you know, finding the most general statement of what you're doing.

52:30 There's an actual place to wrap up this morning, but before you all dash out... Mr. Showbiz, we promised him we'd go back and have dinner in his place and from there we'll go on to the cast, so get your walking shoes. And we're going to reconvene here at 5 o'clock sharp. I'm going to tell you all about Have Ever Lunch. Everything's arranged, buses, everything. I'll tell you all about it. I'll leave the key in the door just for you, but you don't know exactly which one we're talking about. It's where the traffic light is, where if you go up onto the square and turn right, it's the bar right there on that corner where the traffic light is. You go up there, through the arch, you do the very tight 180 degree turn, then turn right, and up the corner where the traffic light is, where you turn right onto the main road, it's that little bar there. And there's parking space near there. We'll go up there in about 10 minutes or so. Let's give ourselves 10 minutes just to draw breath, okay? By any principle, you're right, Michael. Yeah, sorry, I'm listening. I just wanted to say something that I wanted to use. We should not buy them. Well, although I disagree with almost everything else John has said, in particular what he says about any category, I do approve of his campaign for people to stop reviewing for sitting on the federal boards of journals that are published by these outrageous price-gouging corporations.

55:00 But I realise it's difficult about the system. Okay, right, well, it's now the afternoon of the second chance to get a few things done. Can I thank everybody for a superlative exposition this morning. I particularly thank Bill for the quite wonderful tutorial he gave me over lunch. I'm hoping he may be able to say something about that in this session, which I'd like to see devoted, since this will be really the last complete session, I mean we might be able to fit in an hour or half tomorrow morning depending on what people feel like, but it's the last effective full session, and I'll provide useful hints of discussion. The extent to which mathematics can be seen in its development is driven by fundamental oppositions, such as those between the continuous and the discrete, the varying and the static, the whole and the part, the finite and infinite, the one and the many, but perhaps focusing particularly on the opposition between the continuous and the discrete. And the varying and the static. Perhaps you'd like to say a little bit about that, John, and then I might ask Bill to, in specific, in this connection, to perhaps reprise some of the very, I think, for instance, about a principle such as that of lightness is identically indiscernible, which is specifying the notion of absolute or constant, to a purely logical degree, but one might see that notion from and say a little bit about.

57:30 The fundamental oppositions that you see as driving the development. Sure. I suppose I identify as one of the sources of the major. The way mathematics emerged and the form that it actually took. Since many of the basic categories of mathematical thinking were named, divided by their means, they saw, of course, mathematics in terms of oppositions. The very aesthetic, the one of the many, The contiguous and the discrete is something that they, of course, identified the difference in terms of geometry and the contiguous, not very discreet, and of course they, although it isn't very clear since not much has, well almost nothing has come down, and Pythagoras, as we know, discouraged apparently, well we don't know it, but claimed that he discouraged, he didn't write anything down, and he just apparently said he discouraged his disciples in doing so. He would have had a short web of tape recorders. That's right, yeah, that would have been useful. But anyway, we know that for one reason or another the Pythagoreans came to identify a number, the discrete, broadly speaking, although whether it's really Athenism exactly or not, Aristotle isn't quite clear in his account of course, as a kind of basic principle of thinking, of understanding the world.