Panel Discussion

0:00 ...to give their thoughts for an absolutely massive five minutes on what they think is the most important aspect of rugby's legacy, and any general observations they want to make, and then we throw it open. Okay, that's what I was doing. Okay, ladies and gentlemen, we should resume. Well, I just thought it might help to structure the discussion if we did it at the beginning. We could maybe even offer another round at the end, what do you think? Okay, if people would like to take their seats. First of all, I'd like to kick off by thanking everybody for having helped to make it such an extremely interesting and successful three days. I say three days because obviously I count the hours of the marvelous talk on Friday very much as part of the meeting. And I also just ought to say a word to the University of Boston in general and most specifically To the Center for Einstein Studies and much more specifically and warmly still to John Stachel for having done, and Michaela of course, for having done all the work that has made this conference possible. I also, as a matter of record, should thank two sponsors who are unknown to people in this hall but without whom this meeting would not have been able to take place because the Archive for Mathematical Science. We have not been in a position to give any kind of support to it and therefore we wouldn't be here and they are respectively the Schur Foundation of Sweden and the Finnsler Foundation of Moscow and Dr Dimitri Pavlov, both of whom contributed effectively through me to the creation of this meeting and I would like formally to put on record my thanks. Okay, I think the way we should do this, since it's a very informal gathering, is to ask each of our main speakers to start off with a short, I presume an absolute maximum of five minutes, please, but shorter if you wish, just to say what they feel has been the most important aspect of the legacy going forward into the mathematics of the early 21st century. What they see as the key conceptual issues.

2:30 This is a program that he initiated and that others, including others here in this room, have carried forward. And if there are any particular topics that they think we should be focusing on in the course of the subsequent discussion. So if we just ask first Pierre, then Bill, then Colin, Jean-Pierre, and if they could give their thoughts on that. I think if it's okay we'll excuse Sean on that one. I don't know enough about that. And then we'll throw it open. And then I'll take it from there. So, first, Pierre. On that agenda, there are so many things that we ought to go on. Of course, my first comment is something which is striking, but also is confidence. It was not only confidence in itself, it was confidence in maybe we first want to survive.

5:00 I mean, the idea of anthropomorphism, he did not think that. And again, that's what I remember. I mean, things have been developed by him. In algebraic geometry, you have two fields recognized, Galois field, Galois was the first one. The idea is that once you have the notion, you have a one-dimensional case, number of fields on one hand and demand surface on the other,

7:30 and purely one-dimensional, and only in a certain way. And I think the last part of his heritage, I published a paper which is called, But nevertheless, I mean, I was struck by mathematics. You can have a psychonautic experience. Wotanik's father was almost on board, too, and he was born in a place, you know, Gorski, in the center.

10:00 And Wotanik knew very little of his father, and he knew very little about his bachelor's or that extreme. But on the other hand, Wotanik was led, whether or not he was a psychonaut, this is basic for that matter. He was led by a vision that was there, that was motifs. Scriptural knowledge, you know, Neil, Matt, Debra, Debra.

12:30 There is also one of his last unpublished papers, The Long March to Galway. If I were to throw him all over my life, I'd be a prophet. His personal life is a very high crime because it does destabilize his mind. No, that's okay. I think we really wanted to hear all of that from you. I'll hand over to Bill now. Perhaps particularly you could say a little bit more about the centrality of geometric Galois theory. I'm just hoping that some of what you'll say will complement and amplify points, particularly, obviously, in the way that Greg indeed saw logic and set theories fitting into his vision. Yes, well, I agree with Pierre in emphasizing shaping, and I think many people's progression of spatial analysis then to the particular problem of often mentioned, and then calculating the dual space of a polymorphic function, and thereby getting involved into analytic varieties, analytic spaces, and then through analytic spaces into algebraic spaces because in the compact case there are...

15:00 So this large parts of mathematics and geometrical analysis sort of played out through space, through time, and I think it's a very good thing, efforts like yours, to bring out these historical points in a way that's accessible to a larger public. Do you want to say a little bit, perhaps, about the significance of his work in plane topology, and particularly how it relates to the understanding of, really, the relations between our concepts of number and space? It seems to be a very sweeping conceptual consequences of that. I wanted to mention that one particular, fairly technical idea is the idea of the slice category, the idea that given an object in a topos or any more general categories, you can pass to the objects over that. Effectively, spaces that are varying over this space. And this will be a category very similar to the one we started with. There's really one category. There's really this whole family of categories and the way that they are connected. So there's the organic connection between rest and motion, if you like, just in this small, fairly small technical point that you have. But that came when you wanted to understand the rulebook, what I had. The work of who? Kodaira and Spencer. Kodaira and Spencer. There was very much in question. Yeah. Many people were capitalizing, took this idea from him. So once you mention that, that's a very good point. The gap between the imaginary infinity obtained by iteration and then idealization on the one hand... The gap between the imaginary infinity obtained by iteration and then idealization on the one hand... The gap between the imaginary infinity obtained by iteration and then idealization on the one hand... The gap between the imaginary infinity obtained by iteration and then idealization on the one hand... The gap between the imaginary infinity obtained by iteration and then idealization on the one hand...

17:30 The gap between the imaginary infinity obtained by iteration and then idealization on the one hand... The gap between the imaginary infinity obtained by iteration and then idealization on the one hand... And the other kind of infinity which is obtained sort of directly from the real world. Ah, here is a continuum, right, here it is. And yet the first is an indispensable part of our analysis. And yet somehow it never quite achieves it. We see that, for example, in the question of the real numbers in the topos as it has been discussed. The Cauchy reals fall far short of the Dedekind reals because they're too... It's too sequential, too semi-discrete, whatever the gap. Again and again, there is a phenomenon that suggests that the category of topological spaces is too general, not that it contains spaces that are far out, which it does, but the very definition of continuity is a bit too general because even on the unit interval, there are these things which are never increasing but always increasing. Paradoxical examples of the vague and many, many other people, which, according to the ideology which says that the countable is constructive, that's completely confusing because it shouldn't be so. It's very non-geometrical. There is a piano-spelt, space-filling curve. Again, something quite contrary to every geometrical intuition, which in fact is not true as soon as you pass through a smooth category. And then you come to the incompleteness theorem of Gödel, which says that if you have a first-order theory or any kind of theory in which a certain fragment of the theory of natural numbers can be interpreted, then it's necessarily undecidable, which has come to mean that contrary to Grotendieck's confidence and Hilbert's confidence... We can't solve all problems. Something is missing in our potentiality.

20:00 Well, and the recitation of this hypothesis on public television and so forth and so on, the theorem of Gödel has gradually left out the hypothesis which is sort of presumed to be, well of course we should be able to interpret the theorem naturally, but I think not. Another example, it comes up with cohomology. Somehow, cohomology manages to get reasonable answers in spite of the fact that the spaces themselves are not reasonable. There are well-known examples involving sine 1 over x and check cohomology and so forth and so on, Sierpinski snowflakes, blah, blah, blah, blah, blah. There's a whole realm of phenomenon which we learned as students to think, oh, you have to master these because these are the real mathematics that people just don't understand. Things are much more complicated than you would imagine. And the geometric intuition is therefore no guide. That's right. Geometric intuition is no guide. Godendieck, of course, was aware of all this, and at a certain point, after his retirement, he wrote something, a sketch of a program. He hoped to get a grant, which he did, and it did succeed. And he called it tamed topology, so it was describing, well, not the same examples I've just given, but the same sort of... Opposition between reasonable topologies, including algebraic topologies, and this other aspect, according to my analysis, we should be able to describe a category which does not have these pathologies.

22:30 On the basis of mathematical experience, I'll give an example. Meanwhile, back at the ranch, the logicians, there was a tiny minority among the logicians, had the heretical idea to exclude the natural number. This was a long tradition because the logician of the century, which was E. Nightingale, had shown in 1945 that the theory of real closed fields is decidable. All questions about any inequalities between polynomial functions and reveals. All right. Yeah. The theory of real closed fields, as such, does not admit any faithful interpretation. Exactly, yeah. So this was the beginning, and of course, once you know that, you think, well, what if I added the exponential function to it? Would the result still be true? I could generate more and more complicated theories by adding concrete functions that I know I need. I know that I cannot possibly adjoint the sine function on the whole line, because that immediately will define the natural numbers for me, and that indeed is, you know, as the zeros, and that is in fact a valid interpretation of the theory of natural numbers, at least up to the required degree. So then this arrived in the work of abandoned trees of what he called O-minimality, which is a very misleading, strange name. It just means that you can have a category of maps between powers of real, which has the property that when you restrict it to one dimension, All the things that, only things that are definable are the same things that are definable in O-theory, that is, in order theory, finite unions of intervals.

25:00 So it turns out that that has a tremendous effect on the higher dimensions, even though you can have very complicated surfaces and so forth and so on in higher dimensions. If it all, when you restrict it all to one dimension, it's simple, then the whole theory is very, very simple. Particularities, you may have the quantifier elimination, you may have actual decidability, you may have all these desirable properties which are anti-undecidability. These are the finite sub-analytic, okay, actually I will bring you in, I promise, but I had to let Bill run with that, obviously. Finite sub-analytic, you see, and so then this Van de Dries visited Buffalo, and as a prick I invited him because, wow, this is it. I ran to his house and said, look, this is fantastic. He said, yes, yes, we have to, we have to, you have to go ahead with this. So I invited Van de Dries, and I told him, well, Goldenbeek has this program. Well, after that, Van de Dries titled his work also, Tame Topology. Yes, I think that this is important. I already said 15 or 20 years ago, this is a thing that categories should take up, and very few of them. Who else, I think, made promise. A particular example that came up already some time back was... I've described this idea that a general notion of space can be described by a special notion of figure and then the geometry of that space, even with topos. So if you think in a very straightforward way about what is topology good for, and Walter Noll, one of my heroes, one of the purposes is to have continuous paths. You can move from one state to another along a continuous path. And you may not be differentiable, so you need continuance. ...figure shape, and you take endomaps of that as the operators which can express these incidence relations, you get a very nice topos, apparently, but there's one big drawback with it. If you take the square, the integral times itself, it's not totally ordered. You want it to be totally ordered, and yet, if you take the two solid triangles, which are the above and below the diagonal and the square, they're soup in the lattice of sub-objects...

27:30 So it fails to have this property, and this you can trace back to pathological properties of it. So, again, this is still an open problem. Concentrating the whole question of taint topology in one nutshell, to define the monoid of endomaps of the interval, which is contained in the usual continuous ones, But contains, of course, the max and min and the polynomials and so forth, and does not have this pathological property so that its square is indeed totally ordered in the internal logic of the resulting topos. Now, there doesn't seem to be any simple-minded set theoretical definition of this. Which brings us, of course, to... So Grosendieck's solution... Localized to that case is the best one I have so far. Right. Well, we must obviously come back to this further, but I'm going to move on to Colin and... You asked for it. No, no, I did, I did, and I'm very glad indeed to have heard it, Bill. I think, seriously, no, no, no, you have no apologies to offer. As I say, I want to come back to lots of supplementaries I want to ask, but Colin... And then, Jean-Pierre, and then I think we should throw, oh, sorry, I beg your pardon, sorry, Lou, in fact, actually, perhaps we could say Colin, Lou, and then, is that okay with you, Jean-Pierre? And then we'll throw it open, starting with your question, okay? If you can remember it, if you need to before remembering, I think, Colin, obviously, if you can, keep early breathing. Well, yeah, the, what I've, what I've most enjoyed here is seeing that Skolnik encouraged a huge amount of math, even if he didn't. And that's one direction I would take if I was trying to understand more of his relevance to physics, which I am curious about.

30:00 I don't think we talked about how K-theory has gone into physics. We talked about a lot of things. K-theory came up, but we didn't talk about the physics of it. I'm just impressed with seeing the whole scope of things, and I will continue to insist on the unity of that scope. Which is? Which is a nice note to emphasize. Lou. Okay, actually that was very prophetic of what I was going to say. So, of course, there's this constant sense of astonishment for me that Grotendieck had this absolutely fundamental way of finding new ways of finding foundations for things and that he was so He was so completely uninterested in physics and yet there's this incredible resonance between the structures he proposes and ideas that are very natural from a physical point of view. So since I've already talked about topos theory, I thought I'd mention something else, another piece of work of his. Which, again, he seemed to have developed completely independently of any thoughts of physics, which A, has been important much later than what was said before, and B, has incredible resonance with physics. What I would mention would be Grosendieck categories. Grosendieck categories. You mean the AB5? No, no. Things that are called Grosendieck categories. Okay, okay, okay. Do you have a quote? Yeah, okay. Now the things that are called gardenic categories, it's a long list of axioms that I couldn't do from memory, but the idea is that they reproduce the category of coherency of space. They reproduce the axioms of that. So, if you're a physicist and you're a bit pedestrian, you could say bundles. So you're looking at the tensor category of bundles over a space. But the idea that you could axiomatize that, then forget all about the base, and just look at that. So in the first place, in the last two years, this has been the foundation of what I would regard as the first successful attempt to define something that would be a non-commutative variety.

32:30 If you try to do the analog of varieties or schemes, but you don't require the variables to be commutative, I mean, there have been a number of primitive and rather ugly efforts that in my view didn't go anywhere, but there's quite a beautiful idea now, and in fact, my colleague Alexander Rosenberg is, and tell me about it if you ever come back, what you do is, you don't even try to figure out what the space is. You take its category, And you can do enough things with this category to reproduce, in effect, everything you'd ever want to do with the space. You have excisions, axioms, and, you know, and cohomology theories and all that. So it's actually possible to start out with this very complicated axiomatization of the structure and then just say any category with that structure we can then recover all these things that we think of as space-like. Now let me pass to the reflection that anybody who knew about that and also understood modern physics would immediately say, which is that there's this observation due to Einstein that we never see space itself. There are regularities in the motion of matter and in the relationships of matter, and we infer space from that, and this is something that he thought was very important, and that people have repeated many times, but in my view, we've never really known quite what to do with it. So now, what is matter as we currently understand it? Well, elementary particles are cross-sections of bundles, and these bundles are described by different quantum numbers, which basically tell you So, if you like, think of fine monology as being a hands-on way of studying a certain tensor category of bundles. Over space-time. And if I insist that really there's nothing but matter, then all there ever is is statements about relationships between excitations in these bundles and how the tensor operator causes them to come together and form other things. And I could just rewrite Feynmanology precisely this way.

35:00 And this, in fact, is the subject of a paper of yours called Categorical Feynmanology. Or is that somebody else? No, that's me. I thought it was you. That's me, but it's something else. Yeah, I mean, that's the more prosaic observation that phenomenology is really a branch, a tensor category theory. But I'm saying the specific phenomenology as sitting in the spacetime, so taking the geometric aspect together with the combinatorial aspect, is really just a statement about a category of locally, you know, of coherent sheets. And so we now have the possibility of doing physics in a non-commutative space in a way that would make direct contact with things we know how to calculate from the physicists because of this recent construction of Professor Rosenberg's, but all going back to this deep insight of Grokendieck's. So I think it's just astonishing his ability to turn things around. See things by their reflections and other things, and change which is the foundation. And the fact that besides all those other things they had this that we haven't even really realized yet, and that it has such resonances with natural things to ask about physics is, well, I think it's quite astonishing in itself, even if it had only the one idea. So that would be, well, the space-time is not commuted. What does that mean? No, no, I mean, it's more like, in the sense of calm, that you can't think of having a community of algebra. I mean, I don't understand it very well. I mean, as I said, I haven't hunted down Rosenberg yet. We had a seminar on different ideas of Poynton, and we weren't able to bring him in because he's, but, so it's something I still need to learn more about.

37:30 Well, yes, the chair is certainly happy for you to make a comment. Because this is a straightforward extension of the idea of the spectrum of community. You can always assign to the community the category of, let's say, finally presented modules, which is exactly such a tensor category, which has distinctive property that the unit object is a generator, separator for it. You just simply drop that last condition. Another very special case is projective space. Which is not described by a commutative ring, but by a graded ring. But a graded ring also gives rise to a particular category. So you're simply saying that, okay, affine space, projective space, and more general spaces can be described as a spectrum of the same kind of algebraic object. It's just that it's spread out as a category, because it can't be all concentrated in one. That means, instead of having, if you have a spectrum of algebra, he has a spectrum of algebra. So you have to relax the notion of the rita equivalence. The multi-equivalence are the factors which are invertible. And that has been developed better. That's the way Anand Korn is, and in motif theory also. One of the main ideas in motif is that when you have two algebraic varieties, you don't consider only the map, but the multi-value map, the projection onto it, which means that really you are where you invert. One of the things I was hoping to bring into the discussion is the whole issue of how one should treat map space. This is in the setting of generalizations of geometric notions because there seem to be very difficult issues there.

40:00 This bimodule, the con-suggestion, which is to combine modules, is not missing out a great deal in the way the requirements should impose on the map spaces. Because those kinds of maps are not really the space as such. It's something to do with the homology of the space, the extensive quantities on the space sort of thing. It's an abstraction of the space. If you think projective space, I think for every scheme you can construct a category of the sort that he's talking about. It really corresponds still in essentially one way with the idea that... Oh, of course. It's an algebraic... At this point I'm going to bring in Jean-Pierre and then Konstantin, who's been waiting a long time. Thank you for your patience, Konstantin. It contains a lot of key concepts. And it is striking when you read the papers you're seeing here. So if you go back to Albrecht and Klein, as you all know, their basic goal is to define natural inputs as punctures and categories, but what I'm not saying is, you know, another thing is that puncture categories are not defined as natural inputs. More than all, we know that the annual dimension of the population is less than one percent of it. So, if you look at Heidelberg in two months, look, they rely on Heidelberg in the claim to reduce categories, but not in the direction of Heidelberg, except for the original aspects of the diagram. So, the use of diagrams increases, which is really more to the explicit.

42:30 Then you go on to Kaptein and Eindhoven. Kaptein and Eindhoven's book is fascinating because they don't even define that. In their book, it's just the two. You start with, you know it, and then derive from it. Yeah, yeah. So they're not all the same. Right. When you get to the focus, you start with the definition of a category. And then, what would all people turn to now? But there are defined, like, more and more... But also, you have the notion of an equivalence is incompatible with the first line that the criterion of identity is important, that it's not ideal. The notion of idealism is in the Founding Women's Plan, but in 1714 they said, well, in practice, what we need is not the notion of idealism. The notion is equivalent. And in the paper, there is a definition of idealist function, but it does not appear. And he does not use them. So this is an interesting one. So he defines them, but also in key instructions in his context. And that will play, from that point of view, a very important role in the whole study. The other thing I would underline is the topic of categories themselves. There's a very dramatic shift given by that paper as to how you can use categories to do it. It's not only in the background of the language. It is something that's actually a defined axiomatic thing, and then you build on that, and you prove things from that. That was not done. And it's really a new way of using a non-structural, a new criteria of identity. Nothing to do with sexism. And I think this is a key, key step in the whole history. From then on, Bill would be also the same idea, but she responded to that. And then everybody else. I think that piece of the piece is important to something that has the category theory itself. Constantin. Yes, of course. Sorry, whose piece?

45:00 Oh, Gabriel's piece. Yes, yes. Constantin, we're going to bring you in. Let's try and keep ourselves all in the conversation rather than let it break up into... And a series of bilaterals. I think if you, no, I'm just thinking if people all sit at the back it's going to end up with about three groups of people all talking to each other. I'd like to try and keep it as a common discussion. Konstantin, what was just mentioned now. He said that the class of alien objects would be a semi-analytic set, but we know that it's a very bad class because projection of a semi-analytic set is not semi-analytic, so you really have to go sub-analytic, and it seems like he was not aware of that, but what is interesting is that there was a Polish mathematician, Ryashevich, at that same time, who was gradually growing a large block on that way to show the projection of it. Semanalytics says there's no semanalytics, and then he, I didn't even show, subanalytics says, and then he goes back, it has multiplications, like multiplications, you can triangulate them, it has all the characteristics and everything, and it works, and I say it's actually published in France, it's just for here, and it's interesting that... I'm going to give you notes. Mention anything about Goyashevich and he seemingly wasn't even unaware of the fact that the semi-analytic flat is just the wrong kind of a blocker. You have to work with some analytics. So, yes, maybe his intuition was not so good because he was not very good at it. So, he did some of the functionalities, but the theory was pretty hard. And Goyashevich and Tome were, of course...

47:30 I also read the memoirs of Tome, where he doesn't name them, but he says that he's taught When people started deserting him from his seminar, he said at that same time in Paris there was a young bright magician and hundreds would come to his seminar and when I just realized that like how my seminar lived, basically to go to his seminar, I gave up it. He doesn't call me Michael, but it's kind of interesting because he came back to me very long. It's a very interesting point. I don't know if there's any particular reason in the background, you know, Gritendieck's relative lack of interest in, say, real value analysis, sorry, yeah, in real variables. I guess, is this, what I was trying to say is, is this a case where Grotendieck's instinct to always search for generality, natural generality, sure, but still very strongly to search for generality, do you think possibly led him astray? That he didn't find the right concept, do you think? It excluded him. I do not think it's a running thought. Well, what do you think is the source of the... I think it's the fence.

50:00 Can I ask a question directly of Bill, and I hope the answer to it will open up... Again, a further phase of the discussion. You've spoken in print and in, as I've heard you give, about the importance of the recognition of this deep unity between what you identified as separable, unramified, and decidable objects, you know, the unity of those three conditions. To what extent do you think that Grothendieck had seen that, that that was something which was inherent in his... I mean, did you take that to some extent from... And perhaps you could say something about it, because it seems to be very central to you. The so-called unity is really just about the slice, the relativization and the separable in the slice category. The decidable is just to depreciate this, I'm sure he did. The reason I was interested for a long time, and by the way, much part of the objective number theory that Steve Shanuall and I worked out concerns the separable objects, where we're not trying to morph the classification. To try to, as I was describing here before, this quest for finding the properties of the extreme classes of topos is now entirely about cohesion and others that are entirely about variation of discrete sets in some general cases, some kind of. I was approaching this enlarging what was known about these two classes, making them larger, so to speak, by changing the definition, always keeping them either disjoint or intersecting in some trivial cases.

52:30 So this had to do simply with the idea that, well, basically that to be an italimap could often be described in two parts, sort of locally epic and locally monic. So, I mean, the locally epic part, the locally monic part means that it's, that somehow the fibers are discrete, even though... They don't glue together very well. So it's a generalization of Lazard's idea of the spread out version of a sheet. How is it, I forget the words for the moment. Yeah, etal is called flat and unramified. Well, what if you forget the flat and just study the unramified? That will give us a larger category than the category of etal sheets, but still one which is clearly of the, at least of the character of the crude definition. This was the reason that this was in my mind. I don't know if Grotendieck ever looked at it that way or not, whether he considered this a larger category than the category of Sheaves, a slightly larger category, but it also worked out nicely in the combinatorial situations where the passage from the so-called Grotopos of the space, i.e. the topos slash the space, If you think of it as an object, to the corresponding category of sheaves in this narrow sense, which was a quotient, it simply amounts to killing all the idempotence in the site. Is that what you're marking before? Yes, it's based on that. Somehow the lack of any idempotence in the site is an earmark of pure variation. By contrast, for cohesion, you have to have idempotence, you have to have the possibility of degenerating. I guess the reason I put the question- So it worked out in certain cases that this was almost adequate, clearly, which was taken in Joyon and others separately took up the question of flatness without it, without unramified- Without focusing clearly on the unramified, yeah.

55:00 I guess the reason I ask that question is because the thing which particularly fascinates me as a stumble-bomb outsider would-be philosopher of mathematics is to understand more clearly how set theory and narrow sense logic actually fall into place within this wider geometrical picture. You've sort of spoken of the principle that set theory should really be conceived as essential here. And in that connection, which it seems to me is conceptually so profound, obviously your role has been totally irreplaceable, but Grobendieck also, who set the whole thing off, and what I'd like to know is, Grobendieck, as far as I can tell, never really explicitly discussed conceptual issues in language that really would be accessible to philosophers. This is, did he ever reflect on conceptual issues directly in this way? I mean, there's clearly so much profound significance, importance for philosophy of mathematics implicit in his work, but he doesn't seem to, but he himself doesn't seem to, even when he was, you know, thematizing his use of categories, he seems always, as it were, to have thought from inside the subject. What they think about this as they look at what they do. Yes, sure, sure. That's a perfectly good answer. Well, it's because there's centuries of... It's so clear that mathematics is superior to... Yes, yes, yes. Someone trained in mathematics, someone as broad as... Yes, sure. People still laugh at me when I... Yes, yes. Maybe, yes. But it's precisely that great superiority which I entirely agree with you. And, as we say, unprecedented depth and generality of these ideas that would make it so important that they should be got across to philosophers.

57:30 But you felt that was the work of others. We're calling this my assumption that I wish philosophers would be more accessible philosophers, and a lot of it is not that they need more detailed mathematics. What they need is to be willing to take more seriously questions like unity, which hasn't been defined. What does beauty, when you haven't, when you didn't define a beautiful theorem as one whose proof is less than order such and so. Now, I mean, you know, which is a really compelling concept. You know, if they're willing, philosophers, at least English philosophers of mine, and say, these are just, you know, just none of it means anything. You know, he's expressing an attitude. And I think that's a shame, but it's, in fact, it's what's... And it is a difficult book to approach, well, when life is mathematics. You have to do it in big pieces. That's right. I like to pull out quotes from them. Any further thoughts about, I mean, other than the purely autobiographical issues, his is very strong, as to why he never considered the, or appears to have directly considered, the issues in the foundations of physics. I mean, it seems to be so clear that his is the kind of mind that would appreciate that the first principles of math come together as an entwined package, but yet he never addressed the... It's not right to say he never studied. I remember I called him on the phone one time, now studying Feynman, his five volumes of Feynman. We learned it all so that he could discuss more intelligence with us. I stand corrected, but it's what we've been hearing, including even hearing in this meeting, that he had absolutely rejected any... No, well, you have to personally, you know, testament it to the opposite, so I think it's extremely important that that be on the record. I don't know what came of it, not how long he pursued it, but he was certainly pursuing it with his customary zeal, at least for a certain period.

1:00:00 That's extremely interesting. John, could I turn to some comments and questions? Absolutely. I wanted to use that too. I like many things about it, and I'm going to approach it from several different angles and raise some questions. First of all, you talked about defining things empirically, which sometimes I think is misunderstood as pragmatism, and Bohr, for example, has been accused of that, but I think this is not correct. There's something much deeper involved here, and I like to put it this way, when one sets up a theoretical structure, one wants to make sure that the limits of theoretical definability within this structure coincide with the limits of measurability in principle. In other words, using any device which does not contradict the principle of theory you're dealing with. So this is all within the realm of theory. These are ideal measurements. This is not pragmatism in that sense. So people talk about the measurement problem in physics. There's nothing to do with that. This is the question of the limits of definability within a theoretical structure that correspond with the limits of measurability in principle. And this is the way Bohr, for example, introduces the quantum of action. What about limits of co-measurability of quantities? Classically defined quantities may have no limits on their co-measurability, whereas the existence of the quantum of action defines the limits of co-measurability of the corresponding quantum. This is the basic way in which the quantum of action enters into theoretical physics. I didn't see how it entered into the picture that Lewis presented to us, so that's one question. How does H fit in? How does that limit of co-measurability, coincide with the limit of co-definability, enter into the story? That's one question. Okay, so this is a lot to do with why I have the idea that regions, and by regions I'm thinking empirically definable regions, regions that could be defined by some combination of things that observers could observe.

1:02:30 You would try to determine something about the geometry of your space time by probing it and you'd see what different observers observed. So I conjecture that in fact you will not end up with a distributive lattice because making determinations and things about position would make other things impossible. The basic structure of the space time would be a, and that's why I wanted to consider the quantalites, you know quantalites are just Okay, well, is there a geometrization of, if you take, for instance, the free particle, you can talk about things being localized in position space, and you can talk about them being localized in momentum space. So what is all the things that you can localize? Well, it turns out there's a richer family. It ends up being a projected space. There's all these subspaces of the Hilbert space that you could study, but you can't. You know, apply the distributive law to these things. So what I believe is that different observers that are different motion states, you know, different sources moving through the region, would see things that would be complementary. So that's how HBAR would come into it. That's trying to incorporate in the structure. That's why I think it should be what I was calling a quantum topos. Okay, okay, okay, okay. I mean, definitely I'm completely in sympathy with everything you said philosophically. I would have quoted Einstein about no quantity that can't affect the result of an experiment should be included in theory. I mean, that was my original motivation for wanting to remove the geometry of a region which can never escape to the outside. But, I'm sorry, that's an answer about what I think the form should take. The commutation conditions show me how particular measurements work together and how, when you see one, what you can see and can't see from the other.

1:05:00 Well, if I knew that, I would essentially have constructed the theory, and that's not done yet. I've worked out what the general form should be, and I'm looking for tools from general relativity to try to construct it. Put a lot of emphasis on the Pyle's bracket. One looks at the effect of a perturbation of one quantity on the effect of another quantity, and who this one could divide. It's sort of a measurability analog of the Poisson bracket or the commutator, and I think this may hold a lot of future input and find a way to generalize that idea to kind of structure what you're talking about. It may find a way to help us to answer this question, how we can fit this idea into this theory. But now we turn to this measurement and how you apply this in space-time. We know that every real measurement involves a volume in space and an interval in time. How can we idealize those? You might say, well, we can idealize if we're going to a point. It turns out that doesn't work in this, in this, the only real case we understand in quantum field theory. In quantum mechanics, non-orthodox quantum mechanics, I think due to the existence of the, the foliation by the absolute time, one can, it turns out one can generally neglect the time interval involved in the process. But in quantum-like dynamics, as Bohr and Rosenfeld chose, one cannot. This question of co-definability being equivalent to co-measurability, this was challenged by Landau and... No, no, Landau and... They thought they had shown there was a contradiction between the definability of quantities in quantum mechanics and its co-measurability. And the reason they got into trouble was they tried to define quantities at a point. Pointed out, if you look at the commutation relations, at a point they involve delta functions, or, or, uh, D, Feynman's D's, we can say today, take them in a different instance, but there are obviously things we cannot think about. So we have to integrate over a region of space and an interval of time in order to make these quantities meaningful.

1:07:30 And it's also, they, it's interesting, they caught out Heisenberg in a mistake. Heisenberg thought he had shown that you cannot measure all the components of the electrical magnetic field in the region of space. The reason was because he thought you could neglect the time interval involved, when you take time interval into effect, you can co-measure all the intervals, all the components of the electromagnetic field in one region of space-time. It's only when you try co-measurability in different regions that the commutation relations and the co-measurability problems arise. Now it seems to me that this is true in quantum electrodynamics, where we have a fixed background in space-time, things can only get worse if we dig deeper. Pupil, Einstein, Maxwell, or even vacuum Einstein, that the region of space-time should be the basic entity that should be measured. One shouldn't be talking about measurements in an interval of space, as you did, or even measurements of lengths of areas and volumes, two areas, three volumes, as the astakar and loop points of gravity people do. They have operators which have spectra. But it doesn't prove to me that it has anything to do with measurability. One has to look at the measurability separately and so far nobody has done that. But my intuition tells me H to the fourth, the four volume, should be the unit of, the quantum unit, the four volume of space-time in other words, and that quantum of space-time and so forth should be sort of perspectival effects on this four volume as observed from different frames of reference. Again, I don't know whether that fits in with your... Well, no, I mean, I think that's right. I mean, that's why I'm trying to get rid of points. I understand. And obviously all of these are the kinds of... And when you start to include quantum general relativistic effects, these things you're saying are even much stronger. I know you didn't want to get rid of four dimensions, which I like. No, no, I don't want to get rid of four dimensions, absolutely. But the minimum unit that we have to analyze is a volume of space. I guess I should have said hypervolume. There are many different ways of analyzing space and time, which we think of as a quantum process, and from that region we know only a finite amount of information comes out, and everything has to be founded in analyzing that information. What I strongly think is that there are different ways of, there'll be like different bases for that information depending on how you observe it, and those will come up, will correspond to different and complementary

1:10:00 Descriptions of the geometry of the space-time. Okay, so that's the kind of thinking I've got about it. What I heard you saying was that extensive quantities are more fundamental than intensive ones. This is something that has been, already Grossmann complained about this fact, and somehow mathematics has developed a lot about intensive quantities. And whenever extensive quantities come up, they try to translate it into intensive, but really the extensive are at least as important. Again, I think the dimension plays a role here. You forget about the dimension. So for you, intensive is quotient of two. Intensive quantity is a quotient. Is a quotient, yeah, exactly. So this is why you have difficulty in defining sometimes what you thought was going to be an extensive quantity, because quotients don't always exist. Ratios of... Now, one final point. We can only observe the system from the outside. Observing, of course, could be an instrument. And the observer is always really in the future of the... Yes, but not to be a human being or anything like that. Right, absolutely not. It could be recorded on a punch tape in a server mechanism, whatever you like. Once it's there, whether everybody looks at it or not, the quantum process is over. But a quantum process involves the preparation as well as the state. You said I have a quantum state. But it seems to me we'll have to close it at the other end, too. There has to be a preparation. Some process is what connects the preparation to the final registration. I like the word registration better than measurement because it takes out the human possible confusion. So do you agree that a process should involve the preparation as well as the registration? In my discussion, I said we have to pretend that we have a quantum gun that can produce the exact same. The exact same little quantum gravitational bubble over and over again. We have to pretend that in order to be able to have an ensemble interpretation. I have no intention of telling you how to design such a machine. But just think about that, that it produces a little local quantum mechanical state,

1:12:30 and you do experiments, and you have to do them over and over on the same thing, and that's for correlation. And so it comes out the other side and you might want to do that over and over and ask people to do it in different sizes. Where does it come and where do the correlations go? And from that, I mean, the bending of the photons would contain all the information about the geometry of the spacetime. So the images that you see would contain all the information about the, you know, the correlations of them, about the curvature in this region. So then the point would be that you would have to define the geometry just purely in terms of that. Classically, it's a little bit like this thing that Newman did where he wanted to go to scribe and then just I think he made a wrong turn in that. He came up with something extremely difficult and hard to get anything out of, and I have an idea about what he should have done there. But it's philosophically rather similar to that. But the thing is, we have perfectly well defined quantum theories in terms of the signals. And somehow in their tensor product is sitting the geometry, because there's no information about the geometry except all the higher correlations of the signals, which you do over and over again. And we can describe each of those as a quantum theory. That means we have a quantal on each one. So we can then make their product into a very big quantaloid, and finding the right constraints on that is a setting in which one can try to find the laws, the dynamics. I think at this point, I'm going to bring it to a close, because we... Well, before you do that, I think it's incumbent on me to acknowledge that you were the prime mover of this meeting, and any role that we played was truly subsidiary to your role, and we thank you so much for moving us, bringing us all together.

1:15:00 Well, thank you very much. I certainly don't agree I was the prime mover, but I think I probably did have a hand in it, certainly. I didn't tell you you were the unmoved mover. No, no, no, I'm glad you're not. Certainly not the unmoved mover. Was the mover moved? Well, this mover is very moved, actually, because I do think that it has been an absolutely magnificent three days and that all of the talks and the discussions... Not least, this last discussion that we've just had, but we've had all of the discussions, both in the general sessions and one-on-one, have been extremely fruitful and very clarifying and helpful. We're hoping that there is a good possibility publication may come about as a result of this meeting. We've been invited to, well I've been invited to transcribe the recordings which will have been made, then of course to... We've offered them to all the participants to edit their contributions, and we have had an offer to publish them at the University of Chicago in the series edited by Luke Kaufman. We're going to explore that further. I mean, no definite decision has been taken on that yet, and it will be done, obviously, in consultation with the participants in the meeting. Mahalo suggested perhaps that we could take even preliminary versions... If we have the permission of the speakers, I, for one, would be very, very keen to do that. I think it wouldn't be a bad idea to give the speaker a chance to read over it. Well, I absolutely agree. That's why I say, with the prior permission of the speaker, perhaps even after editing... Maybe not the unedited draft, but like a quick edit. That's exactly what I have in mind. In fact, I actually said... Even, conceivably, the speaker said something unbelievably stupid. They would be allowed to... Well, I think the only person at this meeting who said anything unbelievably stupid has been the chairman of this session. I certainly edit mine, so don't worry, you'll all have the chance to edit yours. Once again, thank you for the kind words, John. Thank you all very much indeed. I really think it has been a great meeting. Let's applaud ourselves. We have a table booked at the Jay's restaurant, which is directly across the street from the hotel where the speakers are staying, at 6.30, which gives us just a little over an hour to go back to the hotel and to get ready for dinner.