Mathematical representation of space in relation to other structures in mathematics

0:00 Yeah, hello, could I speak to Bill? Yeah, hi, who's speaking? Is that Danilo? No, it's Mike. Who was that who just answered the phone? I thought it was Danilo. Yes, I'm sorry, but he went away before I could say hi, Danilo. Well, please give him my very, very best wishes. Is this a good time to talk? Okay, obviously I got your email and read it with great care and attention. And wanted to talk to you about what you thought should be a suitable wording for this announcement, because I certainly understand from your email what is defective about the earlier proposed wording. I had a chat with Colin a couple of nights ago. I believe I understand that he and you had quite a long discussion about this recently. I was actually hoping to get a pretty thorough rundown from you on what have been the substance of those exchanges, but would it be a good idea if I called you back another time, when you have more time to talk? Yes, I mean it's obviously a very big topic. If this is not a good time to talk, maybe it would be better if I made a time that's more convenient for you. Okay, just give me an idea, or do you want to send me an email? What sort of time? Okay, I'll give you a call this time tomorrow, when you've had a chance to psych yourself up.

2:30 We do need to put out an announcement, if possible, within the next ten days or so, so I really wanted to get your advice. And, no, clear direction on it, and also on one or two other matters. So I'll give you a call around this time tomorrow evening. Okay, take care. Lots of love to Vanilo and Fatima. Talk to you soon. Cheers. Hello, Fatima. It's Mike. How are you? I'm in one piece. How are you? Oh good, I'm sorry I wasn't able to ring back, I'm a little bit later than I planned to be. It's lovely to hear you. I hope I'm going to get a chance to see you before, well, don't say goodbye that quickly. Do hope that I can get a chance to come and see you sometime in this, within the next year. That's fantastic, you know. Keeps you young. Yes, but healthy tired. Healthy and happy tired. Yeah, I'm not too bad. A few problems, but nothing too dreadful. OK, I look forward to seeing you, as I say. Bye. Hello, Bill, hi. To business. So the essential question is, in what slogans can we hope to capture the essence of Tohoku? They obviously have got to be subtler and rather more inclusive and more discriminating than the defective one that I... The theme is indeed concepts of space and I noticed just as a kind of basis for discussion that in the email you have sent to me you listed five aspects or five attributes specifically which you thought have got to be dealt with.

5:00 you know go through those one by one and one it's rather a bad line can you say again just i don't know why you're very faint yeah i can yeah i can i can just hear you but it's a bad line i don't know why hang on just let me see if i can turn up the sound on this no that doesn't seem i thought i could i thought there was a button on this that i could turn the sound up uh it's strange you can obviously hear me okay i don't know i don't know why it's so faint at this end but i can just about hear you Yeah, it's very strange, but I can only just hear you. I don't know why. Hang on, let me try again. Oh, that's a little bit better. Oh, that's much better. I don't know what the problem was. The first was the extensive quality, which I guess is covered by the notion of space as a carry of homology and cohomology. Yeah, well, the second was the... Yeah, and then the second aspect was intensive quality. Well, that's precisely what I was hoping to hear from you. Well, I wasn't going to necessarily use those in detail in the announcement. Of course, I'm looking forward very much to learning all about them.

12:30 Mentioned in, you know, your list of, which of course is the role of space as as a container of generating figures. As a container of figures and the notion of generating figures and obviously adequacy and co-adequacy of figures as being points with no cohesion at all as a kind of extreme figures with negation sheaves, of course, which shows up in the book.

15:00 I was very struck indeed by your remarks in Haute Bordeaux two years ago. Yes, it was. It was in the Topos Theory Summer School there. This vision of set theory is essentially being subsumed within algebraic geometry. I hope that's something which if we get to this meeting on trends in the mathematical representation of space that you could spell out more clearly for the benefit of your listeners. The other aspect that you mentioned of course was the intensive variable quantity.

17:30 Exactly, yes, arena for motion, arena for variation, exactly, yes. What is this chap's name? I didn't quite catch. Oh, Krauss, yes, yes, I've come across him, yes. It's interesting, I hadn't realized he was an opponent of string theory. I'm intrigued to learn that. Yeah, I hadn't got that impression from reading some of his writings, but... Confusing way.

22:30 I completely agree with you. I haven't read the book by this guy Krauss, This is published last year that you probably know about by this guy Smolin, Lee Smolin, at the Perimeter Institute, also attacking, purportedly attacking string theory with physics, which I think actually has some very good critical points in it, but again, it degenerates into this extremely wild speculation about the possible role of n categories and... Topological quantum field theories and the kind of stuff that our friend Byers is so fond of flourishing around. A great deal of soundbites, but no evidence that they're anywhere close to being able to do serious calculations with these things, let alone that they've sorted out the conceptual issues, as you say, about just what these things, these various additional degrees of freedom which they postulate for mathematical reasons, are supposed to be degrees of freedom of. Whether the background geometry or the, in some sense, the material content of the system, there's a lot of confusion, it seems to me. Yes, yeah, yeah, that's rather my impression, although, yeah, I have to say that's my impression as well, although some of the quantum gravity people, the so-called loop quantum gravity people, I think do have rather better mathematical...

25:00 Intuition, I will feel for, you know, mathematical elegance and the string theory people, but it's a very subjective impression, I think. But can I ask a little bit about your discussion with Colin and what you see as having been the points at issue? I mean, one of the things which he seems to disagree about is the very question of whether there is in fact going to turn out to be a satisfactory notion of a category of space in general. Well, of course, I only got a very slight impression from talking to him of what the substance of your discussions had been. I think he was seeking to defend this formulation of, you know, this determination of the notion of space as a, in the first place, the carrier of homology. I mean, not obviously as exhaustive in any way, but simply as being close to the heart of what Grotendieck was saying at Tohoku.

27:30 Absolutely. It's clear that Grodendieck saw these other aspects. But at the same time, I think there was a problem in that it was extremely difficult to calculate in the homotopy category. That was and still is. Which is why you need a drastically truncated measurement of homotopy in the shape of homology and cohomology, really, to allow you to do calculations. Allows you to do things. Well, above all, of course, allows you to do calculations and parameterize variation in specific contexts. I can tell you exactly where you wrote that. That was in the 1973, your 1973 Bristol Logic Colloquium talk. Every notion of constancy arises in the limiting case of some underlying notion of variation and is determined, its concrete nature is determined by, well I can't remember the exact quotation but I can certainly put my finger on it within a couple of minutes by going to my bookshelf. I think you'll find it's on the first page of your 1973 Bristol Logic colloquium paper, which came out in 1975 as I recall. I'm glad to be glad to be able to cite, glad to be glad to be able to pinpoint your own quotations for you to have the specific determinations of constancy and well obviously not to be able to do anything with them mathematically and so but it's interesting to see the relationship between homology, cohomology and homotopy in that same and also in relation to that sort of same thought which I wish I understood

35:00 I have certainly heard people speak about this. In fact, Carter gave a rather nice talk about it a year ago, 12 episodes or 12 chapters in Grotendieck's mathematical odyssey in the course of Grotendieck's work, and the last one of those was precisely this idea about shapes of space, which he had been pursuing, which Dufton understood a bit more. And this also, of course, connects with the requirement that you were talking about earlier on when we were talking last month, you know, in connection with the criticism of the approach to non-cognitive geometry, that the requirement to take into account maps like projections, inclusions, and pars, and not just isomorphisms. He was saying that there should perhaps be an intermediate class of bimodules having to do with these non-isomorphic mappings, like projections, inclusions, and that without an understanding of those, that intermediate class of bimodules, the overall categorical description of space would be defective, would in some sense be inadequate, and I mean that presumably... And considering those more general examples of maps also, of course, has to do with the way that space operates as an arena of motion. There was one thing you said I remember in that earlier conversation I'd really like to understand a bit more clearly.

37:30 On the historical side, you said that you felt that Klein had had a rather, in retrospect, a rather deleterious influence by laying undue emphasis on the The underlying group of them had led us astray, as it were, in understanding what should be the correct general notion of mapping between spaces in general. This is the long arm of Plato. The idea that symmetry is fundamental and that things like projection maps, inclusion maps, and paths and things which are not an isomorphic are in some sense less fundamental, which of course, I see this as a kind of tension that goes all the way back to Plato and Aristotle. I think it's something to the cast of mind. But anyway, I'm looking forward enormously to seeing you. But we do need to talk a little bit about this meeting because I need to get out an announcement, if possible within the next 10 days.

40:00 And I've got one problem. First of all, I'm going to need to recast the announcement in the light of what we've just been discussing. If you could offer me some suggestions and guidance as to what you think would be a more adequate and helpful set of slogans. Unfortunately, we are looking for slogans here because in the scope of an announcement or a poster or even a short funding application, You know, to lay out all of these detailed considerations about the different aspects and attributes of the notion of space is particular in general, so I'm just wondering if you have any suggestions to offer as to how we should try and concentrate in the form of slogans which are sufficiently discriminating and subtle. The essence of Tohoku in about 200 words. I'd be very grateful. No, okay. Well, I'll just have to do the best I can, but I'll certainly run it past you before we... Yeah. I'll certainly run it past you, but we're going to have to put out something pretty soon. And then there is one problem which I have to tell you straight away, which has cropped up. I will be very frank with you about it. John Stateshaw, who I think is a fine man, and I like him very much, but as you know, he's a philosopher of general relativity by training, he's essentially a historian and philosopher of general relativity, he has got some funding for this meeting, in fact it's at least 50% of the funding, from the Center for Einstein Studies. He wants to have a section on general relativity in the meeting, which is fine. I already mentioned that to you and you gave very helpful advice.

42:30 We obviously need to get somebody who is a first-class general relativist and who knows what they're talking about, but also has the background in relation to Grotendieck's work. Not many people around. John, about four days ago, that he wants to invite Penrose. It's because of Penrose's association with Poulton and other matters that, you know, if he digs his heels in, and if it's clear that we're only going to be able to get the funding to allow this meeting to happen, if he does invite Penrose, how would you feel about this? How would you feel about it, as we're having to share the... Yeah, I was going to say, I mean, I didn't imagine for a second that you'd feel over the moon about it, but it wouldn't as it were push... Things to the point where you'd feel you had to pull out of the thing because if it did i'd simply i'd simply pull the plug on the meeting now and revert to plan b which was really just to try and organize a small series of one-on-one encounters between you and colin and uh and others in the space of the next year to thrush out these issues surprise me bill you surprise me i wouldn't i wouldn't dream of trying to extract such a promise from you yes don't worry i'm not worried about that I'm not even sure that Penrose will accept, but it's just that John has made it clear he does want to invite him, and so I thought I'd better tackle you before I said anything to him. Listen, I'll get back to you within a week or so. A few problems at this end, but I'll tell you about those later. I'm purely on the personal front, but I'm surviving. Give a great big hug to Fatima for me. I do look forward to seeing you. Thanks, by the way, for some very, very useful guidance tonight. I'm going to go away and think very hard about this.

45:00 I can't tell you how I'm looking forward to hearing your talks on this subject. Talk to you again soon.