Discussions / FW Lawvere presentation & discussion, incl. J Mayberry, R Pettigrew, D Bernardini, M Menni, M Wright

0:00 So what kind of thing do you work on? Pretty much anything I do. Is it anything that really interests you? Yes. I'm originally a computer scientist. Ah, very good place to come in from. I'm pretty creative. Yes, and effective topos and things like that, yeah. Are you going to be going across to Cambridge for the meeting for Martin Highland at the end of the week? Oh great, I'm coming along too, so a chance to see him. Yes, I'm looking forward to that. I think we are pretty well, yes, I wasn't too sure about that but if we are. Are there any plans at the moment for how to go over there? I don't think so. I already got a bus ticket but I wasn't sure if there, you know, maybe somebody, if we were going as a body, whether somebody was going to try and arrange to hire a minibus or something like that. Yeah, we'll find out anyway. I thought to be on the safe side I'd better book a bus ticket on the internet because if you book them on the net you can get them very cheaply if you book them in advance. We just go online for the National Express coach services and see if they've got any what they call fun fairs which are they're super cheaper you know giveaway fairs but you can only get those if you book them on the net and I think it's got to be a minimum of seven days or something like that beforehand and they don't always have them available and they only have them for some routes but I spoke to Richard and he said he didn't think there was anything planned about how to get over that which is why I went ahead and did it on the net but that's something that you know things could have changed since then yeah sure it's oh yes a couple of years ago because I remember I got it about 2007 I was a little bit disappointed in it, to be quite honest. Right. What particularly? Well, it just seemed to me a little bit thin on the... It's history. Yes, essentially. It's a historical study of the... It's largely history up to about...

2:30 Yeah, and then it's exposition of synthetic... I think it complements his other book, his primer on infinitesimal analysis, quite well. The primer is amazing. Yeah, that's really good. And that's the book I put in people's hands rather than that one. I give them that maybe as a secondary reading, but not a start on that. I've been trying to teach the philosophers some topos theory in the last four weeks. This is going to be hard. I've been teaching from the bell book, and everyone was getting incredibly confused by the intuitionistic logic. Really? I thought that was the way in, but it's not. Well, I think because whenever they're presented with a theory, they immediately think about philosophically how you might justify it. Intuitionistic theories are always a bit suspicious because it doesn't really correspond to a picture of the world that you can easily understand. It looked to them like it was a sort of ad-hoc movie. Although as anything in logic goes, it's historically more directly motivated by straightforward epistemological concerns and extra-mathematical philosophical ideas than almost anything else. That's right, but now it's regarded much more as a tool, which I think is the right way of regarding it. So you're right, you do then have the problem of justifying it to philosophers. Yeah, because they say, well, why don't you have an excluded middle? What is it about the world that prevents you from having an excluded middle? Then you've got to tell some sort of story about what these little infinitesimal regions actually are. Just glancing at the bridges of Konigsberg on the cover of this, I was reading a very interesting article the other day which tried to justify the view that Euler had actually discovered quaternions. Yes, no, it was very, very interesting. It was in a recent number of Historia Mathematica. I was left quite convinced by it. He certainly discovered something that was to do with quadratic reciprocity that was very close to a special instance of the Paternian formula, which I had not known anything about at all until reading this article.

5:00 No, this guy really wanted, I think he was pushing the claim a bit far, he was saying that Euler really ought to be credited with the discovery of quaternions rather than Hamilton, but he made a good case that he did discover it by accident, at least in a restricted case. Yeah, that sort of thing happens quite often. Easy to read into some historical figure, the implicit idea... But in case of Euler, he obviously had such incredible intuition and range that it's a little bit more plausible. It's also plausible that he wouldn't have noticed what he'd done. Thanks ever so much. So where did you fly in from yesterday? Milan. Is that where you are so far? No, no, I'm permanently in Argentina. I stayed there for a week visiting Bob Walters and Aledio. Ah, yes. You weren't, by any chance, at that fascinating thing they had at Mendrizio at the beginning of last year, were you? Ah, no, that was very... which Bob Walters organized. Thanks. No, in fact, you can see all the... Bill gave a series of three lectures, in fact, on topos theory. There were four, partly four philosophers at that. It was very good. They made videos of the whole thing, so you can actually watch it online. I think I saw part of the lecture on video. Yeah, yeah, so there were three lectures. The last and the last one was a kind of extended discussion session. Do you happen to know Alberto Peruzzi at all? Oh, okay, now he's based in, he's in Florence, but he knows Bob and Aurelia quite well, and he's a philosopher, but he's written quite a lot about... I seem to recall his name.

7:30 Yeah, yeah, he's an old friend of Bill's. He's a very interesting character. I think he's the best philosopher of maths in Italy by a long way, but I think he's the best philosopher of language as well. He's a remarkable man. He deserves to be much, much better known. He does, I thought. It's a little bit incomprehensible why he's not better known. He's written relatively little in English, although he has written a bit, but he has written a bit, yes. He's not that easy to read, even after I've spent, you know, four months trying to wrestle his original text into more or less one-dimensional English. But even so, I agree, I think he should be much better known. I wouldn't have invested so much of my time in trying to straighten out his syntax if I hadn't believed that strongly. I wish I could show you some of the papers which it would be a bit sort of probably unethical to do so as the state that they were when before I started work on them but there was one which I think had the most sublimely complicated nested metaphor within a series of kind of subordinate clauses you know the multiply nested subordinate clauses that I have ever seen in any I won't say in any English sentence, but in any sentence that anybody had ever attempted to write thinking that they were writing in the English language, I can only remember that there was a multiple capillarily distributed fighting front, which was operating in one of the metaphors, in one of the support metaphors, and I had to try and sort this out and turn it into English which analytical philosophers would read without kind of just throwing up their hands in horror. I think I did a pretty good job of it. I think you must have done, because I've read quite a lot of them. I had some tremendous fights. Well, you know, I don't want to claim that much credit, but I do want to say that I really have had some tremendous fights with Alberto. Sometimes spent, you know, two whole days with him before we could get through a page and finally agree that, yes, if he's going to submit English text, this is what he has to submit. So I'm quite pleased if he is as readable as that. Thank you. But I think he's a very brilliant guy and a very deep thinker indeed. And I'm very sorry I couldn't get here for this, but you know, maybe we'll be able to do something another time in Italy.

10:00 Yeah, yeah. I hope he's really had a dreadful time in the last couple of years, which you probably heard from John. About his daughter, and also his wife's mother, and then his wife's sister both died, and he was screwed by his department, he was given a tremendous amount of crippling administrative work, and he's been put in charge, which has been helpful to him financially, because he was really struggling, because Manuela had to give up her job. And so on and so forth. Whatever the equivalent in the Italian system is of the sixth form. Right. And he was put in charge of that for four years, and actually came out, I think, with an extremely good, very, very well thought out proposal. He had to sort of chair the committees and put the whole thing together. And he was actually paid something for that, which he certainly desperately, nothing like the value for the amount of hours he put into it, but still quite helpful. The energy and the emotional energy that's been sucked out of him by what happened to Francesca has just made it very difficult for him. He has still continued to write though. He's writing a big book at the moment on Kant. It's in Italian. I hope it will be translated. I might even try and translate it myself. Which is conversations between Kant and other philosophers. Imaginary dialogues between Kant and all the philosophers, both before and since. That's a good idea. Kant's one of the main philosophers. You could do that. I'm surprised nobody's done it before, at least not on such a systematic scale. He asserted the point to which everyone goes, but who comes before him, from which everyone comes out. I said, of course, it's becoming an uncontrollably long book, as you can imagine. You've got to choose your philosophers. Yes, exactly. Well, it's quite interesting, the 20th century philosophers. In the 20th century, hardly any of the people in the imaginary dialogues are actually philosophers.

12:30 They're almost all scientists. I mean, there's Einstein and Penrose. I don't know. And curiously, there is one philosopher, and that's Putnam, which I think is a rather strange choice. He's a philosopher of language as well as of maths. Yeah, yes, true. Putnam's lecture stuff isn't completely stupid, actually. You'll have a hard job convincing John of that. I don't think John's ever read any of the lecture stuff. No, no, I don't think so. His opinion is based entirely on the models and really on his early philosophy of math, which I agree is a bit threadbare and witty. A bit derivative, I actually think, I say this, I actually think that Putnam came very, very close to plagiarism in that article that he published in the, you know, the thing, not so much Models and Reality, but the thing on, no, this is a modal, no, modal strong, his first version of modal structuralism, which is more or less lifted straight from Zermelo's 1930 paper. And he never mentions Amele. I think there may be, in the second edition, he does actually footnotes Amele, somebody sort of drew his attention to the fact that, but in the first edition of the collected paper he doesn't even footnotes Amele, but the whole thing is just, the whole argument is just lifted straight from Amele in 1930 with that. Putnam praised himself and never referenced anybody else. Oh, is that true? I didn't know that. Well, I don't know that he ever said it explicitly. No, but it's obvious. There is no reference. And yet a lot of the ideas are obviously from other people. But that thing, the modal structuralism thing, is so obviously lifted from some mellow that it's a bit embarrassing almost. I suppose the Zermelo paper wasn't known very well. Exactly, because you wrote that about 1966, and you're right, that very few people in philosophy departments would have read Zermelo. A paper that was published in German in 1913 hadn't, and certainly at that date, been translated. Certainly. I'm afraid it does begin to look pretty, you know, circumstantial evidence, I know. But I think, you know, if I'd been the dean at Harvard, I think I would have raised an eyebrow. It's the kind of thing that the clerk of the course at Newmarket would, he wouldn't actually warn you off, but I think he would sort of say to you, you know, Mr. Putnam, your name has been entered in the book.

15:00 We are keeping an eye on you. It's a mistake, isn't it? I don't think I've ever seen your thesis either, actually. Have you not? No, no, I haven't looked at it. I'll have a look as well. Master Rush. It's one of us. I'm consumed. John Mabry, have you met John before? No, no, no. So he'll come along, not today actually because he's got to look after his brother, but he'll be here for others at the session. He was my supervisor, and it's a system that he created of theory of finite sets that he developed, and then I sort of developed it further with thesis. I see. That's the sort of, I've got what I've got. Turn that inwards. Turn that inwards, yes, turn that inwards. Turn that inwards. Well, it's more McCarty that's really got the, you know, on the warpath about that, but I don't think, I don't think Bill's got, Bill's probably never even read it or even aware of its existence. I don't think he would regard his all that bad as books for philosophers on top of theory go. There aren't that many of them, well, there aren't that many rivals. Well, I suppose there is now, there's... Well, there isn't really now, is there? There's not a serious rival to that, but there are a lot of rivals at a higher level. You know, to Peter Johnson's original book. Well, yeah, but that's written at a completely different level and entirely for mathematicians. And that's very, very hard. At least I think so. Actually, I think if I were now teaching, I would use Bill's book. Yes, yes. It doesn't introduce topos, it at least introduces them indirectly. Until quite late, and then indirectly. But it introduces all of the ideas. Exactly, exactly. And it's at least as accessible to the philosophers as the Goldblatt book, in fact probably a little bit more so.

17:30 I would say a little bit more. But I think also much, well, just, well. You would expect me to say that, but, you know, Goldbach just gets so much wrong, I think. Still, it's a brave stab. But anybody going... What about the graphs? Sorry, well, it would depend very much on the audience, but I think for an audience not of logicians or philosophers, I would rather use the papers on graphs. On graphs, yes, on the category of graphs, yes, on directed graphs. Yeah, I agree. I think category of directed graphs is a very, very good way indeed, actually. ...teaching it from Neff's starting point. I'm not sure I wouldn't use that for an audience of philosophers as well, because the basic concepts are very easy to get across. And it introduces you to what Bill means when he bangs on about cohesion quite early on. You get more of a feel for that in the case of category of graphs. Oh, you actually had him in the pub for a bit, that's ok. He couldn't check into his hotel until... That would have been very interesting, I wish I could have been a fly on the wall for that. I'd like to have got there in time. He's trying to explain the grand scheme to you. Ah, well I hope he's going to explain it more than once, that should be very interesting to hear. He knows how to come to get down here doesn't he? Categorification, the term itself, was apparently coined by our friend Crane. He was the man who originally coined it before Byers, yes, yes, so he lays claim to this distinction. He's been giving a series of seminars in Paris on categories in physics, which has mainly been about this topological quantum field theory stuff, and making all sorts of, I think, rather dubious plays...

20:00 In any case, category theory is not the soft underbelly of science, as a couple of agents seem to think. No, no, no. Well, it's certainly, I think, what they'd like to turn it into. But, of course, they'd certainly like to turn N categories into the soft underbelly of category theory. Well, maybe it is. Still, we won't. So you came straight here from... No, no, you went to Como first. Yeah, I guess so. How are things there? When I left, they were not quite well. Both had some sort of intoxication, so he's feeling pretty bad. I'm not blocking that. Beautiful. You want me to record and sing it by me? Sorry? You want me to record and sing it by me? Yeah, actually, that would be very good. Yes, let's talk about that. Okay, I'll have to think. I noticed it's a two-day thing, isn't it? The 16th and 17th? Yeah, that's right. I'm really sorry. I saw a reference to a paper of Ellermann. I haven't looked at the paper and I don't know how good it is, but apparently the idea is to double-entry bookkeeping. I remember that Bob was interested in that at one point. He was. He also wrote something about it. Anyway, it's directly related to this quadratic extension business we were talking about. I tried to tell him about it, but he complained. But where is the geometry? Where is the geometry? Well, you could do the calculation. Yes, yes. So bad. He just didn't want to hear about it. What was the point about two? You were talking, you were saying something about cohesion. There's an illustration of something with minimal cohesion. I, I, well, maybe I just... Yeah, I mean, there's a, there are two twos, at least in this, in the context of the

22:30 That means that there are two punctors, inclusions of discrete sets into this topos, and the left adjoining inclusion, you know, gives rise to 1, 2, 3, and so forth, that you normally think of as discrete ones, you see, but then the co-discrete ones, which have the same underlying points, but this is a unity and identity of opposites, both of these opposites have the same points, but... But the co-discrete ones may actually generate the category. It's one of the simplest, I don't know, you might say, I said semi-discrete, but it's just, I don't know how to express it. So there's no geometry beyond this dual version of discreteness. But anyway, in particular the two, you see, it's a very connected, contractible object instead of one that splits. So the idea was that this and related things can be used to parametrize becoming and laws of becoming and so forth, just as we do it in differential geometry using d and d sub 2 and so on. In other words, roughly speaking, you wouldn't tell the computer to do something it couldn't possibly do, from one state it has to go to another state which is somehow connected to it, accessible to it, being part of the same connected component, or in this extreme case, the same co-discrete component, would be a way of ensuring that, without too much extra writing. Yes, yes, I see the point. Staying within the component, staying within the connective component, telling it how to do that. Yeah. Telling it how to do that. The sphere in the book might object to working on the Boolean algebra classic fire. Who might object? Bob. So... He doesn't like that category.

25:00 Perhaps it's an attitude. He's got towards me, so he tries me to tell him simple things. So he tried to work with either with a dot with a non-identity arrow as a substitute for a small object. And that he managed to swallow. He sort of liked the idea, but it is an atom, so... Well, you could make it, Cheverie, by taking the classifying topos for... The idea of pointed objects, so you would essentially have not only your single products of that object, formal products of that object, which would mean that it was an atom. I mean this is a way of extending. It might not be that difficult, right? No, I mean, pointed sets is about the simplest algebraic theory you can think of, and in fact the finitely presented objects are the same as the free ones, so there's no problem figuring out what congruences mean and all that. So you just take the category of finite pointed sets and look at it. Covariant set value functors on that, that topos is a... I'll forcefully answer that, yeah. A Boolean topos, not Boolean, but a classifying topos, and it contains your object sort of as a generator in that broad sense of generating. I think these intuitive objections or enthusiasms, or both, that he has, I mean, one should try to channel them into... This threefold distinction that I make between sufficient cohesion, quality, and pure variation in my paper on axiomatic cohesion, because this is kind of the coarsest division, you see, but it's very qualitative. Sufficient cohesion could be background for motion, but it's not the same thing as motion. Motion cannot be background for other things.

27:30 And quality plays a very special role, but it doesn't seem to be either. See, okay, so with sufficient condition, you have the property that the sub-object classifier is connected. And hence any object can be read and then connected out there. That's one kind of thing. On the other hand, if you have just this input, then you have the two, the left and right adjoints are the same. So the components of the truth value object are the same as its points, which are typically two, although it's disconnected. I mean it can be, the object itself can be much bigger than two, but the actual points are just true and false, and those index the same points function equals components function. So this is, at least to me, this seems to be sort of very... All of these have a qualitative distinction, one wants to see which kind of topos you're trying to get at, and of course you can transform one into the other and so forth and so forth. Right, right, right. You classify it from points of things, don't you? Yeah, yeah, so that will of course be, wouldn't that be, let's see, will that come in under the sufficient cohesion or not? Let's see. Maybe not, I'm not sure which of those it is actually. It's so easy to work out. To be worked out. See, there's a rough criterion for, given the other axioms, a rough criterion for sufficient cohesion, given by Grotendieck, was that there exists at least one object with two distinct points which is connected. All right. And so your basic object has got only one point, but if you, and therefore it's... Well, its products also have only one point, but if you take an exponential of the thing with itself in that topos, I have the feeling it might turn out to be disconnected.

30:00 The exponential of a pointed object always has two distinguished points, namely the name of the constant and the name of the identity. And these are often, or these are usually distinct. Distinct means the equalizer is empty. So the problem becomes, is this little function space connected? And I have a situation that might not be in that case unless they work it out. Is the condition for that, for the function space to be connected, related to the role of idempotence in the site? I remember you talking to me a bit in Montpelier about the role of non-trivial importance in the site in the case where you do have this qualitative aspect in there. Yeah. I haven't quite worked that out. I mean, there certainly is an ubiquity of importance if you have this, but conversely, I'm not sure. Because the case of absolutely null cohesion seems to be, the localic case, does seem to be the case where there are no non-trivial importance in the... Yeah, the localic and the group-oriented case, which I think has to be taken together with the localic. Because of the fact that the group representations are locally localic, so it's really part of the same, and already in 1930s algebraic topology, this was well known in a way, that you take a covering space of a space and the fundamental group, all these three things are sort of living in the same category, this group is not a space in the usual sense. It's a kernel of a map between spaces, so I just didn't catch what you said, yeah. This is why we have cohomology of groups. It came out of cohomology of spaces because somehow these groups sort of popped up even if you didn't want them to be there in a category of spaces. So they should have cohomology also.

32:30 Yes, and people like Steenrod have kind of already figured that out. In effect, this idea was there. Yeah, except obviously they didn't use the language of categories directly at that point. But anyway, Grodendieck's idea that the vocally localic is somehow still morally localic, if you like, that immediately includes groups as an extreme case. Yeah, yeah. The GSAS is locally isomorphic to the one-point space, but I'm a methodist, so... He just turned 81 the day before yesterday. I see. I was having a conversation with Cartier just before leaving Paris, and also Christian Husserl. I saw that the... The talks from the first celebration of the ICS are now online. I think not all of them, most of them are online. I looked at one at Mumford, which seemed quite interesting. There were some good talks. The one on nuclear spaces, I think that is online. It was about the only talk which was purely on his work in functional analysis, although Cartier said a bit about that in very general terms in his introductory survey talk. But there was one which was purely about the work on nuclear spaces. Oh, I'm trying to think of the name of the guy who gave it. It was on the, not the very last day, but the next to last day, but that is one of the ones that's on line on that site. It's very well worth listening to. That was the most interesting talk of the meeting. It was an interesting meeting. They had this meeting at IHES for Great Lakes 80th birthday in January, and it was long delayed, but they finally got it. We want to concentrate on the developments that are relevant to what's happening now rather than on the history.

35:00 That's happening now on the moon, isn't it? Yeah, it was happening now on the moon, yes. I'm going crazy. Before we lived, we liked the Schroeder atmosphere. Yes, well that has always been one of the problems with the IHS. I think, you know, Carter would be the first to agree with you. I think in many respects that meeting in Montpelier was a lot more productive. Really? Well, it sounds productive. But no chance at all, it seems, of Grotendieck ever being brought back from... Psychosis and from the grip of religious mania, alas. Who is still in contact with him? Well, Cartier indirectly, because he showed us the letter, didn't he, in Montpellier that he'd received denouncing the conference. Right. So, and that was only a few, so he is in contact. Right. I mean, not face-to-face contact. I don't think anybody's been in face-to-face contact with him. At least nobody from the world of mathematics has been in face-to-face contact with him for quite a long time. When was the last time you saw him Bill, was it late 80's? 89. 89 I thought it was, oh yes. Which in fact was very shortly before his second disappearance. Is that before the Cambridge meeting? Of course it would have been, wouldn't it? Gosh! You mean to say that- You saw, the last time you saw Grosendieck was the same month you saw, first time you saw me. What an unnerving thought. It's a very, very unnerving thought, even that kind of juxtaposition in the time metric scares, makes me... I just realized that recently. I mean, these were sort of two important events in my life in the same year, but it was actually, within a few weeks, I forgot the exact dates. Well, the meeting was in June. Oh, maybe it was after, actually. It was very close. The meeting was in, the meeting was June, the meeting was in the middle of, about around Midsummer's Day, I think it was, about the 20th, 21st, 22nd, 23rd of June, and then we had the workshop afterwards, and then of course you went up to Bangor the following month, about three or four weeks later. Was it that long afterward? Maybe it wasn't even that long. Maybe it was only a couple of weeks. No, I don't think there was any delay. Maybe it was a week or so. There was a short gap, I seem to recall, but maybe no more than a few days. I really can't recall now. But anyway, it was June and July of 89.

37:30 Mentioning Paris, when I was in LA... Oh, yes. Just before you came to Montpellier. ...a philosopher. A Hungarian philosopher who lives in Paris called Imre Toth, have you heard of him? Oh, yes. He gave a talk about Plato and the theory of being in the sophist in relation to Plato's ideas about the mathematicals at Cartier's seminar on philosophy of mathematics about two weeks ago. I didn't get there myself, but... Somebody recorded it, of course, for the archive. I haven't actually had a chance to listen to it yet. It looked interesting, the title, but I'd never met him. But the title looked interesting. Anyway, it seemed like an interesting one. I would actually have studied some things. He gave a, well, Carter obviously thought he was worth inviting to his seminar and to, he gave, isn't he? Yeah, well, there are some dubious characters in the seminar. True, true, true, true, some very dubious characters. That, I agree, doesn't necessarily provide the right criteria, no, you're quite right. But on this occasion, you may have got it right. Anyway, he gave a talk on Plato and the theory of being in the surface, but in relation to Plato's ideas about, well, how one ought to think of, I guess, the notion of... No, well, the notion of numbers and what they call the mathematical numbers, yes, no, the so-called mathematical numbers, not the, yeah, no, no, no, no, no, not things like shapes, not geometrical things, but it's related to this whole issue of priority of being or becoming, which is on this, which I would like to understand Plato's views on better. I think they may have been. Misunderstood. And there may be more real dialectical insight in Plato than we've given him credit for. Talk today. You don't get it right. What? You're giving a talk today. Or is it tomorrow? Two days. Wednesday. Wednesday. The only formal talk so far is my talk on Wednesday. All right. So the idea was we will have a very informal... You will volunteer to give a three-hour talk this afternoon.

40:00 That's the way to do it in the Army. I want six volunteers. You, you, you, you, you and you. So it's an informal seminar. In other words, not polished results if there are any, but not necessarily polished questions and partial answers. Thank you very much for your time. Well, it's always a good idea to start from partially formed things. You were telling me, I remember in Montpelier, that was the thinking about the partial map classifier that actually got you to the sub-object classifier. Oh yeah, this has not been really brought out at all, in particular Johnstone, because he had a slightly different way of constructing the associated sheath, kind of downplays this, also in his second book, too. Well, I don't really need this, but by the way, you could have done it this way. I mean, it sort of doesn't explain. That's in fact what the way it was done. The way it was done, and moreover, a very crucial instrument. That's a shame I didn't know that. Well, what shall I say? Well, I guess I was impressed by the fact that Verdier and company had managed to incorporate a family of things that were supposed to cover something else. Just as one sub-object in the pre-sheet category, a bunch of objects or sub-objects or not necessarily, just maps to a given object that you want to eventually say it's a covering or something, but for that purpose it suffices to look in the pre-sheet category where these things live and just take the union of the images of these things. So it's a single sub-object of the thing that's to be covered. And that's the way... So the fact that this infinite family was incorporated as a single thing and therefore subject to ordinary Hauser-Brake calculation just impressed me, you see. So I thought, well, maybe the associated sheaf concept itself could be similarly made completely finite and just deal with the finite number of things.

42:30 So it has to do, the sheaf concept has to do with partial maps. Partial maps which are defined on, whose domain or definition is it covering should be uniquely extendable to global maps or more generally to maps defined on the thing that's covered. You have to make a single object of partial maps, and then a sub-object consisting of those partial maps whose domains are dense. Well, that involves the map to omega, which is one tittle tilde, you see. The domain is coming out. Oh, so we need this omega. It comes up at this point to classify the domain. Then the capital J is typically called the sub-object of omega, so you pull back this domain map along that and you get the partial maps whose domains are intense, and then you just, so an object is a sheaf, multiplying this J times it, there's a restriction map from J times X into X sub J tilde, and we take something who's... You know, to take a J and restrict any section to that, well, it has to be invertible if the restricted thing is, yeah, so, so there's a, so it comes down to the sort of canonical way of expressing properties in category theory, namely that some natural map that you have anyway is actually invertible, so this natural map from the things that are globally defined on the Js to those that are defined on the subguides. That's the condition to be achieved. And these constructions, so these two or three functors that I've described in this process, by combining them appropriately, you get the thing that you have to apply to an arbitrary x to force it to be achieved. That's again a finite process. You just have these few maps. So that's why... Not only the theory of toposes as such, but the theory of subtoposes as well becomes totally finite at the proper level.

45:00 It also provides exactly the right insight into the way that one should think in this context of the notion of property and logic, just as things which are defined by globally invertible maps. Yeah. So anyway, that was the May of 69, I guess. When I came up with this and exposed it in Rome at the Instituto di Alpimatematica and also in a meeting in Oguwolfak where I remember Mostovsky was there and he seemed to be quite interested in this idea that Boolean value models are closely related to sheaves on Boolean algebra, which are topologies and so on. And then Myles Tierney and I got together. I was at Albrecht Gould's house, and we said, okay, let's have a year in Halifax where we work this out, because he had been coming at a similar, another direction of the same kind of conclusion that we needed an axiomatic theory of cheese, but that was the thing, and so that's never been properly, what I just told you, you know, it should be written down, and it should have been more than sketched, I guess. Well, I was absolutely fascinated when you outlined this to me in Montpelier. You remember the last evening we were there. I absolutely agree. It ought to be written down. Maybe it's one of the things we could make a project for the results of this workshop, at least. Minimal aim. Well, I'm supposed to talk in Cambridge about problems in topos theory. I think there are specific technical problems, but there are also some strategic problems, like consciously fending off the Templeton agents who think that topos theory is the soft undervalue. Well, I must say, in regard to that, that I hadn't heard of the Templeton Foundation, and Bob explained a couple of things to me. I don't consider it to be my fault, so what I'm trying to say is that perhaps there should be more publicity about what these guys are doing.

47:30 Good. I'm glad you said that. I think you're singing to the choir, as he's a favorite. Much better, much better. He's giving the choir further instructions, further exhortation to get out and ring the alarm bells. It's beginning, I think, to be more widely... I mean, awareness seems to be spreading that these people have a very... I'm serious and carefully thought out agenda and that you know they really do seem to have their tentacles into a lot of Of course the agents themselves, especially physics of course, the minimum requirement to be a Templeton agent is to say that it's harmless to be a Templeton agent. You can take money from them, it won't hurt your science, you won't feel any different. This type of thing you hear, you know. In fact, Baez started out that way before he actually, if you look at his week after week after week after week. He talks about this. Well, maybe it wouldn't be such a bad thing to take money from Temple. He talks about that, you know, for a while, and then finally he gets his $130,000 grant to run that in-category cafe and what other various activities he's up to, with Corfield and... But anyway, as I say, that already to claim that it doesn't hurt is propaganda for their cause, because... Especially in times when grants are contracting, people are tempted to say, well, it's just money, and they're not, they claim not to put any strings on it, so, so anyway, that's, that's one strategic thing. We have to at least be conscious of this, and not just pretend it isn't there. Another thing, the other thing is Wikipedia. You know, there's about seven or eight. Entries in Wikipedia have directly to do with topos theory or category theory, maybe more. And they all have serious errors, you see. Again, whenever I survey this for myself, I start thinking, is this actual disinformation going on here? Because how can somebody who knows enough to say, to talk about pre-sheaf topos, for example, say that a topos is any reflective subcategory of a pre-sheaf topos, leaving out the left exactness condition, which is totally crucial, you see.

50:00 So, you know, if somebody's totally ignorant, then you say, well, they didn't realize this was important or something. Now I'm sure there are totally ignorant people who think they know something and immediately rush into Wikipedia, that's a phenomenon that goes along with it, but to give so many details and yet at the same time to give these utterly misleading... I mean if you imagine some young person in Pakistan or whatever who reads this, oh that's what a topos is, he could work for months and be completely confused or become disillusioned just because of little things like that. Hopefully there's a multiplicity of information available so that one is sufficiently conscious of the unreliable nature of Wikipedia in general that one could defend oneself, but still it would cause a lot of trouble. That's of course what the defenders, that's what the apologists for the whole Wikipedia. Project, of course, insist on because it's, for them, it's just like the magic of markets, you see. It's like Adam Smith said, it's a plan. It's an automatically self-correcting mechanism. So it doesn't, you know, so it's in the, but you won't be surprised to believe that I don't buy that. I'm not sure people will buy that now. No, far fewer than a year or two ago, but it's, no, I agree. It is very strange because there's a lot of, especially I think in the articles on general category theory, articles on things like adjunct functors, I haven't studied the articles on topos theory all that carefully because the first two or three I looked at were obviously completely. And I know there's been quite a lot more since then. But the ones in general category theory, they're obviously written by people who have studied the subject seriously and have some clear idea of what they're talking about and are familiar with the literature. But then suddenly at some point you will see something that you just know is just dead wrong and which they would certainly or certainly ought to know is wrong. And it could just be that... You could always be some other person who inserted... Yes, exactly, because the nature of the... The way that they put their articles together is such that it's always subject to that kind of thing, although these days I understand that, I'm not sure if it's true in the scientific and mathematical articles, but certainly on the historical articles, of course they are now effectively refereed and there's some kind of official procedure that you have to go through, filter through before you can edit an article.

52:30 I thought that applied now in the maths and science articles as well, but apparently not. I'm just not sure how it works. Also, for example, there's a question of Schiefto poses the fact that the category of set value achievable topological space has a sub-object classifier, is explained, and then construction of it is given, which is totally wrong, you know, as an espacetal, a space over the given space. So there is a certain space that they give, but it has nothing to do with the sub-object class. It's scary. Just not sure. They fail to use the... The basic dialectic between Spassa-Tallet and Precious, namely the correct Spassa-Tallet, its sections over open sets are just the chief you're trying to describe, namely in this case all open subsets of a given open subset, and it's not like that at all. Or, going the other way, that the fibers of the espaciate delay are gotten by a direct limit over neighborhoods of each point and then glued together. Well, again, if you take this gadget and take this something, they could get the empty set every time or something like that, if you try to take this directly in the case. So the problem is... So they're confused about which limits to take, is that the decision? Well, the thing to take the limit off. Oh, the thing to take the limit off, yeah. All these, all these. So they say, you know, okay, well, it's wonderful, it's self-correcting, but then I don't have enough confidence with dealing with the internet to go in and become... I don't know, maybe I should train myself for that as well as all the other things, but people who are actually knowledgeable about category theory should actually, and at the same time sufficiently confident about the internet and all that, they should actually undertake to systematically correct all these things just as well.

55:00 Well, maybe it was an idea to establish an informal working group of, you know, three or four people who were prepared to vote and agreed amount of their time and energy to doing that and to have a, you know, for them to establish an informal network to decide exactly on what the priorities are in terms of correcting this, like your example of the et al. I think it would be a waste of time, really, to try to correct Wikipedia all the time and constantly. Perhaps it just should be better to put publicly available real information. But where would you put it that has anything like the number of people accessing it and referring to it? And certainly for people learning the subject from scratch, I mean serious people who want to learn it from scratch, especially in places, well not just Pakistan, anywhere, but especially obviously in the developing world. I mean, they're always going to go back first. If I want to have some initial knowledge about who Raufer and Balfour was. And I find that he was one of the presidents of the British Academy. Ah, yes. Which is financing this whole thing. Yeah, as well. But that was over a hundred years ago. It was right at the very end of his life. That was when he knew quite a lot. Not quite. Not quite. Well, Michael Redhead still sits on their committee. Well, of course, Balfour was a very influential, apart from being prime minister, he was also a very influential Anglo-Hegelian philosopher, an idealist philosopher. Remarkably influential. Anyway, that's the first indication. I think you do have to, I would disagree, I think you do have to tackle Wikipedia directly because it's doing an awful lot of damage.

57:30 This is really difficult to put, I don't know, to make Google show a real hit. I don't know, even below Wikipedia... Yes, it will be a long way below Wikipedia, because that's what tends to get most hits, and Google is just ranked on the number of hits that sites get. But still, you're right, putting it on Google is an alternative, and just spreading as much as one can by word of mouth. Even putting an article on Wikipedia saying that it's not trustworthy. Just looked, as it were, at the fixed point theorem. Have you ever Googled Wikipedia on Wikipedia? I never have. It would be fascinating. What does it say? If you put the entry for Wikipedia on Wikipedia, what does it tell you about itself? That would be very interesting. If it were like typical articles on category theory, it would say, this is a complete misreading. People don't like it. This obviously annoys me greatly that those same articles on Toad Coaster, they say things like, it has a reputation for being difficult. Is that technical information? No, no. In whose eyes and who is spreading this? In whose eyes and who is spreading this? It's straightforward to discourage people from looking at it. Yes, absolutely. Whether that can be corrected or not. It's the oldest trick in the book. I mean, it's the trick they're of course doing now with high finance. I was talking to John, his brother, last night after dinner and there was this interview. Was it the president of the Fed? It wasn't the new treasurer. He was interviewed on CBS, I think one of the major networks, and just asked. Finally somebody got around to actually asking the question. The American people would like to know, Mr. whatever his name is, what exactly have the banks done with this X trillion dollars of money that we, the people, have given them? What's happened to it? Where's it gone? Because they're still in the crisis, getting worse every day, they're not lending, they've been given however many trillion dollars, what have they done with the money? Gobbledygook, gobbledygook. Listen, this is all being handled by experts and you don't expect, just you poor ordinary folks out there will just not be able to understand my answers.

1:00:00 I could give you an answer, but they're using all these stochastic differential equations and you just don't want to know because you just couldn't be bothered, your poor head's just don't need to be bothered. It's an incredibly patronising answer that I've ever heard anybody give, a public official give. But it was all based on this, you know, level. The level of expertise that you would need to be able to understand my answer is such that there is no point in my giving it to you. This is a difficult subject. Where are we? Well, that's interesting, but he's well known to be unreliable. He's well known to be unreliable. And his theories are a beautiful corpse. Nobody takes them seriously any longer, of course. Because we all know that nobody works in factories any longer and there is no working class. Not unless you look outside the... It's sort of a window of meta-information from the point of view of the writer of this particular book. Yeah, it's just incredible. Should we go upstairs? John's about to turn around. I was looking up something about community of rings, I think it was. Yeah, of course I knew about community of rings. And they said, well, we, Wikipedia, are introducing this whole new system to discuss all possible mathematical subjects, topology, categories, blah blah blah blah blah blah blah. Terminology like meta-conceptual, linguistics, you know, things like that. But it was a whole system and was supposed to be done by a graduate student at the University of Chicago. They actually followed it and it really wasn't sort of affecting all the fields. The next day it was all gone. I saw this, you see. Unless you're other witnesses, you won't even believe. But actually I saw this, you see, and then the next day I was going back to look it up and I thought, who is this guy? What does he really think he's doing? And blah, blah, blah. Nothing moves anymore. So somewhere they decided, well, after all this isn't so good. Yeah, but they must have decided at the top, because I don't think you can change things that drastically without a fight. No, which of course means that all the claims that are made for the whole thing being just, as it were, market-driven and all entirely self-regulating are very suspect.

1:02:30 You see quite a lot of PhD students now blog about things that they're learning, some of them say, well I was reading this book about this subject and they blog about it, and I think that moves over to Wikipedia, like, well, once it's been ironed out a few places, I could be able to write the answer to that one in Wikipedia. Ah, there we go, there's your ad. There's a personal link, right there at the top. Alright. And then you move. So you hadn't been waiting for us, had you? Yeah, I just turned up. Nobody was there, so I thought I'd go around to Richard's office. Ah, which we only just left. To come up to look to find you, sorry. All right. How's Ben? Everything go okay at the doctor's? Well, we don't know. We haven't got the results of the... I did notice that he's short of pills for tomorrow. The discussion is becoming very... Go on in. Oh, sorry. Sorry, he's an Argentinian category theorist, a friend of Davide Benedini and of Bill's. His name is Mateos. Mateos, he's an Argentinian category theorist. He's a topos theorist, a very nice guy. Yes, kinematics of rigid bodies with lots of interesting things that are obviously to do with connected components. Yes, yes, this is my kind of book, Andrew. And I take your point, John, it might not bring them in, but we might one day live in a society where that's no longer a joke, where people, where engineers do look at a book on, which starts with a discussion of Hegel and say, yeah, that's my kind of book. We might live in that society one day.

1:05:00 I'm feeling familiar with the tradition of rational mechanics. Sure. No, no, philosophical indeed it is. Yeah, exactly. Absolutely. In Italy it's better known, perhaps, than in other countries, this tradition of rational mechanics. Yeah, but in Italy what is called rational mechanics has been rather separate for many years. The people who do, officially, rational mechanics are mathematicians, so they are not really so well connected with actual engineers. There are a few engineers that are attracted by rational mechanics. It's the first modern science. It is beginning. I mean, rational mechanics was the first modern science, the first mathematical science. Galileo's. Yeah, I started with Galileo and I'm second to this book. I think it's a curious one. Well, there's a really interesting discussion of the place of mechanics in the intellectual economy in the preface to Newton's Concipia. He talks about, he thinks of mechanics and geometry as just different parts of one. He attacks the question of why we can't draw accurate circles and squares, and he says the problem is not with the subject, it's with the practitioners, just because we can't practice it. With that much accuracy doesn't mean that the accuracy isn't potentially there so and he sees he says that geometry actually rests on mechanics because the geometrical constructions are made by with instruments and so on okay and he points out that that the part of the mechanical arts is producing motions and changes by means of

1:07:30 And so on. And again, he says the same thing about this. We're limited in our capacity to produce forces, but that doesn't mean that the thing isn't mathematically exact. It's kind of interesting. It kind of explains why Newton used a kind of Euclidean method of exposition. Propositions and scolia and all the sort of critical apparatus that you would have in an edition of Euclid, but in order to expound physics. So that's, I mean, I don't think you get very far trying to publish a book on physics if you did that sort of thing now. But it's interesting how they looked at it. I think maybe he must have probably influenced other people. Thank you. Did you say that Anders is coming today? Yeah, but I think a bit later. I think he'll arrive later. He'll arrive, okay. I think so, but I guess he's flying to Birmingham now. Yeah. And by the time he gets his train, I don't think so. I'm surprised there isn't a direct flight from somewhere. Well, I thought he would be flying to Bristol. Yeah, but I think it must be. I'm sure there's some reason why he'll be coming to Birmingham. They conquer the place often enough. Well you had visions of him coming up the ooze in his longship with his axe. Jumping off right here. Where does the rape and pillage start? Seeing geometry and mechanics as two aspects of a single discipline, which I agree is obviously central to Newton and Galileo, is certainly not dead. I think there have been plenty of people in the 20th century who have taken up that program as the basis of Bedlam's ideas about unified field theory.

1:10:00 But it's done differently because Newton actually mimics the expositional style of Euclid. I mean, he has propositions. I mean, it's just, it's very strange for a modern person reading it to see it done that way, and he really, as far as he's concerned, his proofs are really proofs, you know, they're proofs in the way Euclid studied, Euclid's propositions were proofs, yeah, but the fact that rational mechanics is so little known, in a way, it's... It's connected with the fact that the general public, even the general scientific public, believes somehow that matter is made of quarks or something like that. So to explain every kind of motion, bodies and so forth, you just have to have a sufficiently elaborate, maybe statistical theory about these quarks. It's very, very extreme. This is not really justified by my opinion, but that means that the actual very detailed models of bodies that engineers actually need in practice is considered to be something outside of science, even if it is known that it exists. The fact that in engineering one needs many levels of precision and many kinds of models or bodies, depending on the circumstance, is mirrored in the fact that Cantor, Menon, In practical applications of the canons, people think in terms of infinitesimalism.

1:12:30 Not officially? Not officially, but with a guilty conscience, of course. It's just like so-called illegitimate categories. There are all these artificial boundaries, which means that people's thinking is supposed to become imprecise at a certain point. Rather than, why should not the official way of doing things incorporate this? I mean, even in fluid mechanics, infinitesimal methods. I mean, even when you say you've got a volume element over which the pressure doesn't differ very significantly and that kind of thing, I mean, these are using infinitesimals, you know, and to an outsider, it's kind of puzzling exactly, you know, where the... Model stops in an attempt to, you know, are you looking at a, when you're doing oceanography, are you looking at a square meter, a square, a cubic meter, a cubic kilometer, what are you, what's your infinitesimal number? Because a cubic meter in the ocean is a drop, as it were. Yeah, well, the problem is the use of continua, because models in rational mechanics are continuous bodies. So, whenever you approach the local properties of a continuum, you need some kind of infinitesimal notion, even in time, but also in space, because the basic principle that is already used non-officially is that things can be reconstructed from the infinitesimal. Yeah, all what is done in mechanics, the global properties, even in space and in time, are reconstructed from derivatives. Then, of course, you need the notion of infinitesimals. Yeah, so, I mean, we all know that fluids aren't continuous in the viastasians, but they are continuous. They are continuous. They are continuous. And therefore, there's something wrong with the viastasian picture.

1:15:00 Actually, it's a matter of scale, of observation. It depends on the scale at which you look at them, of course. Yeah. Yeah. I always come back to this question of being and becoming, in a way that the Galileo-Newton advance, describing motion, was the idea that forces act not on states of being, but on states of becoming. So sometimes the very idea that there's such a thing as a state of becoming, this is typically modeled by infinitesimals of some sort, but the general philosophical, difficult in a way, philosophical idea is precisely that. But this is also the odd thing about Newtonian and Galilean mechanics. It's kind of counterintuitive. In a way, I mean, to this sort of earthbound, plain man's view of things, that you change states of becoming rather than change states of being, is what you're saying. Right, right. And so, changes are slow, I suppose, is something you perceive when you change states of being. Yeah. So if we took thousands of years of philosophy, I don't know at what point... That phrase, state of being, or something equivalent, actually came into, you know, it was summed up and recognized, but certainly in the practice of Galileo and Newton, this was becoming a crucial ingredient, you know, that there's inertia, the same idea, that there's such a thing as inertia, momentum, you know, dialectical, the same thing. I finally did something that Davide knows as he might kind of rescue me, but for years I wanted to go to the city of Elea in southern Italy, where Zeno and Parmenides and these characters had moved when the Persians invaded their own state in what were Eastern Greece.

1:17:30 Ionia, yeah. Ionia. So I finally had an opportunity to do that. There was a school where Jonathan Barnes, an alleged expert in these matters, gave me the course and it was very interesting, even though his lectures were practically devoid of information, the participants in this seminar were pretty... that's where I met Imre Toth, for example. Do you know Imre Toth? I think he seems to be an interesting Hungarian philosopher living in Paris. But at any rate, I kind of realized that Parmenides and Zeno, let's say, just roughly speaking, they discovered some of the crucial dialectical moves.