Conversations en route FW Lawvere & Silvana Lawvere

0:00 Openness, that has been very, very helpful. I still haven't really grasped this distinction yet. Yes, naturali for me. Presenti per me, grazie. Yeah, general and particular. General and particular seems much better. General and particular are actually... All of these are qualitatively different, even if they're the same size. It was just completely against the idea of what general is. The general is big, and the dictionary is small, which of course itself goes with the very, very counter, not counter, Phragian point of view. I was just going to say, to the words right out of my mouth, yes, the Phragian point of view, or Plato and Phragian, with the idea that everything is points. So it's just a question of whether they're more... So what I mean, the category of reflexive graphs is general, even though it's quite small compared to most other ideas of believability. I have a minimalistic idea of cohesion, but because of the reflexivity, it satisfies the axioms that I give for general. On the other hand, you can have some huge, for example, seams on Hilbert space. Which is huge in terms of size, but still particular, and indeed satisfies the autumnal accent that I was in. This is in the Bogota paper. Oh, qualitative distinctions between... No, no, that's the... Oh, no, that's the...

2:30 Very short paper. Short? Very short Bogota paper. A sort of gentle hint for Peter Johnson. A general hint for Peter Johnston, yes. A gentle, yes. Categories of spaces may not be generalized spaces, as exemplified by directed graphs. So the reflexive directed graphs turn out to be general categories of spaces, not reflexive graphs. Very contrast turned out to be particular in the sense that it's pure variation over a very, very tiny space-like, single space-like, predictionary space-like. Yes, explain. Do you have that paper? I do have that paper. That contends to be a very elementary model that contains this profound contradiction between the two, that no particular can satisfy my axioms for the general. I have two axioms for freedom. I have two main axioms for the general, which are not true. False, in other words. ...standard examples in particular. So, that refutes the 1970s philosophy in particular. Yes, all topics are general. How general are generalized phases? Well, any topology could be a generalized phase. Not in terms of this more precise... Yes, yes. The contradiction is that all I mentioned was actually in my lecture. Yes, yes, that's right. As well as the third ingredient, which is obviously if you have a general category... In other words, any space, i.e. object of a general that should give rise to another topos, which is the category of the speech sets that vary over that space.

5:00 Or which are acted on by that space if it's really a group rather than a, you know, interlocking regent set. So indeed, associated to every object in a grove or mineral topos, there should be, in that uniformly defined way, another topos which is just... Speed objects can vary over or parameterized by or acted on by that particular object, but this is the general vision of the connection between the two. As I said, I haven't succeeded in fully acclimatizing, but again, within the narrow confines of directed graph theory, I do show that in that Bogota-Wehmer, indeed, from any reflexive graph, you can deduce a category which is sort of like an unreflexive graph, and in particular, you can choose the loop. This is one, obviously, central example of a sort of contradictory directive reflexive graph, but this beautiful procedure for associating a particular topology to that one gives back the category of not reflexive, not reflexive. I really like that paper. So it's simple. I will, I will, I will. I had looked at it, and until now I hadn't recognized its general significance. I hadn't understood what it was that you were dating at in the distinction between the general and particular. It's very powerful, the general opposition and so forth. There's this minimalistic example which already reveals most of the pieces. I should reflect as you do, as you do, as you do, as you do, as you do, as you do, as you do, as you do, as you do.

7:30 How does this affect, at least I think, I haven't got the most general case, the general idea of giving an object a space x, i.e. an object in a cohesive or general form. Very cohesive and very non-cohesive somehow contained already in the Groucho place, in general, and that's part of the idea. Anyway, so an object of the associated particular or fatigue should simply be a space over x, you know, it's part of the slice category. Just as sheaves are representable as pescosa talei, local movie morphisms, so that the fibers are discrete, and actually floor properties, but in particular, the fibers are discrete, so among all the computes, the bistopo, the arbitrary space is over, to give an answer. This is a new Grotopos. In fact, that's the original origin of the idea of Grotopos, which would do that to space. Is it a subtopos or is it a quotientopos? It's hard to tell because the subcategory could turn out to be either. Yeah, I was just going to ask you that question. Indeed, in Grotendieck's chocolate exercises, he has a distinction between sometimes... It's a quotient, and sometimes it's both a quotient and a sub-object, because the quotient has an adjoint of the quotient, and the sub-quotient has an adjoint of the quotient. So, there's an important ambiguity there. In any case, you want to take out some of the huge topos of all objects over X. Thank you for watching.

10:00 As I say, in the general case, it's not enough to say that the fibers are discrete. There should be some more subtle property which lies there. Yes, yes. But in the case of the graphs, it works. You just say you have any reflexive graph x, you have the topos of all reflexive graphs over x, and you pull out the part where the fibers are discrete. And then that turns out to be of a particular kind, and if there are particular effects, it's just a loop that gets back to the non-reflexive. How does the condition for the fibers to be discrete connect with Johnston's quotient-decidable objects? I was going to say, this is connecting up with this unity between separated, unramified, and side objects, which I still need to... I know you've explained it to me about three times already, but the unramified object over there could be... This is a very viable notion of the associated particular over x. Well, actually, more exactly... Yeah, this is really... More exactly, you know, the famous etal maps of rodentism. Yeah, I mean, etal maps over x is another example of the key of a particular story. This is sometimes confused in this course because there's the Gros et al. topos, and for each phase there's the Petit et al. topos. And the Petit et al., this was the original example of a quantified space. In algebraic geometry there's no implicit functions here, so that you'd have to take Riemann surfaces as long as... Oh, I see they're springing right up. That's a good way of doing it. Yes, you were talking about the lack of the implicit function theorem in general. I was trying to explain to you what that was. It's a cell that sort of generalizes when you replace it by the image. In this case, you don't have the implicit function theorem. You get this more general idea of an object that doesn't go along with the function theorem.

12:30 Grazie a lei. Do you have English? Oh, I'll have this one. Don't worry, don't worry, it's okay, it's only... That's the whole reason you have manila folders, don't worry, I'm not... Sorry, that's enough math for just for now. Well, probably not for you. Look, I certainly understand much more clearly now why you said that that old idea, the original idea that was around in the 70s... Topos theory was really just a theory of parametrized variation plus the internal law of the universe. This is something that is really a generalized base in an incredibly narrow sense, but it's not determined by the locale of open parts, because, so I keep saying to people, that Hoppe's Theorists have never actually, or even algebraic geometers, have never taken seriously the idea, well, we can characterize axiomatically what kind of a joke those people did. In any case, one of the unequivalent forms of Joshua's definition is that these are the maps over x which are unramified and flat. Now, so unramified, roughly speaking, means locally monomorphic. Like decidable, it's monomorphic, it splits into parts, right? So it's locally monomorphic, and flat is locally epiphytic. So there's a category of all flat maps in directs and a category of all engramified maps in directs.

15:00 Both of which are of some interest to the intersection of the Atiyah town. And I think both Petit et al., but not the flat, both the Petit et al. and the larger and ramified Petit et al., both are deserving of the epithet Particular. Both are deserving of the epithet Particular, the ones that are larger, particular than the others. Yes, I think I'm beginning to get an understanding now about how this Euletia separated undrammified indecidable objects provides such a key insight into that distinction between the particular and the general separatists. But what again is the definition of the flat case? I'm sorry, I didn't get it. And how does it connect with this condition about the item presence? Well, Brad is also, well, it's a different thing. H.R. is also equivalent to unramified and smooth, at least H.R. is smooth. In other words, a little neighborhood below will really be covered by a little neighborhood above, maybe many times, but it will be a little neighborhood below. Smoothness, flatness, something like that. Flatness is usually associated with categories of linear sheets. So it's a case of the algebraic notion of the quantity of the sphere, which is simply that the tension product, or the left adjoint to change your base, to change your, to the function induced by, to the inverse image function induced by a given mass, you have a left adjoint, which is the tension product. It might happen to be left exact, even though it's left adjoined by nature in general, only right exact.

17:30 Well, that's the definition of flat. It's the case where the tensor product is actually left exact. And that is really right exact. So that implies that they're changing the abelian or linear sheets rather than the non-linear set ones. Rather than to the sort, to the non-government or purpose. Subtitles by the Amara.org community What was that? I'm sorry. Okay, let's just have another one. So the deletions are truth, and that we want to apply to the non-linear theory of the class. This applies to the linear theory, so in some sense it's more remote from the immediate dimension. This is moving away from the question of general and particular purposes. It's quite a deep question. I'm finding the right notions. But can you tell me a little bit about this memoir, document, or however you want to describe it, that Rodenbeek deposited in Buffalo in 1973, this classification of... These are some of the structures that he proposed at that time during his visit to Buffalo, as he was telling them out a little bit while we were in London this spring. Sorry. Not an article, are you? No, no, just a... Just a big sheet of paper. No, I can't read it. It's just a...

20:00 Yeah. As he was talking to Jack Duskin and other people. Well, if you can study this, you can classify some of those with some kind of structure. What is the general point of a structure that can be so classified? Well, now in terms of the logic of sub-objects... We say it's the positive, the geometric, the coherent, or the dynamic. There's four different words for it. It's very uncomfortable. It's a kind of theory of a structure that can be classified. But as I tried to persuade Cartier after his articles and bulletin where he said, well, Kourtney sort of had to go to discover these two factors, and he was good at logic, and Kourtney had no need of logic, because instead of sort of coding every kind of structure in terms of sub-objects, you know, relational systems, and he dealt with them directly. You could have been coding them that way. So, it was clear that in a sense the geometric orthos on Zimper's image preserves arbitrary direct limits and finite limits. Well, then any kind of structure which can be described in those terms will be preserved and hence classified. Now this is the... So, Rodendieck had no need of logic because he could think directly in terms of the direct limits and the finite limits of Zimper's image. So, thinking in those terms... He said, well, okay, the notion of ring is classifiable. Ah, the notion of local ring is also classifiable. Ah, the notion of integral domain is also classifiable. Ah, the notion of Japanese ring is also classifiable. Ah, the notion of this and that. So he had these more or less complicated notions of algebra, but he asserted they were all classifiable. Subtopos and subtopos are the ring classifiers that classify exactly that. So it's just a complete slew of the things that are classified by subtopos as the things that are classified as the things that are classified as the things that are classified as the things that are classified as the things that are classified as the things that are classified as the things that are classified as the things that are classified as the things that are classified as the things that are classified as the things that are classified as the So it's interesting to see that way, that in this sort of complete survey of the model theory of positive, geometric, coherent, dynamic logic, without ever mentioning logic, that's what I find fascinating about it, not looking at it in terms of connectives.

22:30 Oh yes, I think so. You don't even have to be a genius to go to H. Palbert, who, you know, got quite acquainted with what a direct limit is by the end of it, and so sort of think directly in those terms, you know, always encoding it in terms of... Without going through the subject, yes, without going through the whole subject. The logic and the narrow sense. Yes, and then the subject, yes. Excusez-vous. Si, si. When did you arrive at this... The insight about the nature of narrow sense logic was obviously when you sorted out the way that all these structures, particularly the structures you deal with in functional analysis but elsewhere, are naturally... Naturally fit within this interrelationship of intensive and extensive quantities, at the recognition that logic, subjective logic, is really to do with the study of the roots and supports of intensive quantities. And that is still a very unrecognized, I mean, the point is, what Cartier recognized as the result of this meeting, that as a zero to approximation to any problem, everything is a category and puncture, so the problem is defining as one. So in other words, the individual structures are actually categories. Some of those happen to be posets. But to think that the post-set case is somehow the paradigm case of something like this is wrong. Although it is the point of view of Greg Encival in the 20th century.

25:00 Yes, and the point of view that Alberto has fought against so hard in philosophy. The narrow sense is simply in that you view the post-set case as the paradigm case. Money. Money. Money. Thank you for watching. There's nothing particularly ideologically loaded about the Atkins Dough, is there? No, I don't really follow the Atkins Dough, but I do... Actually, I stay away from Italian bread because it's about the one thing in Italian food that doesn't have any taste at all. The one thing in which I am prepared to say the French have got it right. No, it may not be true, but I'm surprised. I struggle terribly with weak wings. You do? You don't show any signs of it in the media. I'm not trying to... I'm winning the battle. I'm not trying to be flattering, believe me. This is not empty flattery. I'm amazed to learn that, actually. I thought you were one of these lucky people who could kind of eat like a horse and still stay penciled there. No, no, no, no. I like you just as you are. Leave it. I'll show Dad. It's pretty shocking to be here. I was just going to show what application and strength it wields. Oh my God! Remember those days? When was that? What year was that? Did you go all sort of en brosse and cropped? Was that a political statement? You emphasized it tremendously by pulling your hair back. Yeah. I know, but it's kind of erratic. That was right when I came back from Japan. Really? You've blotted out those bits. I don't know. In general, I don't register those facts. Yes, because... Well, you've got a big figure. Well, I don't really think about it.

27:30 Yeah, because... No, that's quite right, too. Polo? Polo? Ah, see it. Yeah. Well, you're the last person on Earth that I would expect to judge anybody on body image. I mean, body image. Well, in fact, I did a really, really terrible thing when she was younger. Maybe the second-to-last person on Earth. I'm just kidding. Yeah. Tell me. When she was younger, I used to, out of pure endearment, I heard you were my baby elephant. But for me, you had no kind of baby elephant. This has been a great pedagogical era for generations, that's content-based teaching which has survived. Overcoming this chronic disease in one small way, which is, epidemiologists all use statistics but none of them understand it. Or only a select few. No, very few medics do in my experience. So we want to create the... A course, let's say, or a brief handbook, or whatever it might be, that can help you overcome this by giving a conceptual, clear conceptual knowledge, which is kind of a particular subject on mathematics or algebra. So, you know, I saw a whole different way of doing it, which maybe, hopefully, would work. I was trying to think myself about what this disentangling of the various structures that are involved in functional analysis in terms of the relationship between extensive and extensive quantities would imply for measure theory.

30:00 Oh, you've talked a little, well actually you've probably talked quite a bit about it, certainly in the past I heard you talk quite a bit about this. What was this remark, though, that John Mayberry was so excited by, that you pointed out that, in fact, it was a simple matter of The idea that one can think of a real number as the ratio between the terminals. In fact, we're looking at Euler's material, and I believe that Euler's material actually was what defines real numbers. I'm sure he could clarify that. He doesn't seem to use space control at all, does he? Essentially he's up there, I think. Professors, if you think, Palermo. This is common practice in the mid-late 19th century, as opposed to another, but actually he was, when I first met him anyway, he was working specifically in his third place, which was the Academia dei Dicei, in Rome. Well, probably that's not a member of that academy.

32:30 I'm a certified member of the far-seeing yachts. Well, too bad we've got to change now. No, no, I don't want to come back to work. Up to you. Yeah, I'd spend some of my time. No, I'm sorry. That was stupid of me, wasn't it? I'm sorry. I should just be patient. I just ate without thinking. I'm sorry. Well, no. Dad pointed out you might have wanted a swap because that's the chocolate one. And we took one of each. And I took the non-chocolate one in case you wanted to swap over. And then, just without thinking, because I was listening to your dad and all his fault really. No, I was just being totally absent-minded. Wham! Really sorry. But you would have preferred the chocolate anyway, wouldn't you? No, you're lucky he's here. I would have murdered you right now. No, I'm sorry. I'm sorry, you don't have my permission to slap you. Like you have anything to say. Well, I mean, look, the next few days will be on a Swiss period, so, certainly. Lots of beer and lots of walking. Yeah, awesome. At least they don't charge you for the walking. Although I don't know any Switzerland these days. It's really expensive. Oh, yeah, it's unbelievable. Well, of course, the Swiss, thanks to the Bush's policy in Iraq, the Swiss bank has gone up by 30% in the last year, and it was already... Thank you very much for your time. I only wish I had put the funds from the sale of my house, my three flats, in London, in Switzerland, because I would now be in a much more comfortable position, but I just didn't, I foolishly believed that I'd have the gold, I just wished I had the gold, it's been up by about 30-40%.

35:00 Well, I have 40% more peanuts. It's really never going to be 20. Because the Saudi royal family has shifted even more of their resources into safe haven, into some sort of national standard they already had, so there's a flood of Arab money in the months before. And obviously the perception that the Americans are going to pull the plug on them as the local satraps, as the local... Agents of the... Coffee, yes. Thanks. Yes, si, gracias. Coffee? No, thank you. That their day as the tools and cronies of American imperialism are possibly over and done with, and that pretty obviously the American bush has pursued what's obviously a pretty long-prepared plan B to... ...to make the Caspian Basin the principal American oil province, move resources out here, possibly combined with Mosul, and the Saudis are... If there is a revolution, a reactionary revolution, if there is a kind of popular uprising in Saudi Arabia, in the House of Saud, is history, or if the Al-Qaeda is going to actually take over the holy places, I don't think that even Bush is going to be quite so crazy as to try and send the American army into Occupy Mecca and Medina. I mean, I think it's possible, but then... Well, I can't believe we're preparing plan B. Well, I see you. That's why they've set up these local corrupt oligarchies in Azerbaijan, particularly in Azerbaijan. Azerbaijan is extremely strategically important, and in the Sudan, and in, I mean, all over the Caucasus. It's clearly what the script, you know, that Shevin has is part of it.

37:30 This is why BP has been paid, which of course is now effectively an American company, has received this five billion subvention to build the pipeline, the trans-corpus pipeline, no, the whole... That's the whole reason they had the war in Afghanistan, was for the pipeline. Yeah, and indeed it's one of the chief reasons, I think, why they had the war in Iraq, because the whole strategy is to move the oil extraction business northwards by a thousand miles. Oh yeah, we'll hear a lot more about that. It's like we're hearing again about Azerbaijan now, because in 85 years or so, that was the... The big place where Nobel made his money. Yes, yes. Azerbaijan oil fields, not just, I mean, they're initially the dynamite, but then, Azerbaijan oil fields. John Reed visited the Azerbaijan, at least in the movie, I presume that's more or less correct. He visited there. And Mikoyan, don't forget, he was training there. They made a movie. Yes. Loosely based on another writer, John Reed, who wrote 10 Days of Sugar World. Very loosely. That's Mikoyan. Yeah. He was a counterpart of Herriman in the Lindley's program, which I feel must be something tremendously significant, because economic support for the international provision of counter-revolution should be transmitted with a guide. Of course, it did help the war effort against the Nazis quite a bit. At the same time, Herriman. Mikoyan's autobiography has a long introduction by Herriman. Oh, this is good. This is very important stuff. I agree. Here we have this arch-capitalist praising this alleged communist, you know, greatest human being, Kant. It's amazing.

40:00 They poisoned Stalin completely in a change that was going on with some friends of Eisenhower. And he was the one who spoke at the 20th Congress before Khrushchev. Before Khrushchev to now Stalin and all that this time. Mikoyan was a key figure. So, in this autobiography, I mean, it's incredibly self-laudatory, boring to read the whole thing, but I got to, technically, two volumes. But I've got this, you see, at a certain point, he was, as an alleged communist organizer, an Azerbaijani, an Armenian, and he was, and this is a tie-in from the 1480s intervention in Texas, in the Navy, show courses like these at the Star Wars, the main problem at Star Wars. The main ground of the information is the types, keeping the types. Topos theory is the most perfect theory of types at this particular time, so everyone wants to be able to do nice typed programming languages. He actually gave this topos theory for Star Wars. You see, what I actually felt that I learned, that I grasped, that most people don't grasp, when I actually worked for one of these military think tanks. I actually finished my thesis in this impressive environment. Anyway, what I learned was that most of these mathematical things are not really being used for production of material references. They're being used, they're literally propaganda, both for the public and within the Pentagon, within the military, within the companies and so forth. It's to say, well, you know, this is, you know, it's as crony-esque as possible. We have... The most powerful secret theory. Therefore, what we say, you have to do what you have to do.

42:30 This is the way it's actually used. It's also, of course, an excellent tool for distorting the consciousness of mathematicians. Well, exactly what I'm saying, you see. Naive mathematicians. I actually heard it argued by one. Trotsky, a well-known mathematician, to the effect that while some differential equations are more evil than other differential equations because they can be used for designing missiles, I think this is totally absurd because apart from improved programming and the like, the theoretical advances in differential equations, which are being used in the production of military hardware, are things that were already known a hundred years ago. It's not that further theoretical development in that area has any significant interaction with the actual. It's rather that the high-powered theory, you know, the... To all talk about, you know, topos theory plays that role, as do many other theories within computer science, within the military assembly, within the Pentagon. It's simply propaganda. We have the most high-quality theory. You don't understand it, but we do. Therefore, what we say is the proper course of action. You have to accept without question. This is incredible, this is how it again and again and again, this is the way that it's used. It's fit-phidi-ism under the pretext of science. Exactly, exactly. Phidi-ism but using... You have to believe the prestige of science, whatever, chaos theory, catastrophe theory, any of these kind of slogans that get popularized, they play that role. They have practically no relation to actual production. They have plenty of role to play. The war of ideas, and then the war of ideas in the narrow sense. You have to buy from our company. Yes, yes, the buying for the Solstice and for grand money. Yes, I see that. Oh well. I remember when we were doing this, there was some officer in the Pentagon, very excited about some of the miners' proposal. I went to the Pentagon for another purpose, but he was very excited because there was a germ of the theory of...

45:00 You know, command and control of various levels. You could have commanders at one level, and commanders at another level, and another level, and information flowing up and orders flowing down in a sort of complicated tree, and the whole thing would somehow work. So I had to sort of germ up a picture like that, but again, at the level of science fiction. But he wanted to develop those, and I was quite surprised at why he didn't, depending on the actual military thing. He was highly interested in this far-out theory that would never be implementable as such in time if it ever developed at all. So this was really one of many examples that I saw in that environment that led me to this particular kind of sense of the true significance of advanced theory, so-called. Military industrial complex in particular and Anglo-American imperialism in general. And of course it also helps to keep people from learning about science. By elevating this distorted ideological picture of science. We're almost there, just two more. Sure, 618. 618, yeah. Thank you for watching. Thank you for your attention.

47:30 One of your earlier sets of notes didn't he, a long time ago? Yeah, several of them, yeah. The one that was translated into Italian. Yeah. And he also was my assistant two years ago at a summer school in Perugia. He made the notes for that, which are, of course, not public. I used them. I guided myself. Carboni, who's an old friend and has many things to say. Betti and L'Astoria named L'Astoria, and I'm going to bring up some didactic articles for this journal. I got a teeny because of history and I must have remembered to ask him about it earlier. Yeah. And now I've been finally meant... Okay, sorry, just gotta go and get the... Right, it's both of us, isn't it? Yeah. Here we are, it's this one here. Scusi! Thank you, Mrs. Lena. Is it possible to leave? Yes, together, but two... Yes, together, yes. You leave in 16 hours, please, for Paris. How many days? Just one. Two days? Yes, one day. Second? Yes, thank you. Without smoke. Without smoke? Without smoke, yes. No, without smoke. It actually doesn't bother me that much, you know, having been a smoker, I'm not that bothered, but other things being, well, breath breathing, second hand, well, yeah, if you have a choice, but I'm, well, actually, you'll tell me, but I think the statistical methodology of the, you know, sort of passive smoking is largely driven by the, perhaps, you know. I don't have an ideologically admirable desire to frighten people, but I think there's very little to this level. No, there is a level, but it's much lesser. Yeah, but I mean, it's such a reduced level, yeah. No, but it's not really that reduced. Oh, okay. Well, maybe in that case I should make a point of trying to stay away from it. I think it's three or three percent or something like that of the risk of somebody who's smoking. No, of lung cancer patients that are not as specific and not part of our focus, so. But see, that of course doesn't follow that the path of smoking is the...

50:00 No, but I mean... No, I know, I'm... There's actually growing... Susie? Yes, go ahead, go ahead. I think it has largely dissolved because of the interactions we were able to have here, just because of this meeting. I think I did wonderful. You did. You tell me the fact that he made a mistake. Yes, a maths. You know, he said... Yeah. When I made these, you know, these conjectures, he immediately said, well, that, of course, is true as you conjecture. So he could prove that. He said, well, this other thing is wrong, he said. But then, both of us... It would only hold an appreciative category. No, it wouldn't even hold anything. Well, we were... Both, during the next lecture, in the first few minutes, we both realized that he was wrong, so he came up and told me, and somehow it, yes, it was, and I needed it, but not only that incident, but several other times when we were able to, of course, the fact that he had made this enormous accomplishment of writing this elephant. I clearly appreciate that. Nobody can take that away from him. This is incredible. All these things have really helped. I think we'll be in much better terms. No, no, I know, no, I realize you were never enemies at all, but he's a very diffident person, he's very shy, he's difficult to deal with, he's a very good person. And, no, I thought that was a very good thing. So this was really one of the major accomplishments, in a way, of the meeting office in terms of future work. No, I do think we've done, we have laid the groundwork for all sorts of very fruitful collaborations. Yeah, yeah, yeah. I thought, by the way, John Bell's talk was quite good. Yeah, yeah, I liked it very much indeed. Very well-organized talk, nice and covered a lot of ground, but was very interesting, and I think brings out all the salient conceptual issues. In other words, I was actually disappointed with many of the other talks, I have to say. They actually were much lighter than I hoped for.

52:30 I think, I think perhaps some of the mathematicians may have been a little bit intimidated by the idea that they're going to be... Well, we're speaking to an audience with philosophers, so they kept it very... Cartier's was an example of that, obviously. Right, right. Yes, yes. This is the sort of situation where, unless the philosophers actually do have some mathematical equipment, it's more or less pointless, because they'll just pick up buzzwords and... I'm not saying that any of these will do that, but I'm just saying that sort of as a general, and those that do already know some mathematics won't need to be, you know, talked down to in that way, so. If we ever do this type of thing again, which I hope we will, we should maybe make it a little bit more focused and have it slightly smaller, because it has to do with high-level philosophy as an mathematician shouldn't be afraid of it. Philosophers should damn bloody well do some studying before they go down and drop. At least read conceptual math. Yes, they've got to go and do a lot of hard work in preparation for it. Well, that was certainly true of, I'd say, Longo's talk, but Collins was... Collins was... Collins was... Collins was... I mean, I think... I didn't agree with all that he said by a long way, but it was a good... You gave me the strategic... sorry, tactical mistake in talking so much about the various details about McLean. Yes, yes, I knew that was wrong. I think you should have just been stuck with McLean. The fact was that he couldn't get around to what he thought about Le Verre's physics. Yes, exactly. Which was mainly the topic of discussion. Exactly. Yes, I was a bit... I was a bit disappointed. I think that was just a tactical miscalculation. I mean, MacLean is no doubt great and, you know, he does have many philosophical... He just gave too many details about historical... But, I mean, the general idea of his talk is good. I like these things he says about Grotendieck. Well, Grotendieck said about himself, but... Colin is one of the few who's actually really started them, although he does still, I think, miss the point about how important Gregory Beek's early work in functional analysis is. Functional analysis and the complex analysis and the Grotto post of 1960.

55:00 He's sort of unfortunately still missing something. But I keep chipping away. I mean, being both in the same country, we have long phone conversations from time to time, so I keep chipping away at this, and I think he's weakening, but... Well, I hope very much you could, as I say, if, I mean, you're going to be seeing him anyway, but if I could get you to come over for a few days, a week or two. And it's important because he's one of the few who is actually undertaking to write a book about rugby. Exactly. So whatever he does will be the definitive in terms of the literature. In terms of the literature, exactly. That's why it is so important that he gets it straight. But he's a fantastically quick study and, you know, a very powerful, organizing mind, and he's done a nice bit of material, so he's definitely the man to do it. I'd really like to get him and you to come and spend a few days in, well, it doesn't have to be in Puget, but it's just that there I can feed you and it's a very nice atmosphere and congenial to work in. And I think it would be a good place to spend, maybe just for you to meet one-on-one or with small groups. Half a dozen or so philosophers and mathematicians to really provide us with a study center, one of the few, if not the only, philosopher who has also actually done mathematics, who has actually published papers in serious mathematical journals. This is actually indispensable, I think. To talk about the philosophy of mathematics without having done some mathematics research is actually, in the end, you only have to look at what people, nearly every key, Shapiro people come up with, and it's just completely absurd. That's the source of this, this regress of structuralist, anti-Wren structuralism. They come up with nonsense and they live in a universe where it's acceptable. Well, because they only speak to fellow philosophers, they don't, they're either too intimidated or too... Cynical, because they know that they, they at least recognize that they lack the necessary concepts and correct notions to speak from that point of view. Well, that's new. They have a metal detector and they go through all the luggage. Oh, yeah, yeah, yeah, nowadays. But did they actually make you open it? They normally just put it through a machine. Oh, they did. You've got your cleaning shirt on. Okay. So I'm ready. All right.

57:30 Well, actually, why don't you keep it on there and just put it straight in the taxi? Might as well. Well, I was going to say, why don't you keep that at least until you get to the taxi? Oh, I guess you don't need to. You've got, yes, they've got wheels. Well, I'll come with you to the taxi stand anyway. Alberto, of course, didn't really have time to prepare his talk. What he said about the forms of extensionality in toposphere and how they... Did I like it very much? Thank you. Yeah. No, of course. Does he speak Arabic? Uh, no. Yeah. How snazzy is that? Yeah. Very compact. Um, yeah, I'll be right back.