What are continua?

0:00 What was his thesis on? On play. And actually he organized another meeting on... About Chez Monard. Well, that's interesting. I noticed that there is a... he's giving... No, it's not him. Who is giving the talk on Chez Monard? There is a talk on Chez Monard this afternoon. It is him, yes. I thought so. Yeah, and a discussion, which I... I read a collection of essays in honor of Chez Monard, I think it left me more puzzled than enlightened. What Boucher-Ménard was really about, I mean, I couldn't sense a kind of synoptic focus for Boucher-Ménard's thought, but obviously I don't know enough about it. He lived a long time, and I know that he was very active politically, but it doesn't seem, at least on the face of it, without just a very cursory reading of secondary people writing about him. His Marxism went very deeply into his science or mathematics. He did a great work in promoting logic in Italy. A very interesting figure.

2:30 Did you ever meet him? Yeah, yeah. The first time when I was a student, professor of all my professors of logic. I had the impression that you promoted logic. There was some tension in the microcosm of logic, not just as a chapter of the philosophy of science, but aimed at being embedded within the general view of the world of a certain kind. Well, of course, if it's not presented as being embedded within a general view of the world, then, of course, it tends itself to provide the underpinnings for such a view without really being the metaphysical presuppositions of logic being embedded. Examine. So I tend to incline to Gaimanar's view of that methodologically, although I can see, but my impression is that he was a fairly difficult man, that he was a combative, no, he was not an easy individual. That's the impression I got from reading about him. But clearly he, you know, did a very important work for science in Italy, especially obviously the resurrection of logical studies after the war. What do you think how great the center of logic Italy had been in the 19... before the 1920s, you know, philosophically sensitive mathematics?

5:00 However, well, you know, he liked to be admired, so this kind of... Yeah, I can imagine he was a bit like Popper, I can imagine that. He liked to bask in being a... Not so... Yes, yes, I'm afraid that too. I'm sorry, have some more. I'll get some more coffee. I took part in a congress dedicated to mathematics, and we had some discussions about, small discussions. We started being old and older, and after some, after a while, it was time that we could get on. No, no. Well, he did live to a great age, of course, he did, yeah, but, yeah, this is one of the things I was saying last night, that I have to, you know, record my great admiration for Penrose as a man, as an individual, not just as a scientist and a great mathematician, because although one may disagree with his position, certainly in philosophy and mathematics, I deeply disagree with it. It is impossible not to admire the way that now that he is revered and senior and terribly distinguished and loaded with honours, he is still very accessible and doesn't make a big production of being important. He really is a very modest and unassuming man. As I say, I didn't know about Gemini, but what I have read, I've admired, that it's mostly been the secondary sources, and there was a very good obituary of him in the British independent newspaper when he died, which was what, about 13 years ago now, I guess it was about 1987, a bit later, maybe 1990, early 1990s, anyway.

7:30 There was a very long detailed lecture which is unusual for any philosopher, even any major philosopher in the British press, for a non-Anglophone. It's almost unheard of. I mean the last time that a non-Anglophone philosopher got a major obituary in the Times or in any British paper was probably when Heidegger died. And this is impressive. Obviously somebody in the independent office knew a little bit about logic and philosophy of mathematics in Italy. We also have a conference on the superior normal, also with L'Auberge et l'Angleterre. Please, please, not please.

10:00 ...about his paper. Yeah, we must keep an eye on the time. What is it? Is it 8.45, the registration? I'm not sure. I have to program. Or is it the first paper? Maybe it's earlier. No, I think that's right. I think the first paper for the speech of welcome. Actually, I wanted to ask, I saved the list of his positions, the email I sent him, but I wanted to choose a good moment, a moment not too long. Yeah, you were saying about Gemini, Gemini, I'm sorry, I'm just a soft G. Which of your papers is that? Is that the paper that came from Kant to imply naturalism or is that another? I don't think I've got that one. I'll tell you what we must do at some point. You really should take time to make a bibliography of all your writings, because I really ought to have a, so I can study the order of development, at least have things in chronological order, because I think I still have one or two.

12:30 Missing gaps in my collection, especially of the papers in Italian, which might fill in perspective to that. I hope it doesn't sound like sort of preparation for a deeper... No, no, no, no, not at all, not at all, not at all. But you know, in 50 you can entitle to a fest trip that... Yes, I think I have to do it myself. I have no idea myself what the content is. No. Bill has now done it. He wasn't packed for now. He's doing it chronologically. All right. So I now have all of this. It's very useful to try and get them all bound up this way. It's like trawling through files. That would be a good work. And then I can provide a critical annotated commentary of the time that Lakotoshi's editors provided to him. Peruzzi cannot possibly mean what he says. Or, you know, this passage should be completely ignored. You know the footnotes to Prune Reputation. Incredible. It's a fascinating and charming, very provocative book, but I mean an outrageous piece of editorial, most outrageous pieces of editing, strip philosophy, taken off,

15:00 briefly dismantled and imposed their own interpretation. And then half the footnotes are the editors, you know, Zaha and Walar arguing amongst themselves. We cannot, you know, we cannot agree with each other, but we agree that it's entertaining to read them, but it's not a model for the editing of somebody's work. Having said that, of course, it's exactly the sort of thing that Lakatos himself would have done without. Anyway, I suppose we had better make a move. Do you want to knock on Bill's door, or shall we ring him? He might... I don't want to disturb him, but he's... They were together. They were having dinner together. Oh, no, they were having dinner together last night, so... Okay. Excuse me, Professor Becker, is it possible for me to speak to you for a moment? Yes, yes. Thank you. Yes, is it possible to speak in English? Yes, yes, of course. Yes, of course. Very nice of you. Yes, about a year ago, just after the Économat Sécurieux meeting, I sent you a little note, and I do appreciate how busy you must be, asking some of your publications, which I had great difficulty in obtaining in England, because... And I was wondering whether there was a chance of getting copies from them. Excellent.

17:30 Yes, but I was having some problems with the web page in downloading. I don't know what it was with my system. Some I could. Yes, well, I, yes, that's right, and I, the problem so many people have, I don't have. And I don't have J-Zip compatible. And some I was able to, about half, about half of them. So what is probably easiest is if I just send a letter, I send you a letter of the... Yes, but I am completing the papers in a PDF. Then of course there's no problem. But in the meantime, if you have hard copies or if you have, you know, some hard copies, if I send you... But I can't remember the paper. Well, I had a list, I simply pulled down a list from the website, of course, if I'm prepared, I don't want to take up your time, but no, and in fact the majority, perhaps around 50% I'm also open to download, or else I have copies already, or photocopies, but about half, especially some of the ones in... It would be very interesting. I would really like to be able to study. And you know how difficult it is in libraries now, isn't it? I am publishing the English translation of my first book in English. Bill, good to see you. We were just talking about that meeting, actually, at this very moment. Do you want to come and help? Yeah, sure. I did indeed, thank you very much. You shouldn't have gone to the bottom of writing, I was very tired of you. That's fine.

20:00 Sorry, didn't see you there. Ah, you had actually already been over here for a couple of weeks. That's good. So you've been talking Comand and Milano, aren't you? Yes, Alberta, that explains, Alberta was trying to get hold of you on the phone about ten days ago. We hadn't realised that you'd already left, just to find out what your plans were for after this meeting, and whether that, I guess it might be, oh well, that's... It's bad planning on our part. I don't remember why, maybe it's bad planning on my part. No, no, no, I'm sure it's, no, it's bad planning on my part. We should have tried to, well, unfortunately, I wasn't able to get down any earlier than yesterday because I was busy, um, working my flack out, unfortunately, from, oh, March 1st and then was, uh, I finally got rid of my charge of my calculated reactionaries in Paris. They were very reactionary, yeah. I think I might not get asked to do this one, because they were Catholics, but I couldn't contain myself any longer when we had to come around this place and review it.

22:30 Many years ago, they started a race on his father's car. I'm afraid I could not contain that. Yes, yes. Ah, yes, you mean the fact. I'm afraid I told them exactly what I thought of the... What were your thoughts on Calhoun? On if it's not too sweeping a question. One was called Wasimtun, Wasdol, and Dikontinua. I wish I could have gotten that. I hope somebody was making detailed notes. I always miss out, you know. But the last time we were together you were talking a little bit about unsolved problems on Cartesian. Do you know of any paper about Cartesian? Have you heard of one? In what? Yeah. Do you know of any articles or publications of which they are made, you know? There are many papers with that title.

25:00 Clearly, many people recognize the same concept and use it to find, let's say, a structure to them, a classification of something, you know, a mathematical investigation of this as a subject. In general, it's part of the topos theory. Hmm. The definition is clearly more general. And I can think of one. It's a really rough old paper. It came out in about 1975. I'm just teasing it because it was your own paper and Milano on the generalized, yeah. But apart since then, I can't think of the one with the title generalized logic and metric space. Oh, yeah, yeah, yeah. No, actually, yes. No, it's not. There are many interesting models of this theory, so I know absolutely nothing about them. And do the people who are working on model theory, people like McIntyre and Volkier, who are more and more interested now in the geometric sector, have they not done any work on this, trying to develop the...

27:30 Not that I know of. I've talked a few times with McIntyre. Yes, that was clear from the talk we gave at Paris. That's right. Yeah. It's a very good talk. Yeah. So I thought it was a very big challenge. There is a completely independent subject on the land economy, which, you know, taken in the best light, should be the subjective form of mathematics and physics categories. It's all been naturally absorbed into the funtorial framework, but in fact it is, as you said, objective. Of course, yeah, exactly. So one needs the presentations as well as the other ones. So I put my talk as a kind of challenge. I've developed this so much. I don't want to say... We have to go, I guess. Oh, okay. It's a little distance, I think, isn't it?

30:00 Oh, I see. That's what I wanted to ask. How are you? In general, pretty well. I had to cancel one of my classes in the lab. Oh. Well, no sign of it now. Then it got better, and then it got worse again. I guess today it's probably better. Good. Thank you for your attention. Mindless calculus, or some of the fissures of calculus. Plus calculus, yeah. But also, calculus is actually a kind of object. It's actually a very large one. It's very, very large. It's actually a very large thing. It's a kind of metal. So, again, it's a kind of object. Pressuring the geometry of connected components, functions, both left and right, and the development part of it, in other words, if you add pi zero to the magnitude, there clearly are radically different identities, satisfied by this and that, from the identity satisfied by the user's part, that is, its part, and again, sorry, which part, the...

32:30 You know, the same systems that just leave me out in other kind of views, don't they? Yeah. A system for which there exists a kind of view. Yeah, yeah. You don't have to do anything else. Yes, yes. Does that have to set any real restrictions? That the theory itself can be at the level of the abstract? Right. It is strange that nobody has worked on the kind of the generalization of people's study of that. Is that that? No, just the level of the... What about Gearcroft? The other talk today was on the continuum. Were you, and obviously did you have quite a lot to say about Dedican, where he took the turning board, complete lack of cohesion? I wish I could have been here for that. Perhaps we can persuade you to give it again for another audience. In fact, I am planning to do it one year from now. Okay, we'll definitely get there for that. I've not done this before, but it's a lot of time for any staff to see me. Again, I'm eating a little bit like this one.

35:00 People now are somehow worried about the Foundation. How about that? Right, well, we'll look forward. We'll certainly get to that, but I hope we might have a little bit between time. I'm overjoyed. I'm German. What, um... Well, I'd like to hear it on the way, as it were. Yes, I'd like to hear it. I really, really want to hear about this and get the chance. Well, alas, we've got off at 7 o'clock tomorrow morning. We're not going to have much time, but we'll do the best we can. Talking about, you know, people suddenly getting interested in foundations, when you get a chance, ask Anders about the meeting that they're having next year in Roskilde. I went out there this year, correspondingly to one that he was at last year. This was really just a historical meeting. Well, Mandus wasn't at it this year, but next year they have a meeting on the mathematics of the skills, a kind of historical, come philosophical discussion, it's organized by the Danish Society for the History and Philosophy of Mathematics, on the mathematics of modern physics and its creation, 1850. Well, I think it would be interesting to hear some of the... I want to talk specifically about the creation of the theory of Hilbert space and how did it so easily get accepted by almost everybody, with a handful of exceptions, and Clarke being one, as the framework for the term. Well, that seems to be what... Sorry, I've forgotten the name. Oh, of course, what is the speaker?

37:30 Thank you for your attention. I'll set that up on the photo, I think, on the right. Right, yes. Watch out again, Professor. I'm trying to think, there was somebody who tried to get a debate about going on the Reactionary website, the Bo website, about a couple of years ago, and I'm trying to remember who it was, Colin participated in it briefly, and there were some quite good contributions made, and then of course the panel was closed down. Reads your contributions with great interest to me. No, he was explaining to me that now all the evil things will stop because of the time travel. Oh, he's getting at it. The sword. Oh, you're... Espero! Which I don't believe at all. I was going to say espero, espero. You've been on the effort on this? I've been on it in the sense of, you know, following it quite closely. Certainly I've read many of your posters there. But I was mainly interested in the debate that they had right at the beginning about continuum. The one which, well, Robert Tregesa and Colin McClarty contributed to it, but the guy who really started it, I just can't think of his name.

40:00 Oh, Vaughan Tran, that's right, Vaughan Tran, yeah, yeah, yes, yes, yes, and he made, I thought, some very good and effective points, but he also got slammed on, you know, from a great height by our friends from the Ayn Rand Society. Before they started interacting with each other, they thought it was a good idea to have a conversation with each other on the subject of mathematics and physics. They were pretty much strangers to the true cause of mathematics because they were interacting with each other on the subject of mathematics. Well, of course. That's how they started. Well, let's have a passage of data. Okay. Will we have enough room? Because I don't mind walking at all. Correct it, Professor Minati. Professor, will we have enough room for all of us? Because if not, I'm quite happy to walk. It's easier to go for a walk, isn't it? It's easier to go for a walk, isn't it? It's easier to go for a walk, isn't it? Okay, fine, because I think it might be a bit full in the car. Okay, we'll see you up there, Bill. I'm going to go on foot, because I don't think there's going to be enough room in the car for everybody. No, no, no, no, no, no, you better not. I think it will be a courtesy for the speaker. Okay, I'll see you up there, don't worry. I take it we're going, I take we all know where we're going. Miss Cousy. My name is Lolly. You're not Lolly, you're Michael Wright. Very nice to meet you. De Londres. Grazie. Oh, gosh, I couldn't get Bill to tear himself away. Gosh, I wish I'd known he was talking about the continuum in Milano and Como. No, no, nor did I. I hadn't realized he'd been over here for about three weeks. That, of course, would be why you weren't able to get ahold of him in the States. Oh, well.

42:30 It was a very short walk. I know. I'm sorry, we should have come with us. I just felt a bit guilty about dragging you away from your host. Oh, well. Well, when you see Fatima and obviously Danilo and all that family, please give them a very, very warm embrace from us both. And I'm just very sorry that you have to go back tomorrow so early. You haven't, because you haven't met in a while. No, I haven't, I know, but it will leave me to Adelino. Adelino. Adelino. Adelino. Or he'll leave his mother. I'll leave the seat to say he's grown a lot in the last year.

45:00 Thank you for your attention. There's a lot of things that you can learn from this lecture, and I hope that you will enjoy it.

47:30 The kind of blend that seems to me to be a mishmash of cultural relativism and objectivism tempered by some awareness of the collected sources. Actually, we're somewhat apologetic about it. All concepts of left. The whole thing really means completely disjoint from communism, so, in fact, mainly... Anyway. I think, I think, I wouldn't disagree. I think that there could possibly be truth to what you said. He's a good apologetic, but apparently a lot of people don't identify as arrogant. Well, I think he isn't. It's interesting that in that whole list of people, philosophers, and all that, that's why I left the night, he put in some names, but it's just some names in the previous case. He said it's some names that's not assigned by the world's people. Well, you see, this is, again, as if the county dean of this policy wouldn't have to fall on your knees at the mention of Martin Luther King, therefore you don't want him to be in the ranks of the enemy, that's the part of it.

50:00 But what you said this morning in France is very interesting and very, very significant. The key is that the people responsible for the foam flicked the Simpson theory, which you are right, before they attacked Hathaway theory, they had already attacked McIntyre and the mathematics and the people working on the O-minimal structures. That was their first point of attack. It was quite interesting that that was their first point of attack before they moved on to a general barrage and scattering theory. And those first debates, those first waves of debates, before it became clear that they were not in fact out of the mix, did I think contain something quite useful and interesting. Thank you, wonderful, thank you. I just felt, I just had this feeling that he's just detachively detached from the reality and he thinks that all of these things are really a matter of being organized and scholarly and legal. You better, okay. We'll talk further. Okay. Well, I need to use, find out where the, where the, uh, facility is. Do I have the, um, uh, the BTO?

52:30 No, sir. Answer. Um...

1:10:00 Listen, the significance of, like, the singleton in, uh, the category of sets is a... Because the arity of that, you were saying that, I remember you were talking about the Morali 40 paradox, called paradox, and where it actually comes from, and a little bit about the peculiarity of the single. You have been down in Mexico, I think, with Steve Schaniel working on your book on set theory, and particularly on the fixed point. I mean, it is now. It wasn't before, it was just over there, but I can move now, I think. I don't think there's anybody next door, so... Did somebody come in? Ah, it's busy. No, actually, we're not quite... never mind. Did you finish that? What? On the book? No. No, no, no. You're very kind. I don't... Well, that's extremely kind of you. I'm sorry to disturb you, but I'm fine. No, it's... That's fine. I'm not going to drive it at Nelson. No, it's just I would love to see, you know, yes, nothing. I would love to see, really, a perspective on second theory and at length, really, you know, in a leisurely way, especially.

1:12:30 But, no, I was thinking a little bit more about it. Oh, so you did finish that? Oh, yeah. Well, yeah. Well, that's about it. Good! Well, uh... What do you like to have? What more? Do you actually have a company? Well, I get one, too. This is one of those courses. Was this the work on the text point there? That's right, that's the title. It's only very minor. Could I take that and copy it and return it to you? I can get it back to you on the special courier. I mean, this is your master copy. No, it's not the master at all. Oh, in that case that would be very cool. Oh, that'd be wonderful, I'd really love to. Oh, that's fantastic. Yes, please. And then let me send it straight back to you. Yeah, that's sent back to you, Federal Express. So, you know, you'll have it within a few or three days. Well, I apologize. There may be some minor improvements. Mathematical theory, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra.

1:15:00 This is not directly linked to what is today the investigation on the foundations, even if the vast production of other collaborators has elements that deserve to be considered. I've been here for a long time because I come here as an expert of these things, where I put the work that I have in my school, essentially at the center of the evolution of the mathematics method with the characteristics of mathematics. The characteristics... Classical mathematics on those identified by the elements of Euclid and theorized by Aristotle, at the end of the 19th century, emerged what is commonly called the modern conception. The characteristic, already illustrated in the previous lecture, of modern mathematics is that, in a schematic theory, primitive concepts are considered to have a meaning and axioms are not considered to be true or false, and they are the ones that define the concepts.

1:17:30 The next step is the Hilbert program, in which formalized axiomatics also make explicit the deductive apparatus of the theory of what can be called mathematics. Practically, introducing that predicate, there is ST. The last example is that there is PF. These are two examples that have been elaborated and that show that by unifying science in this sense, we recover a more powerful mathematics that goes beyond the classical one, for example, in the case of the infinitesimists, which have new demonstrative means. Now, in both cases, especially in the first one, the infinitesimals, you mentioned that historically, metaphorically, the infinitesimals were already in the state. My question is, these syntactic enlargements then end up having what you call... Mathematically, it is not understood, but it has some effects. You say that in mathematics, of course, you do not reduce the material need for inspiration, and it exists very well as it happens, but perhaps... This is in fact a practice that the Panoramas do, not zero, zero, they already have everything,

1:20:00 they already have everything. They already have everything, they already have everything. What does it mean? Power? Power. We need a new power.