Lecture

0:00 Thank you very much for your time. Thank you. Excuse me. Sorry, in French. The other day, I had the intention to give a phone call to talk to Alberto the next evening, but it's a problem that happens here. And during the last week, I was absent in Belgium. Yes, if it's possible, if it's not too late. It's okay. If it's not a good time, tell me, I'll do another... Is that okay? Okay. Thank you, Manuela. Thank you. You're very kind. Thank you. Hi, Alberto. How are you? Look, I'm sorry, have I chosen... Please tell me if this is not a good moment, because I kind of slightly detected in Manuela's voice that this might not be a good moment to ring. Yeah. I'm so sorry. No, I was just saying, are you sure this is a good time to ring for us to talk? Yeah, I haven't left it too late. I thought you might already be on the point of going to bed. Yeah, it's okay. Okay. I got your marvelous message with all the details of Pianeta Galileo the other day, and I'm sorry, I left a message with Manuela that I was going to call you back the other evening, but then there was a problem with my phone the following day, and then I had to go off to Belgium, to Brussels for this conference, where I was with Bill, in fact, for, this was last weekend, extremely interesting meeting. In Belgium, it was a meeting of the Belgian Royal Academy in honor of Francis Bosseur and Dominique Bourne, the two Belgian category theorists.

2:30 Well, chief theorists. Bill gave a very nice talk, an extremely interesting talk, on the categorical unity of geometry, algebra, and analysis, I think rather on the lines of his talks in Como in January, which I've now actually been able to listen to because Bob Walters very kindly sent me So I've now watched them all, including your fascinating, absolutely fascinating intervention in the discussion, which of course is by far and away the most important and interesting thing in the discussion. And Bill's response to it. So I've watched those with huge interest and enthusiasm. Anyway, I had an extremely good talk with Bill on Sunday, a week ago today, the day after the meeting. We went off to Mechelen, which is this little city near to Brussels, about 20, 30 kilometers, which Bill had always wanted to go to because it was the place where Charles, the Emperor Charles V was brought up. And we had an extremely interesting and very pleasant day. Obviously, you were very much in my thoughts and in Bill's. There's one very interesting thing which came up, which he told me, which is a sensational development. Two months ago, he went to talk to Jack Duskin. In Buffalo, Jack Duskin, who used to be the head of the mathematics department in Buffalo until his retirement about, I think about 10 years ago, sadly he had a stroke about three years ago, so he's now rather incapacitated and housebound, although he's still mentally very alert. And the reason that Bill wanted to, well, went to see him... It was partly because Colin and I had been, you know, sort of gently asking him for about two years now if he could find out from Jack Duskin if he still had this extraordinary document which Grotendieck had left with him in 1973 when he was visiting the department, which was a kind of memoir of the seminar, the last seminar that Grotendieck gave there, which, you know, I won't take up. It seems to be an extremely interesting record of some of Grotendieck's ideas at that time, especially about how to bypass logic in favour of dealing with

5:00 The structure in the sub-object classifier directly in terms of limits and co-limits and the classifying rings of theory so that you, as it were, subsume logic inside this larger framework. Very interesting ideas. Anyway, Bill had told me about this memoir, this kind of chart or diagram that Grotendieck had left with Jack Duskin and we asked him if he could retrieve it. So he went to see Jack Duskin to get this and he indeed found out that he does still have it. It's quite safe and, you know, Bill saw it. He found out something far more extraordinary which is Bill himself didn't know this because he wasn't at Buffalo in 1973 he only joined the department there some years later although and in fact in 1973 when Grotendieck was in Buffalo he was in Perugia he was of course was in Italy but He had heard all about this seminar that Grotendieck gave. What he hadn't appreciated was that Grotendieck was not there just for a short period to give one or two seminars, but was in fact there for 18 months. Yes, which I had not realized, nor had Bill in fact. Not only was he there for 18 months, but he gave, in addition to the seminar which we already knew about, He gave about 15 other seminars, and even more extraordinarily, he gave three courses, one of which lasted for four semesters on algebraic geometry, the second of which was on algebraic groups, which lasted for two semesters, and the last of which was on topos theory, which lasted for three semesters. They totaled 150. This is a hundred and fifty hours of lectures. Remember this was over a period of over a year and a half. They were all recorded. Yes, and the recordings have been sitting in Jack Duskin's Bureau for the last 35 years. They are now, they are now sitting, that was until two months ago, they are now sitting in Bill's garage. And he has listened, he has listened to some of them. I mean, obviously, you know, with all the other, you know, such a huge, a huge archive that he couldn't have possibly listened, but he's listened to a selection of a dozen or so. He tells me that the actual recording quality is fine, that, you know, Greg Dick's voice is completely audible, there's no technical problem as far as we can tell with what he's listened to, at least, even though the tapes are 35 years old. And he says that at least, you know, some of the material he's already listened to is...

7:30 I'm going to go over as soon as possible with a view to copying all of these. On two special machines which allow you to copy things very quickly at about 20 times the normal speed of a tape and then we're going to see about digitizing them and making them available as a set of probably about 10 DVDs to scholars, you yourself obviously will be high on the list, you know for a wide variety of Countries we've got a list of about 20 people we would like to send the DVDs to and hopefully we would be able to put them eventually but obviously they take up an awful lot of them an awful lot of of web space on the website of the Great Indy Circle that's if they're if their server if the server of the Great Indy Circle is big enough to to host such a large archive if not we might look at doing it on the ...which I'm hoping the University of Oxford may shortly make available to my archive, which is now formed into a trust with Penrose as the chair of it. So all these things gradually moving forward, but you can imagine it's a very exciting development. So I have all my toes and fingers crossed to hope that it will all go okay when I go out. But tell me all about your news. I was hoping to be able to come down and see you sometime. Very, very, very busy indeed. Yes, I'm surprised you haven't gone mad on the with the with the sheer volume of work. But oh, it's the Galileo anniversary, isn't it? Yeah, yeah, yeah. You know, next year is also the Grassman Bicentennial. Yeah, there's, there's, yeah, there's, there's, there's, there's, there's a very interesting conference in Potsdam, which Bill is speaking at, is telling me about in September. Of course, I was just thinking. And I can imagine.

10:00 I can imagine the administrative task of it is absolutely horrendous. I know it's difficult. You must be so... Tell me, when I spoke to Manuela the other day, I'm so sorry. Of course, I already knew about her mother and I'm so sorry to hear about that. I only hope it was a release from her suffering. But I understood from what she told me that Francesca is now quite a bit better and is in Paris at the moment. Is that right?

12:30 Do so, hope so. I mean, you know, I feel so close to you in all the suffering you've been through with her, and is she, I mean, is she in Paris for, I didn't quite understand what Manuela told me, was she in Paris as part of her treatment, or is she actually doing some kind of work there, or? Yeah, she is, she is. Oh, she's back with you at the moment, okay, right. She's working at the shop. That's good. Are you still going for that? Good, good, good, good. She's well enough to have resumed her studies in that direction. Well, please can you give her a huge and very warm hug from me? You know what I mean, at least metaphorically. Better still, even if you can embrace her issues, and say how much, you know, she has been in my thoughts, and not only mine, but I mean your other friends as well, her other friends, thought a great deal of her, and I do hope so, hope that she'll continue to make good progress. I'd love to come down and see you sometime. Did, have you been in touch at all with Bill since, since, since the January Como meeting? Because...

15:00 Because he was saying to me in Mechelen, and in fact also earlier when we spoke on the phone, that he hopes very much to spend some time with you in Italy in the, in next year. Yeah, he was telling me he was very much hoping to come down and see you. This is one thing where I hope I can be of some help. I'll be speaking to him, in fact, possibly later this evening or certainly tomorrow, because I need to talk to him about coming over to copy this incredible, great archive, which he told me about a week ago, and also to continue these very interesting discussions that I was having with him. He will certainly be in England in April, because in Cambridge at the beginning of April, I think in fact the 4th and 5th to be exact, is the peripatetic seminar on sheaves and logic, which is going to be in honor of Peter Johnston and Martin Hyland's 60th birthdays. And he's speaking at that. And then he, I think, I speak subject to correction, I'll check with him, but I spoke to John, to John Mabry last night. He's, John has got some funds for a small workshop in Bristol, which he and his PhD student Richard Pettigrew are organising.

17:30 Which I think if it was at all possible, I hope very much you might be able to get to, but the trouble is I don't know, with all the commitments you have, it might not be possible to travel at that time. He was thinking of having that the week after the meeting in Cambridge, so probably the second week of April. But in addition to that, Bill is talking, and I'll find out from him when I speak to him in a day or so, what the status is. Playing with the idea, because he had thought there was a conference about Grotendieck at the IHES in Paris in January, starting on the 12th of January, which he had hoped he was going to be invited to speak at, but I'm sorry to say, I think it's very bad, they have not invited him because they decided not to have a section on topos theory, which I think is absolutely disgraceful. But anyway, as a result of that, he's now thinking instead that he might go to Italy to, there's apparently a winter school about pre-Socratic philosophy, about the Iliatics, in, I'm not quite sure exactly where it is, except it's near to Elea itself, so I assume somewhere on the gulf of, you know, somewhere around the Sorrento, you know, multi-coast. Somewhere down near there, near to Naples, well not far from Naples. I think the exact dates are, in fact I was just looking at it a moment ago, if you give me a second I can actually tell you what it is. It's about, I think it starts on the 9th of January. Just give me a second, I can tell you exactly when it is. Jonathan Barnes is giving the main talks, and Bill has always been very interested in finding out more about the Presocratics. He's thinking of going. I don't know if he's taken a definite decision yet. Just hang on a second, I can tell you exactly when the date is.

20:00 Yeah, it's actually for two weeks starting on the 8th of January, the 8th of January, and so presumably finishing on the 24th. Yeah, it's a 15-day kind of course, intensive course. So, I'm not sure at this point whether he is definitely planning to go or not, or whether he would actually be able to stay for the whole thing, but if so, then obviously he would be in Italy in January as well, the last part of January. But obviously, you know, you can coordinate something. I will speak to him in a day or so. I might even speak to him later this evening if I'm not too tired. Otherwise, I'll give him a call tomorrow. I'll try and find out if he's taken a decision, definitely, whether to come to the Eliatics thing or not. And then, of course, you know, you can get in touch with him. That sounds as if it might work very well. Well, I'm hoping, I'm really hoping that I am so glad to learn that, you know, fairly good news about Francesco. I'm very, very relieved to hear that. I was so worried when I spoke to you the last time. It did sound grim. We've had so many setbacks in the past, I realize, yes, it's very difficult to be... Okay, I'm surviving. I'm, you know, just surviving by my fingernails. Obviously, you know, the huge events in the world in the last month that we could talk about those. Yeah, I'm just about surviving. I may, I don't know, I'm just surviving. It's ticking over. The archive has acquired some very, very interesting new material, not least from these discussions with Bill and in other things in the last couple of months, and I'm still hoping that we may get some assistance for funding from the German Society for the History of Mathematics, which is basically, yeah, I'm in negotiation with them at the moment and also one or two other possible benefactors, so I keep my fingers crossed.

22:30 But I survive. It would be, of course, as I've said before, and I know it's very difficult for you at the moment, but if you and Manuel ever just want to take three or four days or a week or however long just to come and escape and, you know, come to Fougere for a few days, I have a spare bedroom and you will be so welcome. I can't tell you how welcome. It would be nice, yes. If not, as I say, if all goes well, I hope I might be able to come down and see you for a few days. Fingers crossed. For me too. For me too, very much so. I think I preserve probably the happiest or some of the very happiest memories of my life. So, let's hope they can be renewed soon. Yeah, yeah, yeah. And I think we can keep... Just wondering that. Let me find out what Bill's plans are. The end of January might very well be a good time for me as well. I think my finances might be a little bit on a better keel by then. Although it's very difficult, I have managed to sell the land behind my house here, which has given me a little bit of a breathing space, and I may have some additional funding, as I say, coming in from the German Historical Society before long. I'm hoping so. Anyway, we survived. Of course, at the moment, with all these incredible upheavals in the world and a complete meltdown of the world banking system, who knows where any of us will be in three months or six months' time.

25:00 Hang on, I don't know what's happened. Somebody's trying to ring. Oh, I think there might be somebody trying to ring me on the other line. Okay, absolutely. Take care. See you soon. Cheers. Look after yourself. Hello, is there somebody else there? Hi, is that Fatima? Hi Fatima, it's Mike. How are you? I just got 11 o'clock, so not super late. That's okay. No, it's lovely to hear your voice though. Everything okay? Yeah, see you soon. Very soon, I hope. Hello, Bill. Hi. Yeah, I'm sorry. It was very lengthy, wasn't it? Yes, I'm sorry. There was a lot there. The most relevant bit, though, which was why I was ringing you up, is about the recordings. I just had a lesson, well, earlier today, from my technical guy on how to, yeah, a very, very helpful one, showing me how to convert Audio tape and audio, standard audio cassette to digital. It looks as if you're, yeah, yes, it can be done directly. The only problem is, hmm? Yes, of course you have to do it in the computer and then you can burn DVDs from the computer of course just like you would with anything you know that you've got inside the computer it can be done he showed me how to do it all it requires is is really just a standard standard connecting lead but it seems that your people in Buffalo are correct it can only be done in real time so It looks like we would need quite a bit of time to complete it and the key to it I think would probably be, as I was saying in the message, if there's anywhere at all that we can arrange to commandeer three or four PCs on campus in the computer sciences lab of Buffalo University and borrow a couple of graduate students. Do you think there's any chance of that?

27:30 It certainly doesn't involve any particular specialist knowledge. It would be quite easy for me to show them how to do it if they didn't already know how to convert an audio cassette to a... Well, I don't claim any expertise. As I say, Benoit showed me how to do it today, and it's really very straightforward, and it's something which any graduate student would pick up probably quicker than, because they tend to be much more adaptable than us old guys, when they have anything to do with computers, so that shouldn't be a problem. It's just a question of if we could commandeer the PCs. If we had to do it on just one PC, then obviously it is going to be a pretty time-consuming business. We obviously need more than one recorder but that's very easy I mean I've got a dozen audio cassette recorders here I can bring those with me I've got I'm literally looking at three just on this table in front of me that would be no problem at all they they're they're pretty good yes I don't think there would be a problem the the tapes themselves are presumably one hour tapes are they Because they would have all been recorded at ordinary speed because I know back in 1972-1973 there were no variable speed audio recorders, theirs only came in a few years later, so it should be, what I was thinking might be the quickest way of doing it, as I mentioned briefly at the beginning of that paragraph, might be if I was to make a complete set of copies. And so on and on and on and on and on and on and on and on.

30:00 And then I could bring the copies over here and do the digitization because I can get access to computers here in France pretty straightforwardly or in the UK. And of course, it could be done over the space of two or three weeks. That might, I'm thinking, be the quickest way of doing it. But it's probably best to ask Jack what he considers. The thing is, you mentioned that Jack might be able to help out. I certainly wouldn't I certainly wouldn't dream of asking any payment for it but we would need to because the trouble is I I don't have a tape duplicator which can operate at that speed I have a tape duplicator but it's not anything like such good quality and it only operates at six times the normal speed the ideal thing Which, if there's any way at all that we could acquire one of those, and as I said, I have no idea what kind of funds Jack can make available, that would certainly speed the task, because one of those would enable you, I think you said there was something like 125 tapes altogether, was that right? Right, well, it should be possible to copy the whole 125 in a single day on that machine. So if we made two sets, you're looking at two, let's say three days to be on the safe side. No, we'd only need the one machine and I would only need to stay for two or three days because I obviously wouldn't want to clutter up your home. Great as it would be to come and see you. Or I could stay at a cheap hotel, I'm sure there are such in Buffalo, and we could do the whole thing in about two or three days. Then I, obviously they're pretty bulky, but they should go in a large hold-all or suitcase. And then I could take them back here and do the digitization over here. I'm thinking that that might be the most economic way of doing it. If we do it directly from the tapes that you have.

32:30 In Buffalo, then, well, I'm going to have to stay for at least, I can't see how we could do it in less than a couple of weeks, unless, as I say, we could get, we had three PCs and a couple of guys to help, maybe, say, 50, probably still looking at about 40, still looking at a week, still looking at at least a week's work, I would think. And that's if it's split up between two or three people. Not absolutely certain yet, but Benoit thinks not, although he is doing some more digging. He's going to get back to me probably tomorrow or Tuesday to let me know. This is the chap in France that I use as my technical advisor in these matters. But this appears to confirm what you told me the people in Buffalo said to you. Yeah, it's consistent with what they said to you, so it does seem, I think probably it is the case that you can't digitize them other than in real time, but you can certainly make copies of them to another audio cassette in 20 times the speed, that's not the problem at all. You know it's obviously to be considered but I think that the most expeditious way of doing it probably would be to make copies of the whole thing onto audio cassettes and that way of course you could keep a complete set of them. Jack could have the original spec and then I could bring them over here and digitize everything and with a bit of luck we could get the digitization done within a month or even possibly a little less time than that depending on you know how many PCs I can... ...commandeer over here and how many people I can get to help, but I think for something as important as this it shouldn't be difficult. Do keep a watchful eye on them, won't you? Of course I've got all of yours sitting here since 1989, which of course I ought to be doing the same thing with, but I haven't of course yet got through completely cataloguing those. By the way, they are labeled, these tapes, are they, to tell you which, you know, which are the algebraic groups, lectures, and which are the, not perfectly, but we wouldn't have to listen to each one in real time in order to be able to label it and say what it contains. We would have at least a rough idea of, you know, if it's the algebraic geometry lectures, they're labeled, you know, no, no, that's, that's okay. Well, I think in that case, the sooner I can come and do it, the better.

35:00 Yeah, have a quiet word with Jack and tell him what we're up to. Do you think he might be willing, I mean, I hate having to ask, but of course I haven't unfortunately at the moment got the resources to buy one of these duplicators and then of course we would need, well we'd need at least 125 and if you would have a set of copies, which I certainly would want you to have as well, we'd be looking at, you know, 250 blank cassettes. Which you can buy in bulk, but still, it's still going to be probably about $300 or $400. Are you talking about the size of the actual cassettes? Yeah. Well, of course, I haven't seen them, but if they're standard audio... Cassettes 1972-73 vintage, they should be the same size as those that you can still buy and play today on any standard audio cassette recorder. If they were micro cassettes, well first of all the micro cassettes did not come in until later, they weren't around in 1972-73, those only came in in the late 70s. They're not the tiny ones like you used to have in telephone answering machines, they're the standard sized ones that you just use in an ordinary tape recorder, right? Yeah, yeah, sure, sure. I've got loads of recorders with which we could play those just to test the quality, that's not the problem at all. And those are not bulky, I mean I can bring half a dozen of those too. It might be worth, while I'm over there, just seeing if we couldn't digitize some of them directly into a PC, just to test out how long it takes. I'll keep plugging away at Benoit to see if he can find out any way of doing that, speeding it up, not doing it in real time.

37:30 But of course we don't want to do anything to compromise the quality of the recordings. Oh, dear, I'm sorry, you're still better now, I hope. Oh, good, good. Well, get plenty of rest. And then perhaps while I'm over there, of course, I'd absolutely love to go and speak to Jack. I realise he's a bit frail because of his stroke, so I would love to see, well, obviously, first of all, to thank him personally for having... You know, brought these into the world, as it were, so to speak. And secondly, well, obviously, it would be very nice to meet him. And, of course, I'm also hoping that I can make a copy of this set of notes of Grotendieck's last seminar that you told me about. I'll touch base with you sometime in the next week, but as I say, the sooner the better as far as I'm concerned. A couple of things. I spoke to John Mayberry earlier today, who asked me particularly to give you his very, very best wishes, and I believe he's been in touch with you directly about this project that he has of a mini workshop of a conference in Bristol in April, which of course would be perfect for you because you could come on directly after the workshop in Cambridge for Peter and Martin Hyland's 60th birthday. And the other person I spoke to just an hour ago, in fact he rang me up, was Alberto, who was in very good form and who just finished putting the final touches to this huge pianeta Galileo at Tuscany University's History and Philosophy of Science program that he's now responsible for, and he was telling me that He has some funding which would allow, I think you already know about it, which would allow you to spend at least a week in Florence and he was wondering when would be a convenient time but he mentioned to me that the ideal time for him, and I know he'll be in touch with you directly, the ideal time for him would be around the last week of January.

40:00 So I'm just wondering if you're still thinking about going to the, to that, um, Eliatix thing. No? Well, I hope Karin didn't put you off. I hope, I hope our strange ultra-leftist lady Karin didn't put you off. No, no. Well, if you did, if you did decide to come, then obviously it would blend very, it would mesh very nicely with, um, with Alberto's, uh... Kind of window of opportunity. Anyway, I'll let you get some rest, Bill. Thanks again for an absolutely marvelous day in Mechelen. Yeah, well, me too. Same guy. Hieronymus is just the Latin spelling of Jerome. Jerome and Hieronymus are just the same name. Same, same... Ah, that explains it. No, no. No. They are one and the same. Jerome is just the... I'm not sure what language it would be translated into, but Hieronymus is definitely the Latin version of Jerome. I know that for sure because of Hieronymus Bosch, you know, the famous mystical painter. He was also Jerome, because if you go to the Prado where his paintings are, you see he's actually called Jerome Bosch in the catalogue there. It's definitely the same name.

42:30 That's very interesting because all the references I dug out said that his tutor was, throughout that period, was this guy Adrian, Adrian of Utrecht, who later became Pope, absolutely one of the main architects of the Spanish Inquisition, that wonderful expression of humanism and Platonism, but it's clear that he and Boosleden, well it's my speculation that they must have been very close because they're both humanists and Platonists and they're living almost in... Well, living virtually next door to each other. I mean, they're in virtually neighbouring streets in Mecklen throughout this period, so they must have had a great deal of influence on each other. But I hadn't come across an explicit reference to Boos Leyden having ever actually tutored Charles, but that doesn't mean he didn't. Interesting. I'll do some more digging anyway. I completely agree. Yes. Well, I think there's probably been quite a major industry in keeping it, keeping it on, you know, keeping it unstraightened out. I think so, yes. Yeah, me too. Absolutely. I see that both Adrian and Collett are cited with, quote, are mentioned as citing Piccolo, de la Marandola and Ficino. In their own writing, so there's clearly an entire nexus there of... Yes, I did pick up on that. I think so, yes. Well, they were selling indulgences, so why not the Platonist equivalent of selling diplomas as well? Anyway, as I say, I'll let you get some rest. I'll touch base with you again later in the week about coming over to do the copying.

45:00 Jack is up to it, and it's not going to be too much of a strain on him. If you can sort of float the idea, as I said, I hate to ask for funds, but we would have to acquire one of those machines and about 300 tapes, so you're probably looking at $1,000 frankly, just straight off of that. I have also found that there is an airfare which, from now until February, which allows me to go from, not directly from France, but from Dublin to Boston for only 199, from Dublin, no in Ireland, but you can get from, you can get from, from where I am, from Rennes to Dublin for only 40 euros, but from Dublin to Boston is only 199 euros. Which is pretty good. Yeah, that's one way of course, but that's still only $400 round-trip. I don't think you can get anything cheaper than that, you know, so I'm obviously doing everything I can to keep the budget as low as possible. I'll touch base with you later in the week. Okay Bill, lots of love to Fatima and speak to you soon. Thanks. Take care. Take care. See you. Cheers. This is reflected on the 21st of October 2008, reflecting on the constructed objectivity of mathematics and the cognitive subject by Giuseppe Longo. The article by Giuseppe Longo, first of all, the possibility that one might see limits and co-limits It intends that we should see them directly as subsuming all that can be said, and generalize in subsuming and deepening all that can be said at the level of the structure in sub-objects, the structure in a sub-object classified as the scene and the whole study of relations, the structure of relations within the

47:30 Framework of sub-object classifiers as conceived as the subject matter, as constituting the subject matter of what he terms an erosense logic that's being subsumed within the study as falling into place within the general study of limits and co-limits. The program for bypassing logic as itself is a direct expression of the structural invariance, the kind of structural invariance, the constitution and The recognition and constitution of which is, as it were, the subject of study, as it were, occupies our attention in respectively the epistemological and ontological poles of this notion, of this conception, of this, as I said, entwined naturalist conception of mathematics and of the way that it fits within our conception of the world and of the sources of structure. ...of unity of structure in the world, vis-a-vis the role, particularly, of the knowing subject of the cognitive foundations of mathematics to stress on, and the role of cognition, the role of our cognitive apparatus in the, as it were, constitution. As well as the itself being constituted by in the constitution as well as as it were in the disclosure of these structural invariants. The second point is that the notion of sketch by Peruzzi is discussed in Peruzzi's remarks on the schema essay. Of course the schema essay is very very relevant and connects in all sorts of ways with the perspective being elaborated, explored by Longo in this essay. On the constructed objectivity of mathematics and the cognitive subject, the role for the notion of set, the role in the very constitution of the notion of set as a structural invariant, the recognition of it, whether it's disclosure or the recognition of the notion of set as such a powerful in-rem structural universal, structural invariant.

50:00 The way in which it is constituted and the way in which it comes to be recognized as a structural invariant as reconciled with and in the light of Peruzzi's claim that every ingredient of cognition is the transpose of spatial structuration, as connected with and arising from the recognition of closed and bounded components of a path-disconnected space, But closed and bounded components of a path-disconnected space would just be a kind of figure, would be one kind of figure, having the study of which as a figure, a generating figure for a space in general, precisely because it is a path-disconnected space, would involve the recognition of the incidence relations. Of the figures and the incidence relations. In this case, of course, it would be precisely the case of a space generating figure for a space which is entirely disconnected, which is, as it were, is, when we look at the relationship between the components function and the points function, when we look at the logical meaning of the Cantor construction, would be completely without cohesion or would be just determined by its points. A space which will just be no determined by or live on its points, its bare points. The case, the limiting case, of either complete cohesion, there being no obstruction to the existence of inverses of maps, or further, the absence of any obstruction to the existence of inverses of maps. Being an aspect or expression of perhaps connected with that remark that he made in the Cambridge lectures in 1989 in the Philosophia Mathematica written up version about there being a figure becoming, being capable of becoming any other, he actually says one point being able to become any other, but there's one figure with the associated incidence relations capable of becoming any other in a completely.

52:30 Here, of course, the whole point about the notion of map is connected with the underlying intuition of the geometrical meaning of constructions, without any attention to one particular One point being capable of becoming any other. One figure being capable of becoming any other in a completely arbitrary way, without any attention to the manner in which the motion is parametrized. See the remarks in the 1989 lecture written up in 1994 for Philosophia Mathematica, where the point is sets as a limiting case of the notion of figure, and how the adequacy-coadequacy of points condition as a general. As a specific condition delimited within the general understanding, as a special case within our general understanding of the adequacy and co-adequacy of figures as generators of space. This is understanding of the way in which we see categories of sets as a special instance of a special case of categories of space in general. With the relationship between the notion of space and figure as the fundamental guiding idea, there is the generating notion, the generating concept, this defining structure for a notion of space in general, the defining, and is the rise to and to be analyzed in terms of adequacy and co-adequacy of figure types in the, here we look at incidence relations and figures and subcategories, the relationship of categories and subcategories too. Categories, and potentially adequate and co-adequate, adequate in some situations, co-adequate sub-categories, how the position of points as adequate and co-adequate is as it were seen in this setting as falling into place as a special case, so the geometric meaning of the constructions in set theory as it's kind of involving this absence of all cohesion, and thus giving rise to, for instance, the arithmetization of the continuum a la Dedekind is seen as

55:00 The definitive place within this picture is seen as, I suppose, as it were, how this in turn connects with that case of the atomization, the discretization of the continuum, the arbitrary discretization of the continuum, quote Peruzzi and study Peruzzi's articles, becomes a special case, as it were, of the constitution via structural invariance recognition, articulation of structural invariance. But here this seems to be going naturally with a conception of structural invariant of in rem structural universal. Which stresses precisely the structural universal aspect goes with the geometric meaning of category theory. The geometric meaning of category theory, what confers ultimate unity of structure, is a structural universal, though, of course, I claim it's an in-reb structural universal. Direction of understanding, as it were, of direction of fit within the categorical structure of the ontology overall, within the systematic interrelationship of... There are a number of categories in the metaphysical sense involved in the integration of our conceptual world as a metaphysical unity that's structuring as involving also, of course, the recognition of a category without the being of which is, again, it's not granted the being of anything further in the constitution of our conceptual world as a metaphysical unity that's involved in, as is in the background, in the case of, that's guided by, from categorical constructions, by the geometric meaning of constructions, by structural universals. As against particulars, C. Marquis's stress on the meaning of category-theoretical structures as against, and the view of mathematical entities, which of course it's akin to, especially in the in rem structural universals case, in the case where the stress is on in rem structural universals. It is akin to the sort of role of structural invariance, the status and role of structural invariance within, and the possible, of course, plasticity in our conception of the direction of fit of structure within further structure overall, given the role of these structural invariants and the plasticity of our conception of them and the way that they fit together within the system and hang of mathematics.

57:30 The reintegration of our conceptual, the reintegration of our mathematical knowledge, the reintegration, re-conceptualization, re-conception of our conceptualization, re-mathematical conceptualization, the re-conceptualization of our mathematical understanding of our, of our, of these structural invariants and their interrelationships that one says has the case of Perutz's, sort of, of, of, of L'Orpheus project. For the rethinking of the relationship between our notions of number, space, and logic, and indeed that what has in the rethinking of the status of logical concepts in the bypassing logic project of Grotendieck, this in turn is connecting with the claim that's made in the lecture of MacLean about the geometrical origin or the geometrical roots of logical constructions, the geometrical sources of logical constructions. And how that connects also with the possible metaphysical meaning, meaning for the metaphysics, mathematics, and isis for the way in which we see mathematics as grounded in this kind of structuring activity of the cognitive subject in this entwined naturalist viewpoint or the relationship between the ontological and epistemological aspects of... See the claim about every ingredient of cognition being the transpose of spatial structuration, how this connects with this revival of neutral monism viewpoint as reconciling the epistemic and ontological poles as touched on in Peruzzi, how that view goes with a kind of top-down view of structural universals, the emphasis on the universal, and hence on the direction of fit of structure within further structure as one, not only... Involving bringing in this entwined naturalist conception of the relationship between the epistemic and the ontological aspect of the structural invariance, but also Lucidice, for instance, the way in which this is coming back to the point about the topological meaning of the cataclysmic structure that Lovier touched on in his remarks on the eve of our Florence conference.

1:00:00 In November 2003, I think, in the MacArthur, in the restaurant on the evening of the 17th of November 2003, and the point about, yes, the absence of the case of the absence of all cohesion, the case where we have, yes, we have in the case of the category of sets, the condition of the adequacy and category of points is realized as a good guess against the background of the role of. Figures and their incidence relations as generating figures of space and the role of categories of space and the role of figures vis-a-vis categories of space within L'Orville's overall understanding of the particular relationship between the geometric and algebraic aspect of operations, between geometric notions and algebraic operations, arities, etc. In this context, we can see the case where, as it were, the limiting case of where the space lives on its points as, in terms of, for instance, the direction of fit of further structure within further structure, quay, structural invariance, quay candidates, to be structural invariance, quay structures, quay universals, candidates for being inner M, structural universals, but here, of course, the very notion of universals. What confers unity of structure on physics overall is itself, as it were, plastic and adjustable, so it relates to the conception of the world as a metaphysical unity and a conception of that, the being of which is generous. Within it, of a kind which goes with some versions of monism, with various, I'm claiming this, in this, both this inner and structural universals, quite possibly is connecting with notions of process, with an ontology of the hollow movement type, or possibly as an ontology of the GMD type, as explaining cardinality without cardinality, or…

1:02:30 Numerosity, plurality. Numerosity without plurality. See the kinds of understanding of the particular in the universal. What would happen starting from constructions of the kind? Which Marquis says category theory naturally foregrounds or regards as fundamental, which are essentially an analysis of mathematical entities and concepts in terms of universals. Let us here, which goes beyond Marquis, in rem structural universals, the kind that might in turn be seen as falling into place within or connecting with and understanding of which could be reconciled with a... A universal ontology, a proposed ontology of either of the hollow movement or processual type that would be the ultimate priority of categories of process or of a monistic type, see the point about GMD, rather than having to start from abstract particulars, as in set theory, and of course starting from abstract particulars suggests that the unification of the ontology of our mathematics of the kind that set theory is supposed to provide. Mathematics is somehow the project of which one should aim. So, as it were, the one has a metaphysical, a realist, a realist ontology for mathematics as a whole, into which, within, into which, as it were, our understanding of interrelationship of structures in the physical world, of structural invariance in the physical world then has, as it were, in turn to be shoehorned or seen as a fragment, as a kind of ultimate. What confers the unity of structure and what exists in any way, whatever is, as in, for instance, the multiverse and Tegmark conceptions, one which, and see the remarks in Hilton on platonic atomism in Russell, one of an atomistic sort, one which rests on, as it were, underground on top.

1:05:00 Which Giuseppe Longo draws in the course of that essay on the constructed objectivity of mathematics and the cognitive subject between conceptual and structural invariance and their blending, as it were, their kind of intellectual interplay of conceptual and structural invariance as what are candidates for structural invariance candidates for So, what is the structure of systems in the world and what confers unity of structure and actually provides the very ontological basis in the pattern of direction of fit of structure within further structure, quake, and other structural systems in the world with respect to some overall overarching unifying structure on the one hand. The point is that as regards this antithesis or this tension between conceptual and structural invariance, it's very interesting to look at the Groton-Deep project for bypassing logic and the subsumption of the kind of purportedly, of course, for some ontologically interpreted logics view of The structure of logic, the structure of relations, the structure that's represented in at the level of the structure in the sub-object classifier, the algebraic structure in the sub-object classifier, in the sub-object classifier, as subsumed within, as falling into place within the further structure of limits and co-limits as what the subsumption of one system of

1:07:30 One system of conceptual invariance within another, or one system of structural invariance within another, or one system of invariance within another, where the interplay between the conceptual and the aspects of those invariance is itself something evolving and plastic and subject to the interplay, and between them is itself subject to evolution. Our understanding of the interplay between them within the entire naturalist conception of mathematics and indeed of logic, the position of narrow sense logic with respect to mathematics, is the sort of naturalistic perspective on logic, is itself subject to adjustment, our understanding of it evolves and our understanding of the interplay is... Development and evolution is something plastic.