Mathematical conversations post-ENS seminar: FW Lawvere

0:00 It's about the components function for the fourth or fifth time, but it's with somebody like me who needs that kind of emulsion technique to go over the same ground several times before I really get it all. And I certainly understood about collocation, but I haven't really grasped before. I'd only heard you talk about that once before with Anders. And that was a conversation we're having with him rather than an exposition for the audience, so I think we'll stop there and I'd like to, of course, do a few lectures, do a few lectures, do a little bit more leisurely. Maybe if I ever manage to get to London I could do that. Well, absolutely! Well, you'll have a leisurely... Thank you very much for your time, and I look forward to seeing you again soon. How many people were packed in the room in the last night? Well, to be honest, no, I didn't, but I would think it was around 50. From the time there were four, I would say that there were certainly four people. Oh, you said you had at least 50 people. I want to get back to the 21st century, to the development of mathematics, and loads of questions I was going to ask you about two of these concepts, but that won't be the case. I promise to give you... I'm sure you have a little time, I know you have to be back to talk to Charles Alumi this evening.

2:30 Well, this is now the Senate, indeed. This is now the Palace of Luxembourg, and it is now the official seat of the French Senate since 1945, and the Luxembourg Garden, and this was in fact one of the main execution sites after the commune was crushed, where the prisoners were brought and mass executed, so it's the largest single mass execution site. There's a place on the north side of the same, an inn in Saint-Germain-le-Chez, which is an incredible memorial. That was one of the largest wholesale massacres that took place. And that was where they used to have, that was where the Communist Party and, you know, all French Democrats used to have the big memorial ceremonies every year, almost every day of May, in memory of the Communist Party. But this was the second largest massacres that took place here. In fact, most of them were actually shot down along that terrace there, and their bodies were buried, they were mass graves, in fact, there are graves where the communards were buried underneath the terrace to this day, under the street, yes, indeed, you go to the Fogel-Saint-Honoré, some of the historians read those as many times now, bodies buried under the Fogel-Saint-Honoré. And there had also been hundreds killed in court for fighting. But the ones you're actually executing on that, on the figures that Pierre and his regime gave of themselves, they have litigated between 30,000 and 40,000 of the best-selling figures out there. And I don't think you can say you can count those as being killed in the pressure of the contract. I would like to know more about it. And, strangely, there isn't even a memorial to them here in the Luxembourg Archive. There is a memorial, I think, now, there it is, but they leave this with the, of course, the left-hand side of the Ecole Polytechnique, of course, this area was built in the early 1848 revolution.

5:00 This is the Ecole Polytechnique. During the July days, when they wrote the case for the Royal Royal Monarchy, they were kind of barricaded. A lot of them went into the second term of the story. This was the second round of the revolutionary conference, which Louis-Philippe Stanton had infiltrated a number of agents into the ranks of the revolutionaries, particularly here on the left-back, which in general are evil and significant. Each portal can be joined by other people, apparently sympathisers, who they didn't know, and they couldn't always be sure if they were pleased to come in. And a lot of these people are obviously more than one. So they would tell the proletarians when they came to join them on a varied day, they would say, oh, I am from the Ecole Polytechnique, to explain their problems. They were, in fact, police spies, and so, of course, the students of the École Polytechnique started to get a bad name amongst the revolutionaries. So the ones who were the sympathizers, who were, in fact, the revolutionaries themselves, went down onto the barricades. And whenever one of these police spies appeared, that he was a student of the École Polytechnique, they would say to him, I think it was the mean value theorem. Take the mean value theorem. And of course, it was pretty soon who was assumed that they're called Polytechnic and who was at least Spar. It's a very effective trick. I don't know whether the same thing happened under the same name. Well, at last we didn't have time. But that is attested to. In fact, we've got a little book about the history of the Ecole Polytechnique. Which I read that story. I'm pretty sure it was the mean value theorem that they wanted to say. Maybe one or two of them had their own preferred test question.

7:30 Integrals. Integrals, yes. But anyway, it was a pretty surefire test if we'd have a piece five. No disputing it. Not only for the reason I'm now, I suppose. It is a beautiful morning, isn't it? Yeah. I brought my camera along because I thought we should get some good pictures. Ah, yeah, the pear-shaped, no, the pear-shaped kid. Oh, yeah, the man in the Iron Man. Oh, yes, that thing, yes, the recent remake with Leonardo DiCaprio. Yeah, yeah, yeah, anyway. It's lovely film, but... Oh, why they chose the name. Yeah, yeah. No, no, no, go on, go on, go on. Well, anyway... I think you might have suggested part of the household, you know, the two brothers, twin brothers. That's right, yes, in the story, yes. There's the ultra-reactionary one, who was then replaced by the nice, sweet Teddy Boy Juan, the reformist, the gentle paternalist, as I say, the paternalist. And the names are Louis and Philippe. Louis and Philippe. Lirius, the archer, Hector, is the leader of the enlightened paternalists. It's quite difficult, isn't it? All that happens is that if you manage to be happy, all of these circles, there'll be lots of nice rich men. Nice rich people. Just let's make the rich people nice, and the world will be happy. The kind of Dickens, George Orwell solutions to it. I thought that the name could not have been lost on the public at the time. No. No, that's interesting. I had forgotten that story. I listened to you. No, you did indeed, but I'd forgotten. It's just a little anecdote. No, no, it's nice. No, but I had forgotten that you'd told me about Dumas being a member of Louis Philippe's... I'm always listening to that.

10:00 Well, there certainly are a lot of cultures. I wouldn't quite go as far as that. I would say that I am convinced that there are a lot of cultural conspiracies. I don't want to say culture is a conspiracy. But that is interesting. Anyway, that was an earlier revolution. That was the two phases of the 1848 revolution, which of course was also the trade from the outside, from the inside, but partly because of the development of the production in France at that stage was still not... Something like as highly developed as it was in England, or in the United States, I mean in the main Catholic Center, and such leadership that Witten's class had was largely imprisonment for them, and what there was available was not guided by any body of deep theory. I mean, Blanqui was by far the most influential theory I could think of. It was a radical work within the power of the most important way. But on top of that, they sought to keep control of the revolution under the slogan of 1793, under the slogan of the Freedon Revolution, the Great Revolution. And they in turn, and Atiyah, who was said to be the masseter of the communists, the liquidator of the communists, was also a communist in 1868, so he'd been around a long time, he was quite an old man in 1871. He'd actually been around in 1830, and he'd been one of the chief and most corrupt figures in various parliamentary regimes under the Second Republic. And then he had been in a different position under Napoleon III, rather than the way that he had always been in a composition under Diderot. And he was a great historian. He wrote voluminous about history. He really was a very, very diligent and learned, but essentially a very shallow sort of law. Some people even compare him to Macaulay in the historical sense of it. He was a great seeker after Felton and Clay, even by the standards of the French bourgeoisie of his time.

12:30 But he was one of the people who helped to support the 1848 revolution. They split into so many factions, there were three separate republics and parties, And then, of course, Louis-Napoleon arrived on the scene, managed by Dinter's, actually, it has to be said, even allowing for the greed and short-sightedness of his opponents, I mean, quite, quite skillful. You could actually stand in the sun. Yeah, we could. What a good idea. I was just going to make a picture of you. I was going to make a picture of you outside, but then I got burbling. Yeah, do you want to stand in the sun? That's very splendid. Excellent. Let's stand in the sun. You know, in Germany, a grass mountain is really... Well, you know, I don't think it's radical. It's a reformist. All these things in between are still heading home, so I think we're already going beyond all of that. But going beyond the direction of kind of euclid socialism in the first place.

15:00 In the first place, after that, we lost hope. And then by this class, which is in the fair towards the one level or another, attached to preservation of the regime under Bob C. Cranston and Paul Handel. Is that too sweeping a generalization? Yes, I think it is too sweeping a generalization. On the other hand, I think like many sweeping generalizations, it contains a very important understanding of the general trend, the deep general historical trend. I would say it is. I would say it is because I think that the phenomenon was much more complex and nuanced in different, you know, in different settings and I think that will be much more true of the German. I think on a world scale it's a, yeah I mean I think that there's a general, the general thrust of your I have to think about that. The only reason I say, yes, it's too sweeping is because it's a sound. Too sweeping. But like all the open arms lines, you have to think about it very, very carefully. Thank you for watching. Hugo does remain, I mean, a kind of, you know, giant figure of old, you know, gorgeous radicalism, I mean, that's what he's there. I want to think of some people like, and what about people like, who are, certainly, they never became a revolution for socialism, it was, it was not, you know, no reactionary either.

17:30 In the course of the course of the course of the course of the course Yes, except I suppose that they do seem to be no longer available. But the sense is that there could never be a revival of anything quite like the Gatsby project, quite like the project of the French revolution. Yes, I think that after 1848 that project was understood, it seemed to survive a lot. Yeah, I can see that that's going to happen. The toe-back of the private lecture is going to scholarship now as well as to all kinds of other things. In his case, quite literally, all they do is regurgitate. Yeah, I'd expect them to do that. Well, you're going to have a hundred lectures. Well, no. Well, no. Well, no. Well, no. Well, no. They keep denying this sort of math that you said it was what it was hypothesized to be, but basically that's their whole mission in life, their job. Yes, yes, and then, I mean, they clearly see, unless they become out and out the action, the ideal of the stream, but you can't go back to a religious ideology, the actual ideal of the core of the state of the world. So there has to be something other than that. Then there's a lot of politics and capitalism in the normal technological individualism, which you can easily slide over into.

20:00 It's really confusing for sure, and it just lives, so let's just say, in a wide variety of various forms of knowledge. I didn't think it started at the end of the century. Anyway, it's actually the 12th century, but there's nothing more interesting than that. I was very interested in this paper, and I thought it was fantastic. I mean, even on the face of it, you might think it's rather mistaken. I mean, for instance, it's a lot more than just a walk, and then it's shaped more. But it was, you know, an absolute mammoth idea. Ice, and earth, and carbon, and oxygen, and gas. It was a lot of fun. This is the perfect example of the Paris that was destroyed by Haussmann, you know, the Napoleon XIV's planning chief, the guy who bulldozed all the ancient papier in Paris, and turned the right back particular. ...face of the Grand Boulevard. Not least, the militias thought that there could never be another successful match rising, I would say, with our authoritarian party, the Paris, because there would be very long, clear avenues for the troops to fire down with artillery, and that it would be much easier for the militia to march forward. And of course, Paris is still, what makes me forget Paris, is still probably the proudest group like this, which obviously is a very easy to navigate. And in fact, the balance between those two technologies is much more in favour. And that's the only reason why it's very difficult for us to be starting to lose hope after a couple of days. Because we have to be fast enough to be able to force to lose much more in favour of the big, massive, and certainly fast... The most important part of the development of technology is not the use of the technology, but the fact that much of it is being approached by scientists. And a street like this is obviously very easy to barricade. You only need about three or four benches. Well, one house on the front, and all the houses on the back.

22:30 The toll will keep going for thousands and thousands of years. These teams are guaranteeing that the second M-man, which really was a train in London, he looked at himself as if he was at the opera, garanted at the same age that he'd built a casino in Monte Carlo. Very approachable. He was the architect of the casino and of the opera. And he's just designed to display this incredibly discreetly. Well, I thought we'd just wander down here because it's actually pretty smooth, although we're not going back to this point, going back to the same thing we were at this point. We might go and have a look at that bookshop around the end of the last hour. The author of the Declaration of the Rights of the Woman and the Citizen. Is that the same thing as a declaration of the right to law, or did they make a separate declaration? No, they might very well have made a separate declaration of the right to law. They did, of course, formally avoid disclosure, too, in the 1970s. They were the first government to make a declaration of the right to law. They may not have had any morbid significance.

25:00 It actually did vary, it did vary quite a bit. Well, yes, of course, most of Japan rose against that. But in Waterloo and other islands, they were all discreet. Thank you very much for your attention. That building is the index of all the struggles between progress and reaction in France for 250 years. It was begun as a church, and then it was taken over, of course, by the revolutionaries, and made into a secular building, a temple of reason, and the place where the great... The great men, great men and women. This was 1792. Yeah, that was in 1792. Sorry, I shouldn't have said that. That was in 1792. It actually began in the church in 1765. It was completed under the monarchy. And then became a temple of all the other serbians,

27:30 Jacobin, Radical, Paul, Levin. And then, of course, under Napoleon, No, no, actually, no, I'm not wrong about that. No, it didn't. No, it hadn't existed as a monument to the Great Death, but in fact, Le Grand, the Great Death, and more, and quite a number of others, and a great great great great great great great great great great great great great great great great great great great great great great great great great great great great great great great great In 1873, one of the first things that Martin Luther, who was here, tried to find a way of trying to send him back to the temple of religious reaction. That had to wait for Le Marne. Le Marne was the person who was the leader of, oh, this is a very complicated example, but Le Marne was the guy who succeeded his death. He was actually genuine. Le Marne was the person. He was elected in 1873, and he had totally ignored the monarchy. His program was that he was standing as a placeholder until the monarchy could be restored, and for about six years there was a major struggle by the extreme right to restore the monarchy, which fell apart largely because of the unbelievable stupidity of the monarchy, of the Gaubon-Kennedy monarchy. He was so upset. In the late 19th and early 20th centuries there was no reactionary rigid that he was incapable of perceiving where his own immediate tactical interests lay, and he wouldn't even yield on the subject, he refused even to accept the tricolour and stuff like that in France, so if he took the throne he would only reign under the banner of the Lee of the Bourbons, he wouldn't even accept a compromise whereby the so-called the affair of the flag, they went down to Chambord, he looked at Chambord, he was the claimant for the Bourbon dynasty. They tried to persuade him to accept the problem as though it was his own support, not Marl and all the other topologists and analysts, whereby that he would have the trickle, but it would have to be the superimposed solid, you know. I saw it already when he started. It would have the trickle together. And he said, no, no, nothing doing. I'm never going to accept the flag of the Reds. I'm never going to have any red in my flag.

30:00 It's got to be all white, it's got to be a pure white reaction or nothing. No red, no red in the flag of cancer, I get a red. And they decided, they looked back and shook their heads and said, well, okay, this guy's out for lunch, he's not going to be our, he's not going to be the instrument that our class needs, so we've got to look, we've got to get out plan B. And that was the last we heard of the French monarchy, thank God. But that was as late as the 1870s. But it was during that period, the softening up, the softening up process. ... for the attempt to restore the monarchy, and of course also to re-establish the complete rights of the church, that they did all of this... the pantheon was turned back into a church for about ten years, only for about ten years, they didn't last long, and that was when they put all of their hideously sentimental... ... to Genevieve. Now that was down there, under the park. And then, when the whole project was going on, it just fell apart at about 1879, or about... Again, I think the part, the big capitalist part, by that time was so obvious that, in the saddle, didn't need particularly the Constitutional Monarchy to go on like that. The old, what was left of the old radical imposter protected both the people and the people, so they were quite happy to operate the institutions of the republics, even though there was still no recertification of them, and the revenues were absurd. I think in 1881, Gambetta came back, who had been, honestly, more... Well, it's very modern. Do you know exactly where this route of ADT is? It's very near here, I think. Well, it's straight on the looks of the continent. I guess we've got to run around in a circle then, haven't we? That's my fault. It's alright. I think running around in a circle with you is far more expensive than this.

32:30 Well, that's the truth. Well, it's extremely kind of you to say that, but I think that's much, much more the other way. As far as I think about it, it's far, far faster. Thank you for watching this video, I hope you enjoyed it, and I'll see you in the next one. I'm sure that that was a major influence of his time as a senator, so there's plenty more about him, isn't there? He certainly showed great integrity in almost any of the old line of competitive words by leaders of the revolution. I mean, as I say, he was a huge compromise for Napoleon, even though he would have had great wealth of status by doing so, and went after the political right altogether. And that was, I think, when he did most of his scientific work. And then he came back in when he saw that there was, in fact, no choice between Napoleon as the defender, effectively, of France against the imposition of the Bourbons, the extreme reaction by the Allies. So, just for the reason of pure adjectives, he came back and did serve in Napoleon's last government. All of these forms came up making the theme of the topic right here. These are very interesting figures. I'm just trying to get more about you. I'm just trying to get more about you. It's not difficult. Most I know about you. So I didn't have to worry about much of my class last night. The last few days were tough. Well, you said it. You said some very interesting things about Carnot. Yes, no, I did hear that. Well, that was one word.

35:00 Well, thank you. I was only finally able to verify that. Oh, that was one of the things you were looking at. I was able to verify that he had a method of teaching about mathematics in a pedagogical paper about the teaching of calculus in 1923. And the topic of the variations is what the path and the focus is, but as I've said, I'm quite confident that Adam I would not have said this, so a great mathematician like Adam I would have had a general approach. He would have had a proof that this was a general approach. Other definitions, like the crochet definition and so on. But he is curious that he can publish it. Maybe he had a, maybe he had a general outline of the proof that he didn't have to find the answer. Well, it's possible that it was published long before that by somebody else. It may have been just a common name. It may have been what they call folklore. Yeah, folklore. Something was forgotten for 40 years. I'm going to be studying this for a long time. It's quite reasonably priced, too. Well, I think I'm going to swirl myself and bark on it if I'm so bad. It's much better than the English translation, so I'm planning to use it as a guide. If you'd like, you must have promised to do. Well... Well, I'm not going to... No, not at all. Oh, this is an interesting little book, too. Bill Barkey, Facts and Legends. You know, Lillian Boileau, who we were talking to last night. She's in charge of the archives, by the way.

37:30 Was she the lady... I know she was the lady we were talking to when we first came in. Did she come afterwards? No, that's right, there's an Italian, but she obviously knew from Perugia. She is also one of these people. History of algebra in the 17th century. Obviously, there's a whole lot of stuff there. She pulled it out and summarized it. Actually, her thesis, however, was more creative than that, because she tried to explain her mom, her large calculus, in terms of... Thank you for your attention. This is a very profound original work, and the interest of discovery is a few people, and people have few references to, so there's so much at least that, you know, the one that wants to do something the small way, a small cog in a little wheel, a small column in a little screw, and... Well, yes, that's the difference there. That's crucial, I agree, and that, of course, is why I work on Fermat from the point of view of topos. And, of course, it's why Colin's work, I think, on Cantor and Dedicates from the point of view of topos theory is particularly useful and very, very instructive. And that's a very, very worthy piece. You know, you referred to it yourself, this little essay on sets of points and spaces. Thank you for your attention.

40:00 Oh look it's only 16 let me get that. Would you like a pocket map? Oh come on let me treat you to that. Oh come on it's only 16 euro. It's not going to break the map. Well, he's the one who wasn't at Russia. He didn't come. They wouldn't let him in, would they? Well, he couldn't get his visa. He couldn't get his visa. I think that's the same chap. Yes, but I can't understand that, because he's a citizen with a visa, but according to the little sketch, he's actually a member of the Russian university. So I don't know, unless he's been declared a citizen of the university. It might just have been a... Science is at the epoch of the French Revolution, as far as research is concerned. It's possible for them to research Dominique Flamand's edition of Gatsba? Yes, of course. Perfect. Thank you. You already have the... I already have one. And also the little edition of Valparcano, which is a mechanism in general. Exactly. Does that mean two copies for five? Two copies. Is that possible? All right. I'll get this on my card and you can give me... unless you want to do it with, you know...

42:30 Can I put them all together? Yes. Is that possible? I mean, for a scholarly edition, that's very useful, so to talk about it. Oh, yeah, pretty good. Can you compare the works of people who find it cool? As I say, that was a great reason to charge for my review of the English edition. There's so much more in it than you would think. Well, I promised you the communes. I can always come back to Paris when I look around. There you go, Bill. That's your coffee. Thank you. That's okay. Do you want a separate bag for that? Are you sure? If you want to keep it clean and tidy, why don't you put it in... Okay, yeah, we... That's a mystic in my case, you know. Oh, I might change your mind. No, no, no, thank you. Thank you, sir. Thank you. D'accord. Ah, you're so excited. J'ai oublié mon tête.

45:00 D'accord. I've already seen this book about the discovery of the kind of theory of superpowers on a thing called the Ricciano boy, he's one of these big names in this kind of world, he's kind of a... But my impression is that he is exactly the same. Well, one doesn't have to say that it's informed by any perspective, but it's a lot in that way. Now, this code is an ideal bracelet for a briefcase, except it has a hole in it. Ah, well, the solution to that is to stitch it up. No, no, I shouldn't do that. It's a smaller item to fall out sometimes, so that's why having a bag would be helpful. That's what in English we call a poacher's jacket, a poacher's coat. The reason for it is too obvious. Yeah, you can put it wherever you want. Yeah, exactly. Well, we're back in the Rue de Medici, and I'm sorry, we have done a complete circuit now at the last of the gardens, but that wasn't, yeah. Oh, yes, of course, that's the actual arch of Marie-Thérèse, the famous thousand on the other side. Marie-Thérèse, who of course is also, like Pierre, he married, he was Henri, but it all becomes a blur when you get back to the 16th century. The bloody French monarchy is so complicated with all the intermarriages and the religious wars, that there was a succession of... I can never remember whether it was Henri II, Henri III... No, Henri IV was her son, and then there was one... There was one... No, that's right, Henri IV was her son.

47:30 I think she married Henri III, Henri III. She was the one who instigated the sort of autonomy in Paris in 1572, when they massacred the Protestants. No, no, be careful, yes. Oh, she was a connoisseur, yes. And, well, she was fanatically addicted to power and to fielding it ruthlessly, and obviously her religious ideology was a useful support for that, and whether she refused to go to the University of London or whether she came afterward or before. Sorry, no, no, no, no, I should know, I should know, I'm the one who's supposed to be the historian. I think she came after. This was one branch of the family, obviously, that left Florence and, you know, came, you know, Louis was the prince, or Barad-Louis was the prince. I'm just trying to remember what the sequence went for. It was, this was the period when, well, it was a little after the period when all the great Loire chateaux had been built. That was back in the, mostly in the previous century. And then there's Francis de Tette, the guy who... ...was the trans-Supreme exponent of Renaissance Prince-ship in France, the guy who brought over Leonardo da Vinci, who arrived, of course, with the Mona Lisa in his backpack and installed him in Blois, in the Chateau of Blois, or actually, I'm looking at a rather different little mansion close by the Chateau of Blois. I've visited the Chateau of Blois. Very interesting. The only thing is that it's IBM who sponsored that exhibition of all those machines. I would like to know much more about Leonardo, but I have always been a little suspicious of the Leonardo cult, which is not to make any assistance at all. Leonardo is clearly a very great technologist and a great artist and a great anatomist and a great all-round scientist, but I'd like to know a little bit more about his... In his philosophical position he, and I look back, you know Mussolini was one of the people who did a great deal to build this house of Leonardo da Vinci, the supreme expression of I did indeed visit the Tempio Volta several times. Of course, that was also built on the Mussolini axis, wasn't it?

50:00 But that whole business about the cult of Leonardo da Vinci, you know, the cult that was between genius, is very, very suspect in its fascist association with the great man theory of history, which, as I said, is not like any association at all with Leonardo, but for that reason, I'm a little bit, you know, I want us to be very careful about the source. It's very interesting that Leonardo has always been built up in this way. I wrote a long poem about you when I was in the sixth grade, because I had been influenced, you see, by the reading you were telling me. Well, this of course is why you also became such a great genius. No, I don't think so. I think the reason you became a great genius is because of... Didn't help, didn't help. ...dedication. Thank you very much for your time, and I hope to see you again soon. So you've been around that place, the Clos-Louis-Saint-Blois. Once there, once there. It is a very interesting little museum, quite often. This is a very fine shop and it goes quite a long way back. For instance, it was in this shop that I came across the work of Havai, Jan Havai, the man after whom the seminar room was named. Who was a philosopher of mathematics in the, actually he began his work before the war, in the 1930s, about 1969, sometime like that, and he's a good figure, I haven't, I must be honest, I bought a copy of his on the philosophy of mathematics when I was here about two months ago. I haven't gotten around to reading it.

52:30 But he certainly seems to have had a far more sophisticated appreciation of the range and diversity of mathematical production than most of the people writing in the so-called analytics division who seem to think that mathematics doesn't fit the topic exactly, with a few bits sort of screwed on the edges that we don't really need to learn about. The big theme here is causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality, causality. I don't think reading comms is going to pulverise anybody. Anybody of mine is going to be pulverised by comms and we can actually be doing a lot of this. No, I know, I really like to read what's going on. Did you say that again? And somebody's painted him with this great lipstick. I think that's true, based on the subject of introducing Revere. By no means, you just put it to the base here. The whole of that first paragraph. Well, he's stuck on this for a long time. Well, he's worked on this part of the book. He's got seven different recipes, all in there. I know, I know. I hope you can have a tune about it. Can I let you know, I did actually play back the tape this morning, if you can't come and see, that's because I was slightly late, because I wanted to make sure that the recording wasn't there, which it had, absolutely crystal clear, you know, all your work, you know, recording, apparently, but, yeah, but his introduction went on even longer than you and I had expected, 27 minutes, 27 minutes, right, it is actually on the digital end of the floor, 27 minutes!

55:00 Thank you for watching this video. Now, a very bright, particularly, well, no, I would say a very bright hope, no, hope, given an assignment, is to say, find out whatever you find out can be a cigarette, for all reasons, might be important, for understanding the organisation of all the mathematics that you would have produced, if they'd been very acidic. It's debatable. No, it's debatable, if you could use a cigarette. It's in the history. All these people had come there, obviously because they had heard one way or another they might have something to say, but they didn't have to be convinced that they were in the right place, so it's not bad. What is it? What is he... But what he was trying to say... What he was trying to say was part of my... He didn't mention your political position at all, so if there was some kind of favor or something... Yeah, but the curious thing is he only actually mentioned you towards the end. I think he was supposing to be... He spent the first 20 minutes just talking in very, very general and shallow tones, and that will be significant in the next 30 minutes. Absolutely, quoting all these speakers, and then coming to you, only right at the very end. Well, the last 6 or 7 minutes he did.

57:30 Well, it's not just a recitation of your eargram, it's quoting this long transfer of the long past, and then quite out of context, quoting the adjoining information statement, which was the original one. Well, all these double quotes were very old. Thank you for watching this video, please like, share and subscribe to our channel. One has reached a certain stage, one wants to have more enlightenment, okay, so it's a dual strategy, just for reading, about reading things, so you know, for those authors who are established as great men, Marx. Engels, Lenin, Grasmont, Strauss-Mann. The strategy here is that anything that he says, he should assume as a working hypothesis that it's true. Now this assumption that it's true, if you're working on whatever you call it, this would be a guide actually, sometimes an incredible guide to further investigation, further investigation that we would never have made. I see you're doing this, you see. And so in that way, you work very often. And like the opposite strategy, then everything will come to be true, even if it seems superficially true. You should certainly treat with great discretion. Well, I can do that as a guy. As a first hypothesis. The tentative first hypothesis. Well, I certainly agree. Yeah, okay, I understand what you're saying.

1:00:00 Part of the process. And obviously when something, when it's clear that there has been an immense gain in adopting a particular framework like the calculus, this is a case in point, then quite clearly the correct, the correct position methodologically is the one that, you know, that L'Occitane took as against these reactionaries like Barclay and Thorpe. Overthrow the calculus by showing that it involves commitment to impossible notions, to interherent notions, saying, Allons-y, et le froid vous viendra. So I suppose in that sense, and the same, I think, absolutely, about things in drafts and so on, I think that what you have understood and clarified is because this dual strategy is precisely what these other characters don't take, that the path of skepticism, quote-unquote, which is really a path to fideism, It always has been, since Abelard, that was his basic strategy, via skepticism, or Descartes' complete surrender to any of these people. Oh, Descartes. So in other words, if they're faced with Grassmann or Lenin, they find a series of objections. Everything, well, it could be wrong because of that. And if you refute that, well, maybe it's because of that that it's wrong. Maybe it's because of that. It's a series of objections. This is the skeptics' approach to it. So, of course, this strategy is what we draw, a dictionary. We're the good guys, we're the bad guys, and there are a lot of others in between, but it's something that's established through investigation and not just by faith. No, no, no. When you stated this Christian slogan, it reminded me a little bit. You couldn't read that into that, so I take your point. It's not entirely finished. Oh, we're almost there. So we just have to go straight on up? I assume this is the Avenue Circulaire? Yes it is. No, I see what you're talking about. And in the case of Grassman and Lenin, well of course in the case of Lenin it usually is pretty clear. So even with Lenin you sometimes have to do a bit of digging.

1:02:30 No, no, you have to do digging. In the case of Grassman, obviously, unless you're a mathematician of his... Thank you for your attention. All of these things have cast doubt on various things, and it was only by saying, no, he was probably right about this, trying to find out how he could be right, that we discovered this thing. So it's not that we invented this, I don't think we found it there, but we would not have found it if we had not been explicitly opposing these skeptics and putting faith, quote-unquote, or at least provisionary faith in what the man had to say. You know, I think that was probably among all the things I learned from Hardy O'Bain. The most basic thing, the most basic thing to do, leaving myself to be a Marxist for several years, being a militant founder of the Vietnam War, I still had to completely... The French Communist Party, I mean, the French government was part of it. I thought it was one of the main reasons the Communists left the government. No, but then they accepted it again. That's right. That's right. I think, as I recall, I mean, I don't have the details. They had left, yeah, they had left the government. But now, you see, now they had the chance to get that in the government, and they were accepted into the government. I'm telling you, in an accurate story, in an accurate story, but that seems to be, no, no, no, not at all, I'm the one who needs to find out about these things, because they were accepted into the government, you see, they left in principle, they came back in again, they came back in. Well, eventually, at some point, but it was very soon. And it was all around the Marshall Plan. I hadn't realized that was the sequence of events. Is that David University Press? Oh, okay. I asked because I used to work for Sprinter for a while.

1:05:00 And a friend of mine was the math teacher, Ed Bechtel was the math teacher for the academic press years and years ago. Well, I've also published a stringer, not a full book, but in parts of many of them of mathematics and physics. I worked there a long time ago. It's a small room. You worked at the Flatiron Building? Oh, I did work in the Flatiron Building. I used to live on that square, on Madison Square. There was this old, old hotel, Madison Square Hotel. It's been demolished now. Right next to the house where Churchill's mother was. Oh, really? It was a terribly seedy place by the time we got there. A terribly seedy place. Well, Bill's name is still quite famous on campus even after 40 years. They all have a Bill O'Hare story. And this is going on 40 years. It's been transmitted through two or three academic generations. Thank you for your attention. I suppose this is cheating a bit because this was, you want to get going to Simona Marta, and we will, but what were the three more additional points that you would have made if Anouni had not spent 27 minutes on his interminable introduction?

1:07:30 The Koenig theorem says that the homology of a product is equal to this tensor product, which implies that the diagonal map of space X into X cross X gives rise to a diagonal map from the homology of X into homology of X tensor homology of X to the structure of a commutative co-algebra automatically, because it's coming from a simple thing like diagonals. Commutative and associative co-algebra. Now, it's known that these co-algebras actually form a Cartesian closed category, once you get to see, well, a general leaving aside the closed, in any, basically, in any tensor category, you can speak about the community of co-algebras, and they form actually a Cartesian category, in other words, the tensor product becomes the Cartesian product in this new category of more richly structured objects, because precisely the diagonal map It gives you the extra structure you need in order to form the simple pairing of the characteristics of Cartesian products. So therefore, homology is actually a function from one Cartesian closed category to another one, preserving products. So it's basically a question, can this thing be axiomatized directly as such? Because the usual axiomatization is to concentrate on the linear aspect and ignore the co-algebra structure which is there. All the maps that are induced by actual continuous maps are not merely sequences of linear maps, they're actually co-algebra morphisms. So really this idea, you could consider that the starting point is homotopy because homology is supposed to be an invariant of homotopy anyway.

1:10:00 It's a further coarsening basically. It's essentially a kind of stabilization that's involved. Homology has a certain stability undertaking Cartesian products with trivial spaces. In fact, there's a beautiful proof right on the curve here. It's called Albrecht Dolle. In the American Math Monthly, several years ago, he was showing how to exploit precisely this property of homology. You can get an easy proof of not only the Jordan curve theorem, but something far more powerful. But it wouldn't be true for homotopy. No, that's right. Extra coarsening comes from this kind of ability under product. But in any case, both, you see, all three are really Cartesian closed categories. So the conception should be that homotopy types, even homology types, are still something like spaces, except that they're now qualitative, you see, in the sense that I defined. And so, of course, they're more abstract in some sense, but still they are the remnants of actual spaces made into qualitative spaces, and that's just measuring them by functions in cohomology classes and all that is, of course, an important aspect of it, but cohomology classes live on something, which are these qualitative space-like objects, qualitative objects-like spaces. So this is one point. Yes, that would have been very well worth hearing. The whole open, you know, the whole somebody's thesis should be written. Yes, that's at least an entire PhD thesis. Now this, it's funny because this is known in a way. In fact, Eilenberg and Moore wrote several papers about co-algebras and such. I'm sure we're motivated by this observation. For some reason, they don't take it directly on. Homology is not a sequence of groups or co-algebra or you see once you've realized that it's a Cartesian closed category you could say, well, co-algebra is just one way of presenting the structure. Maybe there are other ways of presenting it. Homological figures.

1:12:30 Once again, he doesn't have to live in a linear setting. In fact, it could be, you know, once delinearized, it could be linearized again. It seems like it would be emotional. Anyway, so that was the whole program, which I thought, you know, it would be worth devoting five or ten minutes to outlining, because there might be people who are capable of sharing it, really, among their own students. Now, a jewelry to that, I suppose, is the observation that, coming back to the homology groups as distances, But actually, the definition of cohomology is itself completely independent of linearity, although it's always made to seem to depend on it. You see, d-squared equal to zero, and you take the image of the boundary of boundaries and the cycles and all this stuff, it's all put in linear language whenever you read about cohomology. Construction has a very intrinsic sense, just in terms of how you might counter it, because it's independent of it. Whether the categories involved are linear or not. So if you have a big category and a small category which is a sub-category of it, that the inclusion should have both left and right edge lines. So it's like the pi-zero and the points. I was going to say it's very similar to the situation with the pi-zero components factor and the points, yeah. In the cases I know of, the pi-zero and the points will not lead to anything interesting that I'm about to say. But it's basically, you see, that because of the concept, normally you think that from the left adjoint you can calculate the maps from it or something, and the right adjoint maps into it. However, in this situation where you have this full infusion, you actually get a map from the right adjoint to the left adjoint. There's a natural transformation going across from one of these... So, just as you have in the pi-o and the pi-x case, but that is a more... For the same reason, because the right edge joint of pi to x is included in x, and the left edge joint, you know, receives a map from x, and you take this composite, which goes to x.

1:15:00 Now, that seems to be in the other category, but because the inclusion is full, it's also in this category, too, because of the fullness. Now, what you do is you take the image of this map, the image of the map, so like, as I say, the picture of points and components doesn't quite tell it because you think, well, of course, every component contains a point, but that's not necessarily true. No, no, you made that point. You actually cited instances last night where that doesn't hold. So the image is a third function. Yeah. So now there's an example. You consider all the chain complexes, sequences of groups with a boundary operator. Among those, there are those in which the boundary operator is zero. It's a subcategory. Well, it has a left adjoin and a right adjoin. The right adjoin is called cycles. The left adjoin is called dual cycles. So the map from cycles to dual cycles has an image that is exactly the homology of an arbitrary chain complex viewed as a trivial chain complex. So the basic move in homology is to take these two opposite invariants, these two sort of opposite invariants, and to find the balance between them, take the image between them, which is why homology has... Albrecht Gould advertised his main property as being half-examined. It doesn't preserve limits, it doesn't preserve co-limits, but it does preserve, you see, those squares which are both pushed out and pulled back at the same time, those can be preserved by homology, because that half-exactness uses that as a basic tool in analyzing homological construction. And that's the definition of half-exactness, it's something which preserves its push-out and pull-back at the same time. The things which are simultaneously both have become simultaneously both under the concrete. So you can see why that should be, because you've taken it halfway between something which is a right edge of your head, totally right exact, left edge of your head and just totally, you know.

1:17:30 Anyway, so there are many simpler examples in chain complexes in which you can compute this. Simplest, maybe the simplest one, is consider directed graphs. So that's a pair of maps, basically, source and target. Edges, vertices, source, target. So there's a topos of directed graphs. Well, among those, there are the ones where the two maps are equal, which means that all edges are loops, right? Because each loop, its source is equal to its target. So it's just like it, yeah. Yeah, okay. It's just like the way you think of... Well, again, the left adjoint has something to do with taking components by zero. It's just a minor variant on that. And the right adjoint has to do with taking a set of loops in any thing. So the homology, then, of the directed graph relative to this situation is simply the set of connected components of the graph, which happen to contain at least one loop. It's an interesting invariant of a graph. Absolutely beautiful. You have the components and you have the loops, but some components have loops. Which just gives you that. Well, it's a simple example for showing, in a totally non-linear situation. Yes, indeed. Completely non-linear. There's no abelian groups. You see, you don't really have to think in terms of... Well, this is a different invariant from the usual homology. From the usual chain complex. But, of course, you can linearize the graphs, too. Actually, I'm just computing that this morning. I didn't quite think of this. Oh, that's incredible. Well, that's the second topic, which probably just emphasizes that it's all in terms of adjoints. Yes, indeed. Once again, clarifying the immense... No necessary dependence on linearity. No doubt this would be one of the ingredients in a thorough discussion of the previous program. Yes, yes. Viewing... I'm really understanding what the problem is.

1:20:00 And in understanding what the precise relationship between that qualitative aspect of homotopy and the... Yeah, the extensive quality... ...courser termination. By the way, that thing itself, I was thinking at the level of two categories, Cartesian post-category with pi zero and points and so forth. Well, you have these two constructions, the homotopy category of Erewitz, which forces the qualitative feature of pi zero equals points. And there's the extraction of the infinitesimal objects, which also forces the same thing. So, actually, consider it in terms of categories of categories. There's all the things, and then those in which . So that's the subcategory, and we've got two adjuncts, and you take the image. In other words, look for the homotopy types, which are actually homotopy types of infinitesimal states. Of infinitesimal states. See, that's again the image. Well, you managed to cram that into less than 10 minutes, so you certainly could have done that. Oh, there's a third one. The second order differential equations which I touched on. You did touch on that, but only... I was going to say that, and then I got it. Well, I've cheated you because I said... Well, I tell you what, let's go after them on Mark for now. You can tell me about the second order differential equations at further points when we sit down over the last copy. Is that okay? Great. Okay, let me just... Andiamo. Andiamo. Andiamo. Mancaremo. And, uh... I've still got a couple of more questions I want to ask you about. One about corn, and one about, well that's uh, and one about kiwi, but they're all kiwis.