Discussions, incl. FW Lawvere, C McLarty, JL Bell, others

0:00 ...operators, one commands the other. The exponential weights are exponential. It's automatically endless. We've seen maybe how to become, as I say, out of our conforming general metaphor. In other words, don't get why Israel is invisible, but the other way around. The quotient. There are two variables for each one. There is an expression for the topology of the theory, and a new expression for the theory of one-mindedness and the other-mindedness. You don't have a better idea what you're speaking about, do you? We will have to speak to this matter when we go back, but there are certain subjects that we want to know. It might just be some drawing of the theory, but I don't think it's very bad. We knew we would have one. I don't see how we can Well, if he's suffering, you're obviously going to want him to suffer, but we're going to talk to the vet before you make a decision. Now, we added this concept of the delta y being divisible by a delta y. I say it's a function of phi, but it's invariable in the context where the upper bound gets different. Yes, as I remember you, as you spoke about very eloquently in Florence in 94, I think it was, yeah. So, seamless principle.

2:30 I think actually in his book, the reference, finally I found it. I do feel because I know how much my mother always... We're very attached to the cat. When we first went to Canada, we had a litter of flies. The cat we had there, we read for you. I know, that's what's so strange. That's so strange. They didn't learn about the traffic enough here. We've got to run over. There's not much traffic there. You end up like your child, sorry, gets hurt.

5:00 I don't think it's all at all. I think he doesn't realize. No, no, he's French, but everybody calls him like that. So he gets converted into something, it's converted from an organism, but also into something which behaves like a...

7:30 I mean, you know, that's not how it works, but what he really made out of it is... And then... I think a lot of things in numbers are going to be a lot easier if you could differentiate numbers from the functions, and get an answer. I mean this, this is a deadly serious point, because, for example, the function theory analogue, the graphic analogue, Rodel can do it. It was first proved by Manning in the function field case, using a derivation, something extraneous to the problem.

10:00 Now, Booyum has got ideas for using... ...solving similar problems in... ...using these in this matter up. I mean, there are issues of trying to take notions from differential calculus and give them a meaning in these numbers that I just made. It's pretty tricky. I mean, I'm probably sure that some of the stuff around Kodaira and Spencer theory cannot be done... The volume has shown that there's a notion in which you can differentiate certain numbers and so on, you know. Just like you would have liked to be able to do when you were learning calculus at the beginning, you know. f of x minus 7y over x minus y, then you have a bump, and then you come back down. It doesn't happen in any way. I mean, he's just basically the denominator of x, and then dy by dx is x. It sounds as if... I mean, I had the lecture about Havansky recently, about something that sounded quite like that, in connection with that thing which is this way of pairing, which goes both in number theory and function theory.

12:30 Yeah, I thought you were talking about that. Well, because that's what they were doing. The reason they thought it was chosen was that cardiomics are more important. Oh, the paper that was based on your talk... No, but it is a really interesting issue as to how far beyond you have to go to calculus. I don't know if there's any paper underlying it or not, but... It's a beautiful principle, as you say. It's clearly there in Hanover. I wasn't there. I didn't know about that meeting. That's the one I have of yours I definitely don't have I would like to study. Is it on the site? Is it on there? No, I don't have that one. Definitely not. It's called Unity and Identity of Opposites in Calculus and Physics. No, that's the recording of the lecture you gave in Florence, which I actually transcribed, but not... ...way of getting the idea across. ...the actual measure of all equations... And, obviously, the principles illustrated in many categories, if you will, presently.

15:00 No, no, no, no, it's not a problem, no, no, no, no, no, it's not a problem. A salad, a croque, or something like that? Yes, like a ham salad with mixed eggs. Eggs, okay, perfect. Eggs, tomatoes, eggs, perfect. Thank you, chef. For me... The brochette includes all of these conformities, doesn't it? No, it's an either-or. It's either the pork or the beef. It's a choice, isn't it? No, no, no. Excuse me, excuse me. You're right, you're right. I'm sorry, my comment is failed. Don't worry about it. Yeah, yeah, okay. No, no, no. It's good. For me, it's perfect. So, I want... Brochette. Brochette, two, four, five, six. You also have six. Six with the salad. Yes, yes, yes, perfect, perfect, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, A red one and a brown one, and a brown one for the lady. Oh, a beer. Excuse me, a beer. Sure, okay. In that case, a beer for me too. A beer too, okay. And a... Three. Okay. Okay. Okay, it's three. It's three. Three. Okay. Anybody else want beer or just stick with wine? Wine. Wine. Wine. Wine beer. I'm going to have the wine as well. Remember the old, actually it's an illustration of, beer then wine leaves you fine, wine then beer makes the question of, don't worry.

17:30 Well generally in Oxford at the logic seminar we did slightly more elaborate versions, Gandhi and myself and Wilkinson. We usually have beer and then wine and then more beer. What is the effect of that? Total oblivion. Occasionally, occasionally yes. The kernel of the map goes to zero, you've got to... It's quite rarely led to total oblivion. I mean, even Gandhi sometimes drove home in that little old Morris Minor phase at that point. I got home all right last night. Yeah, well, we noticed. So at the time, we had our doubts. I do know that route by now. It's a little mapped. Yeah, okay. No, I don't think the sider should make too large a... By the way, we do have another thing with a cork in it too, open tonight for the final. You probably saw. You don't know John Muhammad Ali Macbeth, do you? Sorry? You don't know Muhammad Ali Macbeth. I don't either. Oh, well. He was a student of Gandhi's. Not with a bang but a windpipe, I'm afraid. But then he had problems. There are many strange things about him. I'd really like to hear more about this that I don't know about. I do remember your talk in Florence very clearly on the subject of bringing the variables back together, the Marxist construction, and then also relating it to what Hadamard had done. I remember that, but I definitely never saw a paper that came out of it, I'm quite sure. Believe me, I have a big fat pile of all your papers and I know exactly what's in it. That one isn't. I certainly want to study that, but I don't know. Well, at some point, before we maybe, tomorrow when we're all, everything is, everything else is taken care of. I'm sticking with beer just for now, yes. I'll have the wine later.

20:00 I would like to go through all your papers to make sure, you know, just which ones I don't have, and you can send them to me after you get back home and there's no need to worry about it for now. That sounds extremely interesting. Cheers to a very good morning's work. Well done. Mummy says you can all go and see the castle now, which is the last... Cheers. Cheers made me better. By the way, I hope since John is here, and he's furiously limping, I hope you noticed the, how do you say, the classic British imperialist tactic which I used last night in order to quell the mutiny. You see, if it had been Alexander, you saw the movie Alexander, right? Oh, no, I don't mean the old one with Richard Burton, I mean the new one with Justin Fields. No, no, that's not. He's Colin Farrell. Brad Pitt's the guy. I'm Troy. Troy! No, no, no. You see, Alexander wants his men to follow him over the Hindokush to the utmost end of the Earth because he's convinced that the Occhianas, the world ocean, just lies out of sight. And when he reaches, he's reached the end of the Earth, he really would have copied everything. And they're just getting completely fed up with the whole bloody business and just want to go back home to Macedonia, not surprisingly. Sensible men. But it took them till then, you know, to realise it. So, of course, he berates them all and says, you know, you're coward, you're straight as white, Marlowe would have had you all killed, I'm going to go on alone, even if you won't follow me, you know, I wrote Superman, I'm going to go on to the ends of the earth, whatever you do, and so forth. It's all right, you see, you don't have to lose. Bell says, listen, I'm fed up with all this mathematics. I want some nice woozy general philosophy to talk about. So instead of doing it, can you just use the technique? Of course, of course I can completely understand your position. I'd like the men to understand. There is absolutely no question, no question at all of making the army over the Hindu, of course. Of course we shall make the earliest. These are all practical arrangements to return to Macedonia. In the meantime, I'm sure you have no objection, but you're sending a small reconnaissance force over there, if you could, and only understanding that there will be no major commitment on the other side.

22:30 You're definitely going to take us back, and then of course we'll go back to Macedonia. I'll get to the other side. Well, of course, now these chaps are here, they've obviously got to have the necessary logistical layout, so we'll send a few more. No question at all of entering into any major or long-term commitment, you understand. Go to the other side and keep going and singing. Keep going and keep going and saying, what was all that business about going back to Macedonia? Well, of course, I mean, it's true at that time, but alas, you know, we've had some change. No longer practical, I'm afraid. That's the way they do it in the British Empire. Hey, it works. Of course you can. Yes, of course. Yes, that would be nice, actually. It's very nice. Leo took one this morning, in fact, while we were in the house. But no, it would be nice to get one in here. It's very nice. So that's the way to Crowell Museum, as you'll see. Can I take a picture of all of you? Yeah, that'd be nice. But you have to sit here. Oh yeah, yeah, I'll put it on the... it'll be attached. So this is attached. Make sure... One, two, go on. I definitely don't have this paper. Alas, I would like to study that. And as you said, it illustrates the unity of identity. Obviously, it's very nicely set in the form, because it runs through so many. He mentions the astrophysics of the scale of mathematics, but it's more of a thing in terms of how to provide other ways of a seminar on it, in terms of the mathematics of the category, the other, um, the determination of what is its opposite, I mean, this would be a, it's a lot of institutions, I mean, it's a really very interesting thing, to assembly somewhat like-minded people on similar problems, but within a few decades, to assembly people on... The single problem is that the arrow of short proof is going to be, I mean, nothing is going to be everywhere.

25:00 It's not, of course, a differential calculator, two-pages. I want some people to be involved in it. I have this product that I'm going to change, and we're going to promote it. Just throwing you into the narrow marvels of physics, you know, and I guess the dark machine has a community friend, i.e., algorithms for quantum mechanics. There's a lot of this, constant analysis, but then we introduce the idea that these problems need to be turned off. I mean, we have no idea about the UIO shapes of the x-category. I think people thought the matter was... the proof would probably be... Well, for two young people, two young people, I always forget the names of them, and then the problem is in the hands of math now. It can be divided into some things. Yeah, and which obviously involves paying very careful attention to quantum mathematics. And what does divisibility actually mean? Following the ordinary number of elementary definitions of A divided by B, divisibility by B means that there exists an X divided by B gives A. But the set of all these X's might have a unique outcome. It might be imperative, but more likely, ideally, you can do it. So this has to be a definition. The usual treatment in calculus is that you write down the so-called brain quotient as though it existed. In other words, as though division is a completely, generally outspoken operation, which it totally isn't. ...compared to the probabilistic time-analyzing tests, which give probabilities so high that it is important that we have the stuff that we're using now. And I think that's absolutely, I mean, that's a stumbling block for a lot of beginning students who come to us. In other words, this idea, this diagram of the shape of the category of rings, really does represent his idea.

27:30 Even though Garibaldi substituted it very independently, the number of different forms, perhaps, is quotient. ... would be a function, for example, of two variables, dropping down precisely the diagonalization and setting those two variables equal. Yeah, no, I understand the principle. That's why zero is divisible by zero is completely rigorous, and that's all you need to calculate. That's really to develop the whole... That's how you actually calculate derivative practice, not by limits. No, that's true, and yet people dismiss Marx's mathematical manuscripts and go, oh, he didn't know anything about convergence or limits and theories. He couldn't have known because we now have this... because he didn't know epsilon delta, you know. I think that's what he must have had accrued. What, the dismissal of this? Oh, absolutely. As Bill brought up very, very clearly in this lecture he gave in Paris, which I hadn't realised until now, also exists... The point he was exposing exists in this paper, which I didn't know about. ...completely wrong. I mean, he does... It's just basically spread by Russell, and people believe it because they can't be bothered to study Hayden. Yes. Well, Russell, of course, had to... Russell, of course, had got this very strong sort of propaganda line after he abandoned... The curious thing is that when... if you look at Russell's very earliest mathematical work, which is now available, studied by Griffin... I know, it's sort of... Griffin, he actually wrote papers on the theory of quantity, which, when you look back, you can see he's actually got some quite subtle points there, but he's just missing... He does have some interesting things to say when he's trying to... But then suddenly he suddenly gets into this, because he's spending a lot of time in that book, and yet at the same time he actually does...

30:00 There are a couple of points that he makes on Hegel, which are very interesting. There's a problem with that. Why am I confused? It's before, it's before. Well, because he was one of the founders, he gave some money to the Web, so he was one of the founders of the Web. Well, that's true. At least he didn't exist then. There was, there was, there was, there was. And then the webs, as you say, enlarged into the London School of Economics. That's right, he'd been giving them... No, no, no, no, you're essentially correct, it's just that the name of the institution is James. I can't remember the original name, because it only existed under that name for about two years or so before it came down to the School of Economics. In my book I try to, I take the principles of mathematics and I'm part of it, but what fascinates me is the way that that kind of platonic evidence of metaphysics completely... Absolute praise of Cantor and Dedekind. Although when he first encountered Cantor, he hardly mentions Dedekind, hardly mentions Dedekind. Too much algebra in Dedekind, too much real theory of the structure. He does try to go fairly into philosophy of continuum, philosophy of the infinite testaments, and of course the key philosophy that he's trying to combat... It's something that's actually got coming, well, from Kant and Hegel, and so he actually, and so actually it's quite a useful... But the trouble with Russell was that he was, he had such a quick mind, he was so glib. He didn't, he wasn't a mathematician, essentially, he was a philosopher who thought he could juggle. Get a take on mathematics from whatever propulous object theory he'd adopted. To be fair, he did say Cantor, you know, Russell knew Cantor's... No, he knew Cantor's construction of the continuum, and he did, you know, he knew about...

32:30 And initially he rejected it. He rejected it. When he first went, he thought he had to prove it. But that's normal. I rejected it. By the time he got to write the Principles of Mathematics at any rate, it's according to Kantor that people could use it. And Byrstrasse too, but Dedekind only, you know, you know. But then later, it's interesting, then later when he realized that Kantor had quarreled with his real hero, Frege, he started, you know, he started attacking Kantor. He made many, not so much for his mathematical work, because he wrote the Principles of Mathematics before he knew anything, he hadn't even heard of it. Which is why, he's not following on, he's got ideas of his own, and there's often some kind of transition that's going on. He's completely confused about it. Cantor, Sanket, Ragnosch, O'Connor, Zahler. Oh, but that was, to be fair, I think so was Zermelo. Lots of people were confused about that. No, actually, he's confused. But by the end of his life, he was making the most appalling, scaring, personal remarks about Cantor. No, he compares... The only thing he, in his balls of marble, the only thing he says about Cantor, apart from a completely ridiculous, distorted account, which would disgrace a first-year undergraduate. ...of what Cantor had done in mathematics. It's just to print the letter that Cantor wrote from him... ...to him when he was in the asylum and say, you know, it will not surprise the reader after reading this letter to discover that Cantor believed in fairies and that Cantor was the author of Shakespeare. But actually, no, Russell does understand. He does know about Cantor's conjecture. Because he compares it with Hegel. He actually compares the principles of mathematics. That was much earlier. The principles of mathematics. He makes an incredible conjecture. He knew that because the idealist philosophers that he studied like Brangnett. The course had introduced him to Hegel. See, he'd studied Hegel at sufficient depth to know about Hegel's theory. But it's rather interesting, he says, he discusses or mentions Hegel's notion of continuous continuity and discreteness.

35:00 Which is one of the things we're talking about this afternoon. He explicitly compares Hegel's notion of continuity of a continuous manifold with Pantor's primal numbers. I was very careful, I was very careful, hardly to mention, I mentioned it twice in my book on theory. Sorry, which term? I just didn't catch. The continuum. The continuum. The continuum, I promise, occurs twice. And it puts... Continuum. Yeah, sure. Well, that's because he took it... ...which of course leads Russell to say ridiculous things like, we now know where Zeno went wrong, the arrow in its flight is really at risk, and Weierstrass has solved Zeno's problem, the arrow is, everything is static, because of course the ultimate reality is that photonic atomists are just objects with purely so... The more it's clear that there were two lines, there were two squares, you know, which are as wide as one another over a period of time, not with each other, but within a period of time. Yes, it's very powerful. One was the algebraic line, broadly speaking, the algebraic line. Well, I know, to a first approximation, no, not just, but to a, but no, but to a first approximation. Yeah, okay, let's be more... Okay. Russell, piano, Russell, Quine, and the serious mathematical arts. No, no, okay. On the other hand, these are almost two. In fact, there are histories between the groups. In fact, one or the other. And the other, which obviously is, well, I go back and forth about Leibniz, you know, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah,

37:30 And one should see how much Cantor in fact lies in this line, although he's always represented as lying in this line. He had a problem with mathematics. Then, of course, he claims to have had a solution. He published a paper in which he claimed that... Bon appétit! ...which of course is getting holed up. I thought he was going to get that out of the way. Oddly enough, that problem of course turned out to be really at the essence. Because that was one of the great times when he really had to understand the facts. He can't have been forced to think about how to continue as opposed to arbitrary problems. That's a very important point. But he just didn't have the framework, you know, within which to formulate, because he said it would be very difficult, and it's interesting, because then, of course, once he told that, then he went all the way to the university, where he was happier to, you know, the university was happier to occupy him. It's interesting. We don't seem to... He does seem to have reached the position that he believed that a purely arithmetic ontology, and a very much a purely arithmetic ontology, supplied the foundation for everybody. And if you don't get it, you say something is wrong. But you have to say that this is a really great mathematician.

40:00 He was aware of it, but then... He was obsessed. We became quite obsessed with him. Well, of course, it was related to theology, wasn't it? Yeah. All this stuff about the ordinal speaks, steps to the throne of God, this riddle, appalling stuff. In other words, the root of the order is long-standing, and it can be held at some extent. And then when, um... Which also carries an echo of Plotinus. No, no, he, uh, well, the reason good and liable is bold is... This is actually one of the topics that you specifically asked me to, and you're right, and we will. It's one of the things I'd really like to get onto. It's an interesting fact. When I re-read the Grimlock one, from the other end, from another perspective, my deceptive is very good for this. Madame, merci. Very good. It was okay there? You read Kandor in quite a nice way. But is it really? Well, any or every one of these groups is because we're going to be part of it, by the way. But I have, I read Ireland and Rosemarie, and I think, I can understand how somebody who really wanted to know these things would be very skeptical of a logical system. Something mod, non-educated, and it turns out that the equation has all of the transcribers. Well, yeah, numerical, yeah. This is very good. This is a very popular place.

42:30 I'm asking to see why. It's especially popular when Karine is here, who is an assistant to Perpius. She's on holiday in Spain this week. Oh, too bad. That's better for us. It's a good man. This is a wonderful show. Do you remember us? I don't want to see this happening. I don't want to see this happening. I don't want to see this happening. I don't want to see this happening. I don't want to see this happening. I don't want to see this happening. I don't want to see this happening. I don't want to see this happening. I don't want to see this happening. I don't want to see this happening. I don't want to see this happening. You said something very interesting on this point yesterday, I think, Colin, that you've been commissioned to write this chapter in the book by Paolo Mancosu on Caterpillarian Homology. You said you were going to start the exposition by discussing Kronika. Is that right? By discussing Kronika? Well, no, we do a general essay on that. I didn't hear from him. I just assumed he'd be there. He gave no indication. Well, that's going to start with Kronika. He said, fine, come earlier. I assumed you'd be here this morning. Sounds very... I think it's a very nice line of... very nice strategy for an exposition. When is this book coming? But what's the... what's the... I suspect... I mean, how big a book is it, and what's the organising theme of the book? So, kind of philosophy of real math, but done by real mathematicians. It'll probably take a long time. Who's publishing it? Who's publishing it? The University of California?

45:00 To be honest, I was taken aback by what you said the other night. I didn't know him at that time. He may have known. I'm sure he doesn't, but he is, I think, a careful and thoughtful scholar of mathematics, not just the history. And he's talked to me on general themes of, you know, patterns of unification and explanation in mathematics, and it seemed to me that he knew what he was talking about. I knew him in a different way. What he was trying to do was very different, and then what he was trying to do was... This is probably why he sensibly decided to do something which would explain why he's now doing something which he's much better and much happier doing. Coming as I just began physics, for a long time I couldn't imagine a number that had any value at all. Even though it took me a long time to realize that a number is a function, or maybe zero is a function. Unlike in something like the real world. The fact that you have a complicated structure of ideals and ratings divided by the integers, the magnitude is sort of equally complicated, not exactly an analogy, but the structure of ideals is the infinity rating, something like that, so that in the end one doesn't form the other, we just look at number theory and such, but all this stuff is so big it stops moving. We realize that typically, if we listen to all of them, the numbers arise as values or zeroes out of them, because the persistence of some of them is much clearer, sort of. Of course, it links very naturally to your theme of the continuous discreetness of fundamental opposition driving the development of mathematics.

47:30 One of the things you wanted us to talk about, which I completely agree. Again, you didn't bring your pocket tape with you. We should all be okay with one. Yeah, yeah. I'm sorry. I think the trouble is in here. It wouldn't pick up very well anyway because of the background noise. I gave up. I gave up. I know, but... Well, Canada's really been great. We're very happy there. They've made me good fellow. They've got a very nice department. And they even got me, I think, a lifetime of education. They gave it to me, but I cannot get a certificate. I'm not part of that game. Some people are, and it's been fine, you know, but I've never gone at all. Ray has some people in Montreal. Well, he's not flying out until we're ready. Thank you for watching. This is remarkable because we have not had to do it. Obviously, in our theory, I think it is our theory, but nobody else seems to be doing it, and I think it is your fault. Well, we haven't sort of seen that in the forecourt, so there's a little bit to do with what John was talking about last night. So therefore, you're not doing real chemistry, therefore you're out. You only have one life, you want to do something good with it. It would be irrational. And the thing about the exponential idea, when you go to a semester... I just wish I could get other people to say it that way. I mean, not these guys are actually producing these ideas, that's not a problem, but the kind of people who might actually be able to fund, you know, the kind of support. We're not talking about such a big institute for research, we're just talking about enough funding to provide somebody like you to listen to and transcribe. Somebody could go around recording the meetings, because I can only be in one place at a time, and I don't have to make a living anyway.

50:00 I think with a staff of three or four people, you could do something incredibly valuable. But, you know, everybody's doing it. I haven't forgotten, don't worry. But of course, you're also living in Ontario, which is the problem. Because what I could do once this stuff is on screen, I could sense something. I've been thinking about that since you mentioned it last night. Where the essence of the thing would be brought out. I think that's a great talent. It's very difficult to do. And I think one of the problems is that it doesn't cataclysmically fit in the way that it's based, is what Bill, what Mike was saying, you know, maybe about intensifying, but actually there is this feeling, you know, you know what it is, I used to fight in these battles for many years, that, I mean, just to edit out the ums and the ahs and the uhs, and when people are talking over one another, they can't hear what they're saying. And then I run it past these guys to see as it were what level, because everybody likes a tiny alphabet, let's say. And then the idea is to make it available in the... On a big internet site for somebody like the Royal Society or the Royal Belgian Academy where people can use Realfare or one of these other software systems that now exist to download the software and listen to it or even, although that technology doesn't exist yet, to print it out. But probably with the bandwidth of another five years, people will actually be able to do that as well. But we'll need to get, you know, we'll need to get oral audio and visual pattern recognition systems. At the moment, it still needs a human being to actually transcribe the stuff. You know, there's a huge amount. And of course, there's all the back catalogues, as it were, going right back to the 1970s, including John's lectures in Chelsea and London. And I'd like to... Oh, it's not all of equal value. There's probably...

52:30 I haven't even finished cataloguing it all yet. I've probably got another three months' worth just to finish cataloguing it. But when it is, I'd say, to a first approximation, about 20% of it is of really serious, permanent, comparable value. The other 80% is, well, some of it is just junk, and some of it is stuff that's really purely antiquarian value. There's about 20% of it which is of really... This is why I actually went to see this Russian guy last October in the hope that he might be prepared to put some money for it, like most rich men it turned out that he was just on a private ego trip. I thought if I'm actually going to solve it, it's going to be important. That's Leibniz's principle, and he failed. He's really nice guy, but he might as well have been speaking Chinese to me, whatever they are. Then you can't organize them according to any extension. What do you do? Well, you have combinatorial principles. And these combinatorial principles are not for individuals, but not for individuals at all. No, no, no, not for individuals. Consider all the math. Take two and power. I wish I didn't have a recorder. You can structure an object precisely by choosing among the properties that are there and make them into a structure. Nevertheless, the usual presentation... I don't know. There's a, there's a, there's a... Intrinsic job. Go back, go back to the beginning. The ideology of structure is needed to pass this point. Yes, yes, I know. You know that a given property is a structure, and in the background there's no given property.

55:00 Yeah, but nevertheless... But there are lots of properties. But there are properties of that structure which are immediately presented, which are called intensive... Intentional magnitudes, the thing you actually see or feel. This was a candy at the time. Nevertheless... Okay, but since a lot of it is actually presented that way, when you're up to the level of abstraction where you're dealing with individuals that can't be identified immediately, in other words, by the fact they're red or green or wear a hat, in other words, there's local ways in which you don't have to consider the class as a whole problem, there's even a single problem. Okay, but my point to these distinguishes any two given points. So what is your point about the last lecture today? No, my point is that the way you then have to organize it is actually common for... I mean, you must consider maths. You're forced in that situation. So that was my point. There's nothing else you can do about it. And you're saying that? There's nothing else you can do because the objects aren't presented in this local way. That's my point. That's exactly right. Well, the only real way of doing that is category, which is essentially a... Well, it's just extensionality, isn't it, at that level? Yes, it is. No, I ask. I'm not trying to delay it. This is one reason why I was pressing Bill on this issue of how you express extensionality at the top of something. No, but look, to explain why, to explain the matter in general as to why it is. That you need category theory to organize, to present this theory of science. It's quite difficult to explain. Yeah, of course. Dualizing is the truth about it. Double dualization. You know, the, the, the adjunction map is, is, is marked. That's all it is. Yeah, but it's marked. Yes, that's the crucial point. The problem is to cut that down and make it suggestive as well. Okay. You know, so which function, which measure... And when you do that, you know, we're going to...

57:30 And when you do... Sorry, can I, sorry, can I get clear about this? Well, it's what you teach them then. Yeah. Right around explaining to them how difficult it is. I didn't say how difficult it was. I told them how simple it was. But I want to get clear on... You know, the corollary of what Bill's going to say. You have to get as soon as possible to the idea of quotient space. This is absolutely, this is absolutely... And once you've done that, you see the way... And the existence of... They won't understand anything. Actually, introducing the idea of function space is not difficult. You can motivate that very easily. It's just their own intuitive understanding of space itself. But the point that Bill just made, the point is that when you specialize to the category of abstract sets, then of course there's this monic. ...which is automatic, so that's the point, that the components just become the point, as far as I'm concerned. Yes? Hang on, I want to make sure I've understood clearly the point of the bill. This is the case where... Yes, this is the one I wanted to ask you about. All the business it has to offer is a Cartesian product because the customers are divided into roles. So this is a fundamental... You know, law is high school algebra, but because of the fact that it's an isomorphism of sets and either functions, I have to give you a completely classic, clear explanation of it. So, anyway, isn't that an everyday experience? Yeah, anybody can understand the point about stacking the dishes. Some Chinese, in other businesses, different kinds of dishes. Oh, sorry. Some Chinese restaurants say eat in or take out. Others say eat in and take out. This is the fundamental. So how do you explain this from the point of view of the clients or, you know, either I want to eat in or I want to take out. On the other hand, on the level of what the restaurant has to offer, it's a product because they have to deal with every possible map from the client set, all the clients, into the kind of meals they have to offer.

1:00:00 A Giddey-Math space's exponent is divided as a sum, itself is divided as a product. So the sum, the order from the point of view of science is the product from the view of the restaurant, yeah. Explicitly explained, not just in terms of words, but in terms of clear mathematical operations. But the basic operation is roughly the same. In fact, it takes us in one instance from the... And in the case... From the goddamn first day, man. Yeah. And in the case of, and then you, and then you introduce the category. Don't tell me what I think. I'm in the philosophy department. I don't do, they don't do this kind of stuff. I have no idea in which department you're traveling. I think it does. I think it does in practice though. I'm sorry, but I think that the orders of magnitude are more ignorant than the ordinary peasants in the fields. Okay, but nevertheless. Well, if it's that bad, I agree you have a problem. Well, I don't really have a problem. To bring it back to the point that the bill is making about the co-discrete, discrete, co-discrete spaces of Cantor, the, this is, I'm never going to be able to get out of this. This was because of M.O.D. that the Chinese were not going to take. Yeah. Or take out. Yeah. That was driving to Watertown, New York, the home of the Ellis brothers. The wrong M.O.D. I saw this sign saying eat in and take out. I've puzzled about that for a while. They're driving down the street, you know, the freeway, and I realize, well, of course, from the restaurant's point of view, exactly right. It's a set of maps. What they have to offer is any map in that book. And what's the meaning now of any of them? Any sort of map, you know, a pair of maps. Sure. On the other hand, short to hand, on the other hand, short. And when you're introducing this, as you say, on the very first day to the very first students of the class, you can introduce it so easily, you don't even have to call it the function space that's going to throw, you can call it, in that context, you're just motivating, calling it, if you like, the demand space, I mean, the space of all possible demands where they find, and then make them understand that, of course, it's a mathematical problem, a very basic one. No, but I'm just speaking pedagogically now.

1:02:30 But I would like to understand a little bit more about the theory. Yeah, exactly. I'm doing something. Who said it was unsolvable? Who said it was unsolvable? We'll get to more detail. Anyway, I really want to understand the lack of answer. I really want to understand the point that... I want to make sure, I just want to make sure that I'm, I confess that I don't know if I say it, but I'm reasonably happy. Hobbit says that a function is just an order there. Thank you for your attention. I still just want to get clear. Sorry, sorry, John. I just want to get absolutely clear, just to test my own understanding of how inter-functional space, that's why it's really begun to fight. It's like John Paul Jones, you know. I thought you were an emeritus, sir, I've only just begun to fight. I don't know what this is. I wish this wouldn't make too much of a fuss about it. Yes, yes, I understand that. And the root of understanding that I actually traveled down through that transition repeatedly throughout these years. There's also the gratification of writing a book. I'd like to do that in a variety of ways. It is true, but what was that? Once a clarification of some kind has been done, then it's true, you do wish to, whether one is doing it in the right way, that may or may not be the case, but something that you've understood, you do wish to show, the understanding, in some way that's intelligible to other people.

1:05:00 Now the question of how correct, there's another separate issue there. Jesus Christ. Oh, that's gone, I'm afraid it's gone. Can I just ask those one question on the, just to test my understanding? So in the context of this construction, with obviously taking the function space, the understanding of the function space, taking absolutely center stage in the exposition, you introduce, you introduce them to... Countess Lauter-Einsen, Cantorian, Cardinale, abstract sense. In terms of the restrictions on the maps from the punctual space into the... I mean, just explain to me again how you would, in explaining this, I mean, I think I understand it, I hope I do, but in explaining it to kids who are taking it for the first time, how you would explain the specific case of the Cardinale. Well, no, I'm talking generally in terms of the function space construction. Yeah, sure, whatever, wonton soup, whatever. There could be a combination, you could figure it out. There's a certain small number of offers that are possible. I mean, we get quite a lot of them. So what the restaurant has to do in Britain is to be in a position to, to every client that arrives, in the back of the room, to send a meal. You know, our health department, we had no way of knowing what to do with an arbitrary math. No, it's not just that. No, I don't mean that. I mean that simply those are how you train the professionals. Now, actually, clients are divided into two parts. One takes out the other.

1:07:30 Carnivore theory in action. Yes, yes, exactly, yes. So therefore, we see that one of these maps is actually divided into two parts. These are typically saying, for every ethan client, for every take-up brain in the angiotensinic system, and who are under-contaminated, and these pairs of options are equivalent to simple functions that evolve. So, m to the power c1 plus c2 is equal to m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 times m to the power c1 plus c2 The other is the hand representation, where the hand on the field is actually the ball, hence among the math. You explained this, of course, as the proprietor of this restaurant. Well, I think you should have it the way it is. That's right. Thank you for your attention. And the way in the concept of the Lauter Einstein, the abstract system, you're motivating the understanding of the way that the points function, the structure of the components, you can then...

1:10:00 You can first, if you like, think of an interpretation which is reflected in the relationship between the components. I think it's absolutely a beautiful way of looking at it. But the space of science, as you said, is divided into two parts, so it's not connected. Maybe there's two parts that are connected. Exactly. In other words, maybe there's no property in the suit of a category, maybe there's no... No, sure. Well, maybe there is, in which case... This depends on the exact determination. I'm sorry, I was only asking you to raise your voice because of the background noise. Well, it would actually, but it's probably a bit late for us to move now. We could have done that. Actually, that would have been a good suggestion. We could. What a good idea. Let's do that in class. But in fairness, John, this is exactly the sort of thing that philosophers in a philosophy class should be able to get straight away, because after all, it's precisely... The issue of, you know, which is the more fundamental, the geometric or arithmetic, which is the how can one express everything. Because they're given all this propaganda from Russell that, of course, everything is out on the arithmetic and that there is only... You have to have yourself this purely logically-aggregated notion of object.

1:12:30 When it comes to the language, it might be deceptive, we pick up an awful lot from language more or less automatically. We don't understand the language at the deep level at all. Which is why we have always... We live in a world, we actually live in a world with a spectrum of details. I've been working ever since I learned the language. We'll find back something more like this example. Which is why beginning philosophers are always thrown by the logician's assertion about and or. Well, that's not how I use it, you know. It's precisely a Cartesian product. Yeah, well, I've thought about this too. Not some. Not some. I am abstract. Well, I am abstract. But anyway, we have... A suggestion has been moved... Sorry, sorry, sorry. That we move outside for the coffee, because it's nice and it's quite warm. Yeah, yeah, yeah. Oh, I find it warm. Bill and some of the others would like to go outside for coffee. I know. We'll help. First of all, let's make sure there are enough seats. We've been battling for years, aren't we, on the same side of the same page. Hang on. Sorry, excusez-moi. Excusez-moi, c'est possible pour nous de prendre le café dans le terrasse? Oui, oui. Parfait, excellent. Yeah, no problem, coffee on the terrace, no problem at all. We're okay here, aren't we? Perfect. Let's have a coffee. It's an absolutely fascinating discussion. But coffees, we all want coffees. We all want coffees. Is there anybody who doesn't want coffee? No, I don't. You don't? Okay. Cease. Cease, madame. Do you have a glass of wine? Bill, can you sit down and tell us this? You know, literally trying to satisfy my needs. Yes, yes, yes, sure. But, I do know an example. Yeah, the long chart.

1:15:00 Well, actually, it's a porpoise that has this other kind of structure. Ah, okay. Namely, J.L. Sweeney. Yes, yes, yes, yes. He has this differentiation, or puncture. It's actually puncture. Which is, you know, causes... But he has assigned to every species a formal power state, so it's an objective deflation of formal power states, but it does satisfy precisely the Leibniz rule in terms of tensor. So the derivative of the extensor one, or his tensor, whatever you call it, is the product of species. It's not the Cartesian product. Even though it's in a phocos, it's a very special phocos. There's other hom and tensors. So, of course, the direct sum is the Cartesian product. When he speaks Boschian. Oh, sorry, back to where you were. I read it. He's going to come up with his first campaign to see if he's going to take us to a point. So this is small. And there's a little bit of the... Everybody has to figure it out. It's one or two elements. Yeah, yeah, sure, sure. It's quite, it's quite, it's quite shaky. And when I say hom, it's not the exponential. It's the hom that's adjuvant to the exponential. Which is a convolution. The domain of the function category consists of permutations. So you consider all possible ways of taking the sum to... The permutations, domains, and convolutions, and such convolutions like, it's actually a such a thing as the ground base and the structure of the region, and those structures are probably from the category of, as I said, the work that they do, that they cooperate with each other.

1:17:30 Oh, you're looking for a code, you know. What's my beginning code? So, a function. So, that's the... Well, the C is for all the functions. So, a structure is a function. So, the C is for all the functions. So, this is the only case in which it goes further. I don't want to cut across what you're saying now. With respect to this palm, which is adjunct, they use it for tens of times. And the palm is adjunct. And the differentiation is never done. Anything but a palm. Thank you for watching. This is obviously extremely important and I want to understand it better, but I just want to ask you, when I got into the conversation back there on this whole issue of the priority of function spaces in the exposition, you just discussed with John a point about, not about the Leibniz rule, but about... There was some claim that John was making about the identities, there was some claim that John was making about the identities, there was some claim that John was making about the identities, there was some claim that John was making about the identities, there was some claim that John was making about the identities, there was some claim that John was making about the identities,

1:20:00 There's the truth value of them all, and if, in the internal logic, it's the case that if you had two figures of the same shape in any space X, there will exist a property, i.e. a map to omega values, so they're not indiscernible. You could discern the relation. The distinction between these two figures by means of an intensive quantity. That's what aliveness is all about. Now this enriches to a general statement that the object X maps monomorphically into the double dual. Namely, if you consider all possible maps from x to omega as an exponent on omega once again, that's why it's a double dual, there's always the canonical map from x to that, but that map is actually a monomorphism. Meaning the two figures, the two X, if they are different, we will see that they are different. If you store it on the level that it should be stored, it's time for another dialectical claim. Oh, a fine point. Crucial. It isn't always true. Namely, instead of the object, you have this double exponential, omega to the power of omega to the power of x. I don't know if you see what I'm talking about. Let's draw a picture. Yeah, I'll put that somewhere. Sorry, Colin, it's okay. Oh, actually, I've got a microphone. How about that? This is a subject of the canonical math like this.

1:22:30 Right. So what is it doing? It's assigning to every figure an X of some shape A. It's assigning a certain thing, a form, a canonical of X, is actually the... The name of a certain map from omega to the x to omega, from the x to this to this, and it's that map which takes any property, any map from x to omega. Of course, in the case where the figures are specialised to points, reduced to points, you've got it's going to be satisfied automatically. What's going to be satisfied? The identity. No? Okay, I haven't understood any of those. Fine. It's all... the best way to look at it is in terms of, you know, we have a Cartesian closed category with just some funny object to make it, so that it turns out that if it's a true value object in our world, it's a topology, then it would be a conjuction. Yeah. So this is really just the evaluation, also not an ad. So, the point is that if X, X1, P, and then, you know, a story of 2C. That says exactly the next one and the next two are indiscernible, because the fees are the mechanisms we have available to discern. Yes, sure, I understand that point. These are, well, in the language they could just be the predicates of the language. Yeah, if you think of it as a form of language. Well, a formal language could present precisely... Could present in this, yeah. So, okay, so then Leibniz's rule is this implies x1 equals x2. Yeah. Which is, you know, the statement of this map is a monomorphism. Right? This is just what you get by applying this map to each of these, actually. So if they become equal after supposing them as if they were already equal, that's the definition of monomorphism.

1:25:00 But now there's another one which may or may not be true. For omega. You see, you could choose some other object, some other, like real numbers, like physicists say you measure things, you don't just talk about true and false, whatever. You could choose another type of object and you could ask the same problem. Can I distinguish arbitrary figures by means of measuring with respect to arbitrary properties that are valued in that other object? Right, which in the case of thinking of it as predicates in language, just be, you know, are there... Yeah, there's no need to complicate my presentation, but the point is that's exactly how you always talk about it to philosophers, that's the problem. I'm thinking pedagogically. You can now see that there is another version, you can say a kind of refined mix, which is true in some topologies and false in others. Let's say we're working in a topos E. We define the set of lamps between two objects, for example, an omega. This is the point. We're really in a context where we deal with... There might be totally lacking cohesion, in which case they'll be just like abstract sets, so we can consider the function space, but we can take only the points of it. So intuitively, these are just the maps, literally from X to Omega, never mind how they might be parametrized by other methods. This is the usual notation, just the set of maps, yes. Well now, precisely in this situation, we can exponentiate spaces by sets, right? Where S is in U. X is an E. We commonly talk about taking the s-fold product of X, you know, like a Vespa 2, for example, which is a quill, you know, so this is defined to be just, well, it is X to the power of something, but it's X gamma upper star, you know, because this is, you know, this is a gradual character.

1:27:30 So say that last bit again, because I'm going to catch what you're saying. This is a geometric morphism of topos. Therefore, there's the left adjoint of the points function, which is like the insertional. This is a Cantorian idea. You can strip off from any cohesive space something like this space of pure points. On the other hand, any space of pure points can be considered as an extreme case. So what this notation means, more precisely, is you first have to put it into the same world as x in order to conform the exponential. So this is what we get. We get raising to the power S as an endo-consonant V applied to any x looking at another plane. Right? If you have any x raised to the power S, this is the genesis, so it's an endo-consonant V for each S down below here. Precisely defined that way, and commonly just denoted this way. It's like when you have variable quantities and constant quantities, usually you use the same name, and among all the functions there are the constants, so the constant function 2 is just called 2. It's not called, even though, precisely, there is an occlusion homomorphism in the domain of the constant laws. Well, there you don't have to worry about the additional possibility of the parameterization of additional variation. You know there isn't any further variation. You've just stipulated that it's going to be. Yeah, so the notation is equal both. Or in other words, you use the same symbols, lambda, for scalars in all sorts of different vector spaces and different algorithms. Anyway, okay, so, this is always a set, yeah, so I just can catch what the Lou used to say, this is always a set, yeah, this is always a set, yeah, so I can apply this globalization in a different way, I can just raise it to the power.

1:30:00 In other words, I'm just using the discrete part of the first function first. Right, right. Right, that's the point I was at. Because of the nature of the adjunction, any object P, if we look at the space corresponding to the discrete space, the point of P... That's the sub-object of P. It just contains purely the points of P, and the other interesting figures of P don't factor through, they don't belong to this part, okay? So you have that, and so therefore, therefore you can restrict along that, so restriction along that sort of map... Sorry, I'm going the other way. Restriction along that sort of map would take you from omega to the omega to the x, back there. In other words, if you talked about these functionals that are defined are really the true functions, folks, then, of course, these functions could be trivially applied to functions on the street park, so there's a sort of trivial math there. All right. And in relation to the identity of undiscernible, that's... I'm getting to that. No, no, no. So the basic Dirac map into the double dual case. Right. There was one which involved the English people as the originators, but there were others. Again, though, you see from plenty of statistics, you have this idea that, well... If you have two distributions, which are point-like, and they're different, well then the price must be different, or that's liveness again. And so there is a natural transformation here also. This exists, not that it doesn't exist.

1:32:30 But the point is, you see, this is destroying some information, probably. The fact that this one is always a monomorphism might be destroyed for this one. This one might not be a monomorphism, because this is a degrading process, so the degraded version of this discerning might not be so discerning anymore. No, it might no longer be, well, which means you might not have the predicates in your language which allow you to discern.