Discussions, incl. FW Lawvere, C McLarty, JL Bell, others (contd.)

0:00 No, that's not the right answer, but I think it's a good one. And that's the code I gave you. I could use that one.

2:30 Thank you. Real numbers, Hilbert space, operators of Hilbert space, and way beyond that, we can measure them. If you go down to the lower level, with two, you can measure all finite, all finite. In other words, any function around two personalities in a finite domain, reserve, say, even union and intersection, it is determined by a point. What it's called is an ultrasound. Thank you very much for your time. So you, you know, functionally have just satisfied the planetary operations that are, you know, up to the stars. If you didn't compute, commute, follow, there's an instant one you follow. Take the plane or any, any space equipped with a non-negative measure, like area, area in the plane. Now, you can, there's a functional assigned to every, let's say, measurable factor.

5:00 To every measurable set, we assign a judgment. True if it's got positive measurements. False if it's got zero measurements. It's non-negative in any case. Many of them, if you know, lower dimensional sets for sure and lots of other totally weird things too, they have zero measurements. And then there are the nice fat ones. If you want to contain at least one nice fat piece... So true means to be sufficiently fast in non-magnetism over two years. Now notice that that does preserve... I have no zero in that one. Positive. You've got the dichotomy between zero and positive and it's not the same thing. I prefer to say positive because that persists into other contexts where you don't have a... But notice that that does preserve finite union. To say that the union of two sets are positive measures is equivalent to saying that at least one of the two has more measures, okay? Because if they both have zero measures, then the union has zero measures as well. And using the fact that the measure is a sub-addict, there's no way to... A union is substantial if and only if one or the other. The concept of being everywhere, sorry, of being, of being a pump, because a point has this property that it's in a union of two sets, if and only if it's in one or the other. But the points, as the actual points, do indeed satisfy that big union. But this concept of these similar natural generalizations is the idea at the time.

7:30 Yes. Fascinating to see in how many directions one does see points as acting as either the kind of special case or the beginning of some natural generalization. This is wonderful. I would like, obviously, to keep those and study them. That was the second page. Oh, they go to the castle. I thought you might. I was going to say, son, you're not going to turn around to your daddy this evening and say, I didn't go to the castle. You really threw a tantrum. I know. You kept me talking and no one. You really threw a tantrum. No. Bill, thank you. You earned your trip to the castle. This is fantastic. Really, well obviously I wish this could go on for another three or four days at least, but one of the things we should try and do is to... Let's go over this material. I'd be fascinated to get Colin. You've really got to look at that paper. Yeah, yeah, I'm sorry I didn't know it. Well, that's my fault for not having chased it up. No, I never had that. I definitely did not have that. Well, you didn't. I promise, I would remember if you had, because believe me, I've got... I've even had them bound, believe it or not, so I do know exactly which of your papers I've got, and that's not one. I've got most of this from the notes I made in your Florence talk, but I still would like it in... No, I remember, in 94 I got your Grassman paper and... There are three or four others, but the calculus paper wasn't one of them. I didn't realize that you'd written it down. Well, certainly send me that when you get back. And there's loads of other stuff as well I want to compare it to. But this is wonderful. Sorry, this is... The paper has not only a complete treatment of calculus in two pages, but also... Also some very interesting remarks about physics. And the theory of vector field differentiation. You mentioned van der Waals multiplication as illustrating the hot and cold adjoints. Well, the hot and cold adjoints are just the extreme cases. I said I have to find the van der Waals multiplication.

10:00 It will explain how one passes over into the other. There's a map which shows how the hot states will go over into the cold states. But, you know, the beginning and end of the process. Because you're absolutely right, the process actually takes place at the casting and whaling point. Condensation of steam was, of course, an excellent illustration. Engels used it, didn't he, I think? Of course, yeah. Most people who attempt to do mathematics, this is an excellent example of quality changing into quality. This is equality, not in cold, it's equality, also in my formal sense. There is a quantitative description of the dynamic process through time as render balls, but also in the same paper there is a discussion of Knowles, Knowles' concept of bodies as render balls. No, I definitely haven't got this paper. I really would have liked that. You have the usual notion of topological space. I really do want to study that. Is that actually on your site? The surface of a ball, that's the point. Contradiction, as precisely accounted for, has two, has always got two faces, the inside and the outside, and each of those has its own face that looks outside, participates in all the interactions with the contact forces, but looks inside and has things like mass and thought. I mean, I just can't wait to read the start of this paper. And of course, Boundaries is just a really great paper and I don't understand why nobody has read it or makes use of it. Well, one person's going to read it very soon. Now, son, go see that castle. Why do I do that? You go straight over there, son. And you buy a ticket in that place just across by the bridge.

12:30 Yeah, that's where the little light is. All right. Sell them in there. I'll see you at 5.30, Bill. Oh, you haven't actually started yet. No? Okay, good to see you. Oh, I got loads of copies of that. I made about six copies. Of this? Oh, you saw them, did you? Of what? Of this. Oh, are those the ones that I just made, or have you just made some all yourself? Just four of them now. I thought I had made some before. Oh, okay. Well, there's probably even more sets than I thought in that case. I was just going to do a set of them for everybody. Okay, let me go upstairs and get the recorders ready. How did you enjoy the castle? Well, I loved it. It was really great. I must visit it some day. Hi, guys. Has everybody enjoyed the castle? Well, I'm sorry to have kept you waiting. Oh, Leo, sorry. I'm sorry. I took those with the intention of copying them. Is it okay? I had to give it back to you. Well, no, that's okay. But I was going to copy them now. Is that okay? Okay. This whole academic publishing house is now...

15:00 Well, talk to Dolph Gabbay at this UKCL out there. Well, I'm going to say something fairly brutal at this point, which is that not for the first time I hear an academic whinging about the outrageous cost of books. Published by, you know, the major academic publishing houses. Why the hell don't you do, you know, what John Byers has been campaigning for on his website for years now? First of all, why don't you support these new publishing houses like KCO, like Polytechnica? Who is publishing a book with Polytechnica? It's better to have a book published by Berkhauser or Kluwer than by Pollinger, but that's... All right. Okay, well, it was a particularly unfair thing to say to you, but as a general observation, I stand by it. Yes, well that's what I wish everybody would do. Good.

17:30 I'm on the board of a series called Categories. He's coming to the meeting in October, so you'll meet him there. You'll have a chance to judge for yourself. I don't know. I've not met him. I've corresponded with him. He's very young. I'm very happy to say I haven't seen the final thing yet. I mean, they only cost 40 euros. I mean, it's really, you know, a 300-page book is pretty good. No, we want to get the, the, the, the, the copyright out of, out of Donald and Marko. We're so very, we're so very, we're so very, we're so very, we're so very, we're so very, we're so very, we're so very, we're so very, we're so very, we're so very, we're so very, we're so very, Well, although I disagree with almost everything else John Pyatt has said, in particular what he says about N categories, I do approve of his campaign for people to, you know, to stop reviewing for sitting on the editorial boards of journals that are published by these outrageous price-gouging corporations. But I realize it's difficult to buck the system. Right, well, it's now the afternoon of the 16th, and I trust everybody had a pleasant break this afternoon, and a chance to get a few things done. Can I thank everybody for their superlative exposition this morning, and particularly thank Bill for the quite wonderful tutorial he gave me over lunch. I'm hoping he may be able to say something about in this session, which I'd like to see devoted, since it'll be really the last complete session, I mean, we might be able to fit in an hour or an hour and a half tomorrow morning, depending on what people feel like, but it's the last effective full session of our discussions, I'd like to devote to a discussion of broader general philosophical questions.

20:00 John submitted a list of topics which he thought might provide useful hits of discussion at this meeting and the first of his suggestions was that we might discuss the topic of the extent to which mathematics can be seen in its development as driven by fundamental oppositions such as those between the continuous and the discrete, the varying and the static, the whole and the part. The finite and the infinite, the one and the many. But perhaps focusing particularly on the opposition between the continuous and the discrete, and the varying and the static. Perhaps you'd like to say a little bit about that, John, and then I might ask Bill to, specifically in this connection, to perhaps... I'd like to reprise some of the very insightful things that he was saying to me at lunchtime about how one might think, for instance, about a principle such as that of Leibniz's identity of indiscernible, which he thought of as spasticized by the notion of absolute or constant, the purely logical ingredient of the notion of object, as the Phrygian would see it, how one might see that notion from the categorical standpoint and indeed from the point of view of this. General opposition between the continuous and the discrete. Perhaps you'd like to lead off and say a little bit about the fundamental oppositions that you see as driving the development of mathematics. Sure. Well, I think if one looks back at mathematics of ancient Greece, particularly, of course, the Pythagoreans, who, I suppose, identified as one of the sources of the major problems. The way mathematics emerged and the form that it actually took, since many of the basic categories of mathematics were named, provided by the aliens, and they saw the importance of mathematics in terms of opposition. Various statics, one of the main... the continuousness of the screen is something that they, they, they of course identified the difference in terms of geometry and the continuous number of the screen, and of course they, although it isn't very clear, it's not much, well it's not as it comes down in our language as we know...

22:30 We know that for one reason or another, the Pythagoreans came to identify a number that is discrete, broadly speaking, although... Whether it really happens exactly or not, Aristotle isn't quite clear, and he isn't, but the account comes, of course, to Aristotle, as a kind of basic principle of understanding the world, and we know also that when there are geometries, the continuous, in other words, There is an analysis of the continuous developed. They prove that at some point the Pythagorean theorem is proved, but not called the Pythagorean theorem. And this slide, of course, as we know, will dance to the problem of the commensurate. Because it seems that they had claimed that quantities could be reduced, if you like, in the case of measurement, to the comparison of total numbers, and this seems to be an important aspect of their philosophy, which is claimed because it's part of that discovery of the... In the basis of harmony, you know that music, of course, is part of it. Later on, it was part of mathematics, which became part of the prodrivium later. It's often said that the first crisis of mathematics, you know, this is a traditional notion within the three crises, but the first one, in any way, which is also plain, is the inter-clash in some way between the continuous and the discrete, the idea that continuity, in other words, continuous quantity, seems that, in other words, that one can't measure it, the ratio in which character is reduced to... To the greater the numbers, in other words, to the discrete numbers.