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

0:00 ...of predicates and all that. So you're degrading this ability to discern, which might possibly destroy the fact that you can discern. So sometimes it does and sometimes it doesn't. That's the part I'm trying to do. But very often for these Grotto people, they're still... This still distinguishes, yes, it still acts, yeah. So there was this is monomorphism, for example, in superficial sets. Right. As it happens, and you have to calculate what omega is and so on and so forth. It's not obvious, but it happens to be true, or it's true for a reflexive graph. Yeah. It's true for all graphic topology. You know this bizarre theory I have about the hierarchical structure of the internet and all that, which I recently found out is not so bizarre because it's almost identical with the problem that all the big guys take very seriously, namely the hyperplane arrangement. Well, I would love to understand that. He did mention that, but I had no idea there was a connection with... I had no idea that there was an internet connection with... And therefore obviously has implications for the kind of organization of data. Oh yeah. Oh yeah, definitely. That's right. That's right. In other words, if you want to distinguish between two pieces of data, it suffices to consider this space of predicate as a discrete space. You don't have to consider it as a fully cohesive space. Even though you're in a cohesive world, the fact that x and omega and omega to the power are quite totally cohesive, the exponent... You can ignore that on the expo, but it's not a monotone. In other words, this version of Leibniz fails, and there are many examples of failures too.

2:30 We could talk a bit about them, but I think you should give this exposition again when we get back, because this is really so, well, this is very, very central to the... ...concerns that any serious philosopher of math should have. And if we're going to get anything serious out of talking about these very general positions of John, of the continuous and the discrete, then this is the kind of thing we need. I'm really sorry I didn't bring in my recorder, actually. This would have been very useful. But I think I've got, if I can keep this, I think I've got, I think I can reconstruct this. In fact, to test my understanding, I will actually send you a resume of this to make sure I have understood the key points. Categorically, it's the notion of a co-generator. Ah, I see. I don't know how that will ever be stated. A co-generator is an object such that you get a monomorphism into the double dual. But then to make it more precise, the internal generator is an external generator, whereas the generator itself is in the category, but the condition of that co-generator is here partially external, where here it's totally internal. Right, so it's like internal sounds stronger, but it's the one that's always true there. Well, I've often heard Alberto in his expositions on this subject of choice and extensionality in topos theory, which is one of the ones I want to get around to talk to you about, make this point about the tradition. Well, I've dug out the paper, so I'll show you when we get back. This condition for one to be a generator is, of course, a different condition in the kind of internal setting of the topos than in the case when it's a subtopos of some enveloping. This one you can sort of, it really is kind of lower order. I mean, this idea that the use of the discrete exponents, when that works, or in general the question that it poses, is just what I was saying. In other words, if you have two maps, the purely scientific bit of what you're saying constitutes anomalous multi-sensors.

5:00 The question is, can these be separated by a mass into some fixed object? If it's a fixed object in the category, what do you think? Is it the case that these being different applies to the exists of these, for which these are different composites of this? I certainly want to understand this better. This is some of the most, for me, this is some of the most exciting, you know. ...exposition of the whole week. I think I've understood that. So this, the fact that it sort of rests entirely on this diagram is more nearly correct if you use the discrete x-axis. Yes, and there... Whereas with the other discrete one, it's a little more subtle. It isn't just a set of maps from x to omega. You're going to have to consider maps from other parameter objects as well, p times a. P times x, P times, I know they're a similar diagram, but quantify over all possible parameters as P, well that's the significance here, and that sort of shows you why it's stronger, because you've got, for all P you've got this condition of equality. Well, of course, these are easier. Not of course, but it's easier. Whereas if you only have it for P equals 1... I don't know if you've noticed this, but this is going to be very confusing for me to reconstruct if you notice that by this kind of definition, could you possibly do that on another page? The set of points is just, you know, gamma star of y. There's nothing but e of one y. The set of maps from 1 to i are the points of 1. This is a trivial special case of this definition. It's compatible with this idea that the points are the maps from 1. Just as you generally get an object of view, you can totally treat it as an abstract set in contrast to cohesion and non-cohesion. I just thought I should write that under this. Yes, so this is the case where you... And this connects with the adequacy-cal-adequacy.

7:30 Yes, it does. Cool. So it's just the case where points are adequate and cal-adequate. No, it's only half of adequacy. Ah, okay. But now we come to the measurable cardinal. Because... Now do it on another page because I'm never going to be able to read this. Yeah, okay. Because I want to study this and test my own understanding. I'll never do it if I can remember the order. Thank you for watching, please subscribe, like, share and comment. Are you not going to do a couple? No, we'll go. Okay, but I want Bill to finish this. And then I've got to go and send my passport to Russia. Okay, well, you can join. That's the entrance down there. You know where the entrance is, don't you, Angus? Angus, the entrance is down there, exactly, yes. You get the tickets in that little... Where the kids are standing there, that's where you get the tickets. We'll see you back at 5.30. Okay, right. Just finish this, Bill, because I absolutely want to... You know, of course, that he was the son of a Viennese banker. Stanislaus Ulam. Ah, I didn't know he was the son of a Viennese banker. Brought his brother into the U.S. And his brother became an anti-Stalinist expert. He was a huge brother of Stalin. Oh, I've come across, I never knew this was the same family. Adam Ulam, yes, I've never read it. Adam Lua. Didn't he happen to know that he was the same guy? I must talk to my friend Peruz who's writing the book on the Cold War about this, he's very wonderful. Adequacy is the dual of it. Let's sort of, continuing here with adequacy, let's say co-adequacy.

10:00 Adequacy is the dual. Right, so the thing is this. It's x. We know Leibniz Principle is valid, that's the first step. Any object, any space x, is part of the space of all, you know, judgments made on arbitrary measurements. Think of these as arbitrary. Sorry. Not a bit. The castle closes at six o'clock. The castle closes at six? Well, I've asked them to be back at 5.30, so it should be fine. Oh, okay. Yeah. Well, no, they've got two hours to finish it up. What time do you say it closes? Six, that's okay, they've got two hours to do it. It's only 3.30, not even that, it's about 20 past there, I think. No, we're okay. Thanks, thanks. Sorry. That's why we're going to have the free time to go round the castle now and then have the evening session after. Have a late-ish dinner. So if I think of these functionals, the capital C, as I said, they're sort of judgments made on arbitrary measurements. Yes, yes, sure. You could say, well, of course, if I started with a point, then the corresponding judgment, well, a delta, a giro, the corresponding judgment must be rational, must be very, very, very rational, just because I know I started with something concrete. So now the interesting question is the converse. If such a judgment is appropriately rational, Does the thing really exist? This is the converse that I've perceived the point. Is the point really there? That's an issue we're going to have to address. So how discerning does he have to be? Different meaning of discernment. How discerning do these functional things? This is second-order discerning, actually. I have to be, in order to guarantee that you really saw something, that particle wasn't just an illusion, because this, the full double dual would allow for all kinds of illusions, and this is huge.

12:30 Tantor's theorem says that omega to the x itself is always bigger than minus. And now we've even done that twice, so it's twofold bigger. So it's a double, yes. Twofold bigger. So mostly, from this point of view, mostly it's irrational, in the sense, well, I mean, in the other sense, of course, it's rational. I suspect the people who say that quantum particles violate the identity of indiscernible would be terribly excited at this point and think that this was the machinery they needed to explain that. But, yeah, we won't go down that particular... So this raises the idea, you see, that by considering the functionals, which are suitably to be determined, suitably rational, this is the subalpha, the minimum requirement is that delta factors to it, you know, so this is still delta, basically. If hom is a strong enough restriction, then this will even be an isomorphism. Sufficiently rational, it will, in fact, be uniquely deterring a world. So, it becomes of interest, what does this mean? It's a restriction, it's a restriction. Well, notice that there's a kind of, an obvious kind of restriction. In other words, on the trivial side, anything coming from here will obviously have the following. Any algebraic operation on omega itself, even an i-ary one, which is a good thing, maybe i is 2 or 3 or 4, an infinite set or whatever, but for all such, you see, you can consider, you can consider, the thing is that omega to the x is automatically an algebraic structure of any type of omega itself is, a product of algebra. I'm going to show you how I can apply this to theta on the space omega to the x. And then having done that, I can apply my function on here. On the other hand, I could apply the function on the first.

15:00 I would take the one to omega, but I would still depend on the parameter i. And so I could consider this square here. Now this will be, this whole, this will be commuted if c equals... You know, the Dirac at some point or the other. So, you just, you just, you've got functions of two variables, and you evaluate, first in one variable, then the other, or the other way around, you get to the same value at the end. It's complete cryptology in the algebra of Cartesian closed categories of functions. So, the idea is to consider, you know, the, call it HOM sub those data or something. Meaning that I'm going to consider all those functions as Psi, such for all theta in this big theta, this thing holds, for F of theta on Psi, F of theta on Psi. I call it F for Fubini. It's just like Fubini. The Fubini question of integrating a function of two variables. One way is to integrate one variable first. ...trailing along the other parameter, and then integrate with respect to that parameter all the way around. So the two sides of the Kolbini theorem are not always equal. It's a unity and identity of opposites, in fact, in a literal sense. But therefore those particular elements for which it's true have a great interest. Or in this case... Those elements on one side, such as to all elements of a certain class on the other side, you have the Fulvini relation. So this is... I never thought it would get that way. These are all obvious questions. Once you see, oh, Cartesian closed categories, okay, these are just elementary, most elementary, but the fact that you, it's really the only account, you only encounter them after you've slaved through, oh my god, Callaway additive measures, oh my god, you know, all sorts of counter-examples and so on, but they're really very elementary things.

17:30 Wouldn't it be wonderful to write an introductory textbook on differentiation and integration starting from Cartesian post-categories and just doing it right? Wouldn't that be a fantastic project? It sure would. And the thing is, you wouldn't have to be a cutting-edge category theorist, do you? You'd just have to understand. Well, good. Good basic understanding. Right. But it's the point that you were making with John, and it's the point that even philosophers should be quite able to get at. You start from understanding function spaces, and then you get to the very end. But of course, starting from, you know, starting from constant sets, I mean, you're, of course, you just have so many layers of... There's an ulam of x. Yeah, I was going to say, where does ulam come in? There's a statement that there exists a psi, even inside you. Which is not. Which for all x, t is not equal to pi over x. And the traditional case, which was, you know, in response to a serious question raised by Banach. Banach did most of this, and then Ulam did a little bit, and then Tarski did a little more. Somehow the mathematician Bonhoeff got left out of the commemoration, but Ulam and Tarski turned it into this question of abstract safety, known as measurable cardinals. This means that X is measurable in their sense, if you touch your eye, which is precisely not measurable in my sense, because here is a measurement C which failed. So the idea is that actually this was in the case where this was how it's known. That was the traditional case. So these C's are what's called measures. They always use the term measure, including this negative aspect, that it's not a point in that context.

20:00 They always mean that there exists this thing which is bad, and it's not really a point. Bad from my point of view, and good from their point of view, because they want to go on up and speak. So, it's a fairyland. Yeah, yeah. But you notice that even, suppose there are, suppose this is two, like, essentially. You don't, there are... Uh, you guys are gonna need the key, aren't you? Yeah. There are these ghost elements. Oh, hang on, wait a minute. I'm sorry about that. So, you wanna, you wanna go back into the house? Yeah. Okay. Do you wanna just lie down for a bit? Oh, you need to make the call to your vet, don't you? Well, if you can, or do you want to go back to the hotel and make it? No, we'll do it in your place. Okay, sorry. Is that okay? Yeah, it's okay. Well, you know, I mean... It won't be long. That's okay. I mean, sure. In your circumstances, I'll of course make the call. Just leave the key in the door. I've got to go and get my passport. The coffee so I can fax it to these Russians and yeah I'll come back in a minute I'll say. Well we're going to reconvene at 5.30 so yeah okay. Sorry don't continue. Can you excuse me while I respond, you used the loo. Yeah okay. Would you like another coffee? No it's okay. Excellent. There's binary operations. Okay d'accord. Oui c'est bon pour vous? Okay d'accord. What do you want? The visa? Here. Here. Okay. Okay, sir. Thank you. Thank you very much. Do you have the address? Okay. Yeah. This is binary operation. Then you can measure finite, arbitrary finite sets. It's sort

22:30 of a typical thing that, you know, the Ulam coordinates are way above the size of the, of the... Right, which is why it's such a misnomer to call the measure of a car. Oh yeah, especially in the off-road because it's just a phenomenon that you jump way up. Yeah, yeah. Do you understand French? No, this is not the French card, but I think it is. So normally it works, no? I have another card, if it's not... Okay, it's easier. This one, I know the code. I was hoping I could use that one, so I got that, so... Okay. Yeah, sure. Okay, well, I'm going to take a moment. Okay. Okay, merci. No, that's okay, because we had a good meal, didn't we? It was 7,000... 117, so that's... Thank you! So, planning would take care of it then, eh? More than take care of it. More than take care of it. Right then. Well, thanks very much. From countable all the way up to measurable. In other words, countable succeeds in the usual circle. It succeeds to measure real numbers, Hilbert space, operators on Hilbert space, and all the rest of it. You have to go way beyond that to find something that countable can't measure. That's sort of the fascination of it. But if you go down to the lower level, with two, you can measure all finances. All finite. In other words, any function on truth values over a finite domain, which preserves, say, even median intersection, it is determined by a point. What it's called is an ultrafilter. These functions are like ultrafilters, but an ultrafilter on a finite set really is just a point.

25:00 But now the idea is we lift that up. The ultra is ultrafied into having also I-ary operations where I is not necessarily finite and then we can measure a lot more. So, merci, merci, je vous laisse. Merci. But there are interesting examples nonetheless of things involving infinite sets where you, you know, functionals would just satisfy the finite-ary operations. So there are, you know, we need to compute, commute on the assignment. One is the following. Here's an easy one. Take the plane, any space equipped with a non-negative measure, like area, area in the plane. Now, you can, there's a function that assigns to every... Let's say measurable set. So we're in a category where all mathematics and mathematics are negative. So negative to the x is negative to the x. For every measurable set, we assign the judgment. True if it's got positive measures. False if it's got zero measures. It's non-negative in any case. So many of them, as you know, lower dimensional sets for sure and lots of others. For the weird things, too, they have zero measures, and then there are the nice fat ones, and the ones that contain at least one nice fat piece, so true means to be sufficiently fat, and non-negligible if you wish, and I notice that that does preserve, to have no zero measure, yeah, a positive, yeah, well, okay. If you've got the dichotomy between zero and positive, then it's the same thing. I prefer to say positive because that persists into other contexts, so you don't have to symptomize a dichotomy. But notice that that does preserve finite unions. In other words, to say that the union of two sets has positive measure is equivalent to saying that at least one of the two has all. Okay? Because if they both had zero measure, then their union would have zero measure as well, using the fact that the measure is a sub-additive or a teaser additive.

27:30 So there's no way...so it's...a union is substantial if and only if one or the other is substantial. So the or is preserved, but it's only a finitary or, of course, because as soon as you start looking at countable sums, you get them to... You know, these are paradoxes of measures, too. That's finite. So in that sense, the concept of being everywhere, sorry, of being, not being everywhere, being somewhere, is a generalization of being a point, 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 point... The actual points do indeed satisfy that they're big unions, but this concept of being substantial or some set that's at least somewhere non-trivial is a natural generalization of the idea of a point in that way. It's fascinating to see 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 thing. I want to 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 live through a tantrum. I know. ...kept me talking and, oh well. 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 go over this material. I'd be fascinated to get Colin. But you really gotta 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 mean 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.

30:00 It starts off by saying, you know, Mark really had something to say about calculus. I've got most of this from the notes I made in your Florence talk, but I still would like it in a... I took care to write a text, way back in 94. Now I remember, in 94 I got your Grassman paper, and three or four others, but the... The calculus paper wasn't one of them. I didn't realize that you'd written it up. Well, certainly send me that when you get back. There's loads of other stuff as well I wanted to go through. 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 with respect to a function, not with respect to a variable. But also the discussion of van der Waals equation. You mentioned van der Waals equation, it's illustrating the hot and cold adjoints. No, well the hot and cold adjoints are just the extreme cases, which I said I have to find an analog in van der Waals, which will explain how one passes over into the other. There's the map which shows how the... Hot states will go over into cold states, but the beginning and end of the process. Because you're absolutely right. The process actually takes place past the willing point. Condensation of steam was of course an excellent illustration. Engels used it, didn't he, I think. Of course. Most people who attempt to... This is an excellent example of quantity changing into quality. This is equality. Hot and cold is equality. But now there should be qualitative description of the dynamic process through time, as van der Waals found. But also in the same paper is a discussion of Knowles, Knowles' concept of bodies that can undergo... No, I would have, no, I definitely haven't got this paper. I really would have liked that. Beyond the usual notion of topological space, I like the description of a body.

32:30 Well, I really do want to study that. It's not actually on your side. Is the surface of the ball part of the ball or not? That's the point. That's the question. That's a real contradiction. Sure. That's precisely accounted for. The ball itself has always got two faces, the inside and the outside. And each of those has its own... The face that looks outside participates in all the interactions through the contact forces. The part that looks inside has things like mass and volume. Oh, it's wonderful. I mean, I just can't wait to read the start of this paper. And of course... I've just convinced myself that this is really a great paper and I don't understand why nobody has read it or made use of it. Well, one person's going to read it very soon. Now son go see that castle. You go straight over there son and you buy a ticket in that place just across by the bridge. Yeah, where the little light is. Sell them in there. I'll see you at 5.30 Bill. Just gotta go and get my passport copied.