FW Lawvere

0:00 Now this is a very, very categorical or geometrical algebraic type of thing. It means that hom sub 3 or 3 to the x comma 3 equals x. Given a set x, consider any process which, to every partition of the set into three parts, tells you which running element it's supposed to be in. It does this in a consistent manner, namely, any of the 20-second permutations of the 3-ohmic set. It's compatible with that. When you commute these indices, then the Goosley choice is similar to commute. Then, of course, if you had an actual element, this would arguably be true. So the question is, is it such that the combination is true? Any such procedure of measuring can only measure an actual element. Well, that's the definition of a finite set. Nothing to do with Zodiac. Yeah, I'll stop for a minute. No, it's not about building, huh? Yeah. Anyway, anyway. So, axion. Accent of the axion of natural numbers exists. Axion. There exists a category. I mean, it existed in the sense of being an object in the medical language. Yeah. Look up the equivalence. Classified as just these sets, among all the discrete categories that there are, those 7 acts, 1, 2, and 3 are the ones that are felt. If you go to the next book, consider instead of 3, the natural number. Or, better, counting sections. The 20 of the 20 are counting sections. And that's a real mathematical object, fixed work. No, other than that, I'm not going to comment with you on it. It has all these operations, like atomic sections. See, ring theory is all just atomic sections and just guys, but it's like you could describe this thing in various ways, but it's basic high school mathematics. You'd say it has one object, and it has various individuals. So, instead of hom sub 3, call that thing, I don't know, r, hom sub r of r to the x comma r. In other words, you've got something set up. Suppose that for every function from X to R, think of it as a complex plane, any map from X to a complex plane, then this oracle will tell you which real number is the value.

2:30 And again, this should be invariant with respect to all continuous smooth algebraic things that preserve atomic sections into itself. So if you follow your function about any such thing, then specified fake points should be similarly moved. Well, I suppose x is such that there is no such thing except an actual point in x. And one attempt to call X measurable, because you just see how to measure its money. But in fact, it's the opposite. It's the first, you know, it's less than the first measure in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, in the, Instead of being really measurable by the process of straining things out, these endomaps are used to test such oracles. So again, axiom, there should exist a category, it's got to be a topos and so on, but it's a category in the universe, an actual object, which represents exactly those X, those disputes. Now there'll be others. The universe is particularly enclosed, and you can take the category of small sex to its own power, and that's the ordinary type of random convergent meeting and so on and so forth. So, this is the outright definition of small sex. And the assertions of the category too. It's unique. The topos of finite sets and the topos of small sets are both uniquely defined by this sort of algebra, geometry, duality property. Which is absolutely fascinating. And so you don't have to worry about all this kind of mystical class set distinction. You're still talking about classes, but there's no iteration. Yeah, exactly. It's not the mysterious difference between classes and sets, exactly.

5:00 No, I'll show you the distinction. With classes, we can have subclasses as a universe. In fact, we can just use it. If you talk about classes of x's, what you ask is what they represent. But we just demand that certain ones be represented as well. No, no, there's always that distinction between elements of the universe and some classes of the universe. But when I say universe, I don't mean something like V, because in... See, in Veril, it's very, it's very disconcerting, because in Veril Bernays' set theory, V is not the actual universe of a model, it's an element of a model, right? It's just an element of a model. The universe is somewhat bigger because of the universe. I mean, the universe of a model is that actual theory. So, this is what I... This should be worked out without hierarchical iteration. Completely idealistic. Only gods could do it. In fact, set theorists used to talk to each other that this demon was more powerful than that demon. It's true. Well, you were saying that even Grotendieck had fallen into this trap of basing the diverse notion on iterations of size, thought of in that way. Well, you were saying that you actually have got a suggestion for a foundation in the, you might call it an orthodox foundation, a sense of foundation, which is quite different. And where presumably all these maps can be... Well, you'd have, given that this is precisely based on the algebraic and geometric properties, you'd have a kind of semantics, the ultimate semantics of this construction would be equations, I think. Yeah, yeah, yeah, yeah. Yeah, that would be much more... See, I really don't like this whole Audi trend, small, even the word small in a different way to... I think Bernays himself... I was really happy when Royale and Murdyke revived what they call algebraic set theory, a way of talking about some sort of pseudo, I mean, it's a V which exists as an element, but of course does not contain all small elements.

7:30 I mean, actual set theorists don't even take that way. They know, of course, that B and the power of B should exist. And that's no more inconsistent. If there exists a model of general Bernays, then there exists a model that you have B and the power of B. But with the usual, full, lab-declared properties and not some truncated one. Well, this would be a foundation very much in line with the spirit of your remark, I'm at Francis Borsola's meeting in that little place in the Ardennes, where he had... Ah, Haute Bordeaux. Yeah, Haute Bordeaux, that's right. But really, one should think of set theory as essentially a branch of algebraic geometry. Oh, yeah, yeah, yeah. Oh, yes, yes. So there's this role, so I had three... And I had some objects that are called comic sections, or theory of rings, or complex planes, or some specific mathematical objects. But you could take any math, you could have a similar idea, you could take any object like that and do double dualization into it with respect to all its endo-maths. I mean, you see, when you talk about ultrafilters, it's the same sort of thing, except they use binary applications in their intersections. It's a general phenomenon, you see, that if you talk about A-ary operations on the object V for these purposes, you might equally well take the set V to the power A, and only endomorphism is that. So the modality of endomorphism, you can always think in terms of endomorphism, you just take the dualizing object to a slightly bigger. That's why I took three instead of two, you see, but you can use two, but then you have to talk about binary alphabets. Yes, of course, but you have to strain out the real ghosts, you see, and you want to strain them out. Yes, so it makes better sense to choose something that moments might be bigger.

10:00 And Isabel, I got the whole idea from Isabel. Yeah. His idea was you could use catables, the natural numbers as his test on it, but again... If you use two of the natural numbers, you can use reals and reals. It's like conic sections. Well, you see, it's actually a rather, I think, stone's observation. You don't need to use all pertinents in the math. Just addition and multiplication. That's why ring theory became so important. Why it became important for analysis. But although, in some sense, the natural algebraic theory to do this testing is all continuous maps, all continuous in-area operations, or all algebraic ones, or whatever your category is, in fact, the algebraic ones suffice even in the smooth and less Milner style of the field. Milner showed that points of the ring C-infinity function, in the sense of ring theoretic points, are the same as the C-infinity theoretic points, which seemingly are much more restrictive, I think. Did I say that too fast? For setting up this dualization, the first thing you think of, you have a dualizing object, is to use all end-of-maps of it, or maybe all binary operations of it or something, in order to test the maps from X into it. But it may be a non-full subcategorizable thing, but ring theory is enough even when you're... Your ambient theory is actually much richer for this purpose, for the definition of homomorphism. Of course, the definition of the ring and the definition of the C-infinity algebra is much deeper than the definition of the ring. But the definition of homomorphism... Yeah, it's just dependent on the ring. Yeah, it turns out to be... It's just dependent on the ring. Yeah, which is basically... Yeah, so the addition of multiplication is enough. And so, as I said... I think the most shocking way to say it is that it's a conic section. No, that's really... And then you get in the plane. Rather than thinking of the line.

12:30 You can think of the line, fine, but then you have to bring in binary and ternary operations. Quaternias would be even better. Quaternias are great. With Maxwell. Of course, yes. Actually, it's a very interesting idea, using quaternias. If I can just fact-track about, to ask you something about Isbell, one thing I really would like to, do you want to just have a, do you want to have a small glass? We won't get mad. Yeah, Isbell, John Isbell, the, this connects with my earlier question about your remark about set theory, that the fruitfulness of giving set theory is essentially a part of algebraic geometry. I'd really like to understand more clearly this issue about adequate and co-adequate sarcasticity and how the category sets were picked out by the condition that the points of the things which are the adequate and co-adequate points, yeah, can you go back to explain this again? This is sort of circular because the adequacy really makes sense with respect to the background. Yes, yes, yes. That's why I was saying subcategories other than that. So it's hard to show any natural backgrounds here. We want to talk about the adequacy of a subcategory. Typically it's a small category A mapped into a big category X. And you want to say this A, or this vector, is adequate for X, or co-adequate, two opposite possibilities. But this has to do with their being based on some background. Both of them are based on some background. Now, I don't know which paper that I talked about this in the last lecture, but what kind of a nature this X is? The really neat thing is that you just copied it today. You can find S inside X. So concretely, let's think of X as being the smooth world or the analytic world or the algebraic, this type of thing. The point is that there will actually be some very small a, one or two objects, which are our favorite figure types, let's say, so we're figuring in some x as a map from one of these a's.

15:00 Well, and let's say x is Cartesian closed also. Well, so we're led to look at those objects s for which s to the a equals s. All figures are constant. So those are the discrete objects x or s. You can use that as a back, so it's really there's an opposite between the A's and the S's based on the Galois connection, you know what that is, based on the equation S equals S to the A. There's a natural map from S into S to the A, the embedding of constants. The crucial connection is that's an isomorphism. So intuitively that means that S thinks that A is connected. Well, A thinks that S is discrete. That's a good way of explaining it. You can look at it both ways. Well, I don't like... Perceives is better than thinks. No, because, you see, thinking, you might cogitate and figure out something. Anyway, they used to say thinks, and I insisted they say perceived. Or witnesses, perhaps. It's more direct. Just the direct... Anyway, so... So, you can start with a few, very few A's in that way. There's a sub-channel. Like this. Is it possible to read a little bit? Oh, you've got it out. That's great. This is in that thing you printed out today. Right. I'm sorry, which I haven't had a chance to look at. No, no, of course. It's outlined there. But it's in another paper that's published. Yeah. So we don't need any external set there. It's internal. Provided the x-sort of thing is rich enough, you find this discrete part of it, you see. And that marvelous thing is in algebraic geometry, you don't get sets. You get the Galvatons. It's cheese on fields alone. There's this whole craft about algebraic sets. The algebraic set of a scheme or in one general algebraic space is not an abstract set in the sense of Cantor, not quite that pure, precisely in the sense of Galois, it's a functor on the field.

17:30 It's a functor on the field. There's points on this field and also points on the other extension field, and these all fit together functorially and satisfy drastic sheath conditions, namely that any extension of fields is a covering. Right. And the case of where it's a set, where you've got sheaves on the non-pronged space, because this is a special case of this. No, no. I mean, I'm saying if the world consists of algebraic spaces over a non-algebraically closed... If the base field is algebraically closed, then it does turn out to be cantilever set. Right. But if it's non-algebraically closed, then you have non-shearing on field extensions, and you consider all those... I think Rotby made a mistake in using the word point, not figures, in general. Yeah. Well, of course, in a way, he corrected that mistake by saying, of course, points are never merely points. You know, this was probably what he was getting at. No, no. No, see, there he actually meant these real points. Oh, I see. Well, I... No, he meant, actually, because, precisely, the field extensions have a lot of morphisms. Yeah. So, this, this, this function is not something cumulus. It's, it's, it's for every... Every finite field extension is set, but for every morphism a field is an induced map, and so a partitioner for the other morphisms allows a field to happen. But see, this doesn't have to do yet with curves and even infinitesimals. It's just points. Points in one sense, naive in a very narrow way, but richer than cantons, which is how it should be, I think. This gets rid of a lot of the unnatural things about how algebraic groups aren't really groups and all that stuff, because they are groups. The unnatural step, and the way that most people talk about algebraic count, is in effect to take not that functor on fields but its direct limit and call that the points. You mash these together, but since that category of fields is not directed in the categorical sense of directing, it is maybe in the sense that two things can be amalgamated. But not in the sense that two arrows can be co-equalized. You need both those clauses for the categorical definition of direction.

20:00 And so the direct limit doesn't preserve even products. And that's why they say algebraic groups are not groups and all that stuff. They remain groups if you take just the underlying set in this sense, but not if you go further. Because the only way to go further is to take the direct limit. If you took the inverse limit it would make some kind of sense, but then these would be, you know, these would just be rational points. If you want all the points, it has to be a function, not a continuum. I guess so. Whatever you mean by richer structure is above that, of course. No, I haven't understood this before, and I'm just going to try to get my head around it now, because it's dizzying and very interesting stuff. I do see the point about the Galois, which has got the Galois structure inside. It's the underlying thing. For example, the pi zero function preserves products. If you go down to that level, but not if you go further by this direction. To get the good intuitive properties, you need that category of Galois sets instead of the pure sets. Instead of the pure abstract sets. The case of the pure abstract sets, so when you talk about the case of the adequate. No, no, no, no, no. Sorry. It's because of digression. I'm sorry. I could. No. You take any reasonable category there. Category of cohesion. No. Cartesian is always kind of good. And a few objects A. So it has a great parameter. You can take the spectra, the dual numbers, the tangents. Or you can take the line, which is really just the end of the line. Just a very simple... By the way, please, please, will you have at least one mouthful of that for you, I guess? Please. I've got to taste that.

22:30 You don't provide food. Oh, okay. I will ask her for a sparse meal. Okay. Because I really do want you to taste it. Yeah, I'm good. But, sorry, please, go on. This is, you know, this is synthetic, isn't it? So synthetic means that you... Put yourself out in the world that you imagine and try to write down axioms that describe what you imagine and then you might find that you have to change it, see, that basically it started from the objective vision rather than some building up process that you later figure out. It's not like any notion of algebra comes first. I think even the notion of human algebra came before the thought of history. Propositional calculus is embedded in order to present the abstract algorithms that were implicit or semi-explicit in Boole or in classical logic. You have this experience leads to this vision. There's nothing mystical about this. This is the way thinking works. The experience leads to this vision and then you try to make it explicit and you go, oh, it's an algebraic structure. I can even present it. The Boolean algebra is a good example. In other words, the kind of the Boolean algebra is the abstraction of the propositional calculus. Yeah, yeah, it's not graded backwards. Just the opposite way around. It's the presentation. The propositional calculus is the presentation. Yeah. And that's why there are many different propositional calculi, depending if you tell a check or stroke or blah blah blah blah blah, but only one version of the algebra, you know. Many presentations through the world life. What is one of these levels? What did I say? This is the basic vision of what the planet is like, meaning all the objects that we have, not just one space, it's a category. According to Bernoulli, Moriewicz, Voltaire, it has to be Cartesian flow, it's so-called. That's why we have exponential space, we assume that. If it didn't have that, then it wouldn't be mathematics.

25:00 Too bad for a category of topological reasons. Okay, so they're in this setting. Now, moreover, and that's another feature about such a vision, I must tell you about them, Cantor's notion of self, Dedican's notion of self. According to Emmy Nurture, who actually wrote down, did I tell you this? I actually wrote down, she was actually obviously agitated by this question, which was after Penrose. And so she talked to Bernstein, somebody who actually had known Cantor and Dedekind, what did they think a set was. It's fantastic. It's so fantastic. I didn't know this at all. This is very, very interesting. It's been available there for years. This is incredible. But then, adding to Cantor and Dedekind, as cited by Merton, by Bernstein, there's a postcard that Max Zorn sent me once, which I was very... He was obviously sympathetic to my project and he came across this quotation from Hilbert. So the point of the norm that Hilbert stated is that to define a set you do not have to have all the elements first. This is what he got. You have the real numbers. It's a conception. Or any power set. It doesn't necessarily... You have the idea of the power set, and you have some examples in mind. You have both. We know a few examples of real numbers. We also know, you know, that it's a completely weird field or whatever. These are the two things we have, you see. And then, okay, let me get into, oh, Cauchy's sequence was, oh my god, that it concussed, whatever. We have these two levels of... The objective is vision and a few key examples. We know a lot of more examples like that if we look at it more closely, but even a very few of those are key. Zero, one, minus one, two, e, and pi. That's something I really need to know. Anyway, so the same idea with such a category, a vision and a key category. There are a few examples. Now this might be the objective idea of motion. Tangible, representative. It might be just that. Or you might think it's an interval or an arrow, an abstract arrow.

27:30 A category of categories works very well. Just an arrow. A few, let's say 3 or 4, 1 or 2, whatever. Things that are clearly examples. So you let these be A. Now, consider all the S's for which S to the A equals S. Just those things. That's a subcategory of the smooth spaces and those free ones. We're going to use that as a background. We'll use that as a background. Now, see, from this definition alone, it has all sorts of marvelous formulas. It's automatically got sums, products, and explanations just from the formula. Look at that formula to make a substitution. You've got to have sums, products, and explanations. And it also preserves any projective limits that might happen to exist, so modulo the adjoint function theorem, which is a reflective subcategory as well. So there's a reflection, the pi zero concept, which automatically preserves products because of where the thing is defined. So there's a reflection that preserves products. Now, Cantor's bold idea was it's also a crow reflector. Every space has its subspace of pure points. It has its pure points which you can constrain as its subspace. The part of the space without any cohesion, just think of it as another object, not yet a discrete object, but mapped to the given object of the universal problem. So that's it. So now we have the unity of opposites. But again, thinking about it in terms of the properties of this as a category of space, it's a category of space. So now we have the set-up to consider the adequacy. Consider those areas that we started with, or any sub-channeling, or even any factor for a small category. Any more correctness, we'll take an internal category and dispute things and talk about an internal category. Never mind, just say two or three objects, whatever. A few objects. Are they adequate for the whole category? What does it mean they're adequate? Well, given an object X, we can define the pre-sheaf with value that's not set somewhere, but with these discrete objects.

30:00 A pre-sheaf on the category of these chosen A's. The obvious A. Namely, two of the A's. To every object x we want to associate with the general object, Hilbert space, interdimensional sphere, any object perhaps far from those that we've thought of so far, x. We define a functor from the a's, the s's. So take x to the power a. Now take the Cantorian part of that. In other words, the discrete set. The discrete set corresponding to x to the x. So now this is a functor from x, the category of all the x, the original, our ambient category, into the category of s-value pre-sheaves on the x. I mean, this x and this d x. Is that a full functor? If so, it's adequate. In other words, can we detect? Well, you could ask for it because it's faithful. Faithful. If we have a map from X to Y that induces a natural transformation, you know, between these two factors, whose values are S's, and matter with respect just to maps between the A's. Just with respect to maps between the A's. That's the thing. So this might be faithful. In other words, can we tell the two maps between some interesting spaces are equal or not just by looking at whether they become, they look, the discrete version looks equal after you test it with some A-shaped figures? Yes, yes. You see? Isn't it clear? I mean, it's so clear. Yeah, now I've really, I have actually gone... It has got two or three steps, and it's very clear. And I see exactly how the case of, you know, the accuracy of Koenigsegg points, and that falls into, you know... So then, but then the really, I mean, the faithfulness is... Fall into place. Faithfulness is the incredible thing. This is the much stronger Isabel adequacy condition. Namely, suppose you've got two spaces, x and y, and then you look at the... Discretized versions of the A-shaped figures in each are being induced. Suppose you have a natural transformation there. Here, naturality is quite crucial. It's natural with respect to the maps between the A's. So the maps between the A's are called adequate.

32:30 If, given such a natural transformation, there is in fact an actual map from x to y that induces it. So that we can now go, turn around and present the whole thing by defining our category to consist of certain pre-sheets on A with values in them. Which would correspond somewhat to the use of practice, except we've got this S, which there are lots of different choices for S. Now, co-adequacy, you see, you just, you consider figures and incidents related. Consider the geometry of incidents and figures and incidents related. But by the way, I've started actually thinking of it more in terms of vibration. It's the same idea I used to do, but the geometry of an object X. Consists of figures, which are maps from the A's, and incidence relations, which are commutative triangles. And then, any map in the category is continuous, quote unquote, meaning that it induces a map from figures to figures that does not tear the incidence away. Now, so the dual situation, when we consider figures and incidence relations, we consider function and algebraic operations. Between two A and A prime, you may have a map A that calls an algebraic operation. It will operate on any map from X to A and you will map from X to A prime. A might just be the cube of A prime or something. Ordinary algebraic operations are included in the commutative triangles of the other sort. So that's the algebra. On the way of operations, a map from X to Y induces a homomorphism in the opposite direction, obviously, that's just, you know, on the homomorphism, I don't have any functions. And so co-adequacy would mean that there's a, you know, these A's are enough to detect everything about the arbitrary object and maps between them in algebraic terms. The algebra and the geometry are both parts of the master geometry. The co-adequacy aspect is really centered at the algebra and the algebraic aspect is more centered on the equations and geometry.

35:00 So the typical algebraic geometry typically isn't. Algebra is not actually co-adequate. I mean schemes are only locally affine. So the so-called affine spaces, the algebra is A, will be co-adequate among the, via the spec function. It will be co-adequate among affine spaces, but not among all schemes or in the whole. So that's sort of the typical thing that, at least the original notion of figure that you thought of to describe the thing, the original adequate notion will not be co-added, although you could always start climbing, you know, build up something where it is, by using more operations. Yeah, but you'd need to put more. But coming back to this memorabilia, that notion of finite and small, in terms of pre and conic sections. This is precisely a co-edit grade saying that in the category of finite sets, three is co-edit, one, two, three, again, and in the category of small sets, the natural numbers of the real numbers of the complex, something like that, is co-edit. In other words, small is so big that the algebra, the high school algebra, that's co-edit algebra. This is more like Koch with the way the speed sets themselves behave. Even within that, there's room for co-ad. That's right. So to answer your question, you see, in some sense, yes, to talk about adequacy within the category sets is vacuous because that's already in the background. But co-adequacy, by contrast, is not at all vacuous because you can find that in algebra, you know, a system of algebras. Spaces that serve as function types within the category of sets. But that now is getting into this business about measuring sets. There's nothing mysterious about the idea that you could measure them. Whether the word measurable applies to the objects in this category or to the category itself is a totally different thing. Typically the set for this one is the category itself. Whereas it should be whether the objects themselves are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing, are doing,

37:30 I was very much helped actually by that. I was very much helped actually by that. I was very much helped actually by that. I was very much helped actually by that. I was very much helped actually by that. We'll give you credit for that. The lecture seems to be making a strong impression on the academic physicality. In other words, I have to think of the right words. It's very much in the spirit of the traditional idea of the field extension. Of course, it's a non-trivial thing to do, the extension, but so are two hundred properties are kind of barriers, not a problem, but in a way, it's not a problem. And that's the scene that's meant to love you. Yes, it's just a bar I caught in there. Can you explain what's out there? The bar has this notion of... Ah, like a... You take all of that and it's covered. Oh, I see. All of that, all of these maps are epimorphic. You see it's the... The field's reversed. The field's like one with one of them. Oh, I see. Sorry. We've been walking all day looking for you. Well, there's no other way to get around. Oh, you can just join the site. Oh, yeah. We're destined to be epics in the topos. The homosorphic? In the topos. Is that what I have on here? Again, this is very... We'll know what's on there. It's a conceptual...

40:00 Tomato soup. We need... I'm sorry. Thank you for your attention. Thank you for your attention. Yes, exactly, which again makes your point about Sinclair as being a kind of branch of algebraic economy. There are a number of different types of mathematics in the field of physics, such as quantum mechanics, quantum mechanics of the universe, quantum mechanics of the universe, quantum mechanics of the universe, quantum mechanics of the universe, quantum mechanics of the universe, quantum mechanics of the universe, quantum mechanics of the universe, quantum mechanics of the universe, quantum mechanics of the universe, quantum mechanics of the universe, quantum mechanics of the universe, quantum mechanics of the universe, They stopped on the street, and I don't think I've ever seen them.

52:30 Well, we did basically these one-on-one with each other more than that. They used to, they had one every single year in 1968, where you'd be a scientist, and then when Boyd Yee was here, he didn't like it, so they let him be a scientist. Well, why do you want him to do anything? Everything's fine. Yeah, they let him be a scientist anyway, but now there is no person who does all the research. But, instead of using topology, we can use two of these. And those all are adequate. Now, what, what, typically, again, typically in the case of a proton decarbonizer, we can have a pair of these in the street background. The street background is just an ordinary set. But then as Colin showed, you don't have to use all management sites. You can find very, very thin categories. All of these have been mapped onto the site, namely, you can have any category with no end in the street, supermodel, and so the one that's in the C-train often congratulates and then he saw a children and then he passed. Any site, any Courtney Turkle has a site in the sense where the site is not in the category, it's a story that maps into it. Oh is that right? You're all equal. The generic figures of those self-intersections are only, you can compare them, but you can't, there's no need to confront it with self-intersections. I was amazed. I thought that the whole... Confront it with self-intersections in a non-trivial way. But, he holds to the old view that... It's the easy way to figure out... Which is a little bit surprising. Mile one. And then, if the opposite happened the next day... I've been saying, you know, that if you didn't have a science, you wouldn't have to have any problems. That's why it happened. There was no material in the science that was meant to be internal. And I'm just so amazed. It's so kind of you.

55:00 Oh, really? Yeah. You know, there was a discussion that they wanted to see. They didn't leave. All these great things happened. No kidding, I could try to listen to it, that could go right, and you know, we might be able to use it. One of the things that we're trying to call now is what should you accept? How do you know? Well, I would be saying that there's always something that you do, that you do have to have, and that you're trying to accept. And Peter was saying that. No, he launched it because, and this guy talks about Khrushchev's secret speech in which he's going, well that had some impact on him, but what had a bigger impact, and he never wants to mention it, is Khrushchev being overthrown in 1964. And then Mao says, oh my god, they're going to do that to me, because I'm no more powerful than Khrushchev was. I need to be as powerful as Stalin was because they denounced him after he died, but they couldn't touch him when he was alive because he was powerful and I'm not. And so that's when he got the idea of the cult of personality and all this. And the very next year, he started laying the groundwork for the Cultural Revolution to encourage Stalin. And he said at the time he was treated like an old Uncle Edward, you know, with respect, but he wasn't relevant. There was a guy who, as chairman of the party and chairman of the Politburo, never attended Politburo meetings, led one, chaired them, unless they were talking about purging somebody, but he was rarely okay king anyway, and, you know, didn't want to be politician. He had no interest in policy. He didn't discuss policy. He was never there. He couldn't care less what the policy was. He cared about power. ...and getting it, and holding it. And that's what the Cultural Revolution is all about. Because he had been marginalized, he wasn't growing in the country, but partly it was because he was getting a lot of money. He had no interest in governing and he had no interest in power. I agree with that, but how is that different from what the guy said? The guy said he wants the Cultural Revolution to keep evolving the vision. So people say, oh, he wants it because he has no power anymore. And by, you know, the very fact that he had to go outside to party. To get people like his wife, who had been, they really took the page and curdled us, super, super, super, super, super, super, super, super, super, super, super, super, super, super, super, super, super,

57:30 I'm not sure how they happened to do that. It was just incredible. So the actual words were Western democracy, of course, the Persian tyranny. Oh, yes. Wholeheartedly jumping into the war preparation. In legal oversimplification of super-undescribable files, versions of these are the identical enemies. Well, as my friend Mike, who met me recently in New Zealand, there's got to be possibly the worst film I could have ever seen in my life. It was so grotesque. It was actually quite common for different people, depending on a culture, to take it to Western India and be adorably surprised. My friend might do a bit of a trance here. He wrote rather an interesting piece shortly after the surge. The first paragraph is just this passage from Thucydides, where he describes how in the Ecclesia they voted for the search of Jamaica after the Syracusan expulsion in Korea in 1921. They had either to cut their losses or to call out the war. They had to kind of, you know, double-arm them, and of course they designed them double-arm, so they were sent out further at the same time.

1:00:00 That is not, that is a not true story. I was going to revert it. She did not wear blouses, she wore mouth suits, and she was never alone with Habermas. And he just points out to them, and the citadels, and Habermas spent a lot of time with that girl at the same time. She was taken around, not by John Key. From the end of the debate, it was quite clear that Habermas, I reckon, was going to be the Athenian empire in the West. Next on the list is the question, does one need to be left on the line of the law? The countries have not had a lot of relations nor arguments with some Chinese politicians who are qualified interpreters. I think it's one of the things you've been told me very well. I only wish I'd learned the math as well as I've learned mathematics. And I was about 25 minutes in, I was the worst guy, his name was J.P. Peckham, and it talked about the run-up to the Nixon visit, how it came about, and it talked about the Nixon visit, and the plane that everybody was trapped in, and the optics on the plane, and all the reporters were in it, and there's some major reporter like Walter Cronkite at that one meeting. In 1972, he shot an olive across the room of a scholarship. Utterly unscrupulous. Liked it. Cantilever. In China? Non-standard. There are many cosmologists that reject the standard of physics. His own position is to make it clear that in the beginning it was the people who also rejected the physics. There are certainly sensible things to say about the flimsiness of the evidence. But I learned that. There's one thing I learned from this talk. There was a French scholar, the old... ...predicted back in 1896, and a couple of Swiss astronomers in the early 1930s, who predictorized it on the background temperature.

1:02:30 Should the impact of the crisis of one of these two countries in the 1930s produce the same size prediction? 2.8 percent. The present estimate is 2.7 percent. These were the predictions from the view that the red chip is not... Sorry, no, the background. The red chip is not... No, no, no. Well, this is a separate issue, but this is the point about the fact that... But it could have been a little... The background radiation is not a constant. It's a cosmological origin. All the poor people in the Big Bang camp who were addicted to the background radiation got it wrong by, in most cases, a little more than a minute. For instance, Penzias, where is Penzias? I don't know about Christ, but I don't know. You need to tell people how good the dishes that you can sell and you can't let the chef come in and do it. And so anyway, there's this young woman who was convinced that they'd find it around 50 degrees Kelvin, which is over an order of magnitude out. And none of the people got it. They all predicted it must be above 7%. When they detected it, she must have shown me something in the picture, because she said to him that she thought she was complimenting him, and I knew that she thought she was, but she said that in the moment when she said, oh, you look much older now, you didn't have a picture, you're right, and she said, you look like about 58 in your picture, and you look like about 68 now, and he said, well, actually, I'm 75, so I think it was a little complimented, but anyway, after we went down... Well, no, the 1973 prediction. But I think Guillaume got it within about three days. He was the reporter. Guillaume. I made a good second note. Very, very interesting. So this was extremely interesting. And he also made the point that all of the people which were sent... They met a woman. Like art, like art, like art, like art, like art, like art, like art, like art, like art, like art.

1:05:00 This was part of our speech, and he was speaking very rapidly in French, which I was very much like to understand. But the gist of his answer was, well, I didn't let you see me because I wanted to concentrate on the computer and have a strong observational reason for this, but we've had a lot of conversations, and I don't really know much about the secret of his work, the secret of his work, I really don't know much about the secret of his work, but he was a good person, and he was a good person, and he was a good person, and I understand a lot of the point about the secret of his work, which is that he had a... He was saying that at the time that he wrote, that was very little within the observational evidence. But the observation evidence now is not very long-term, so therefore the answer was not very long-term, although it took a significant amount of time to be thought of at the moment. I absolutely agree. I had a very good talk on science conference. Very interesting. That's right. Thank you sir, I gave a talk. He gave the talk, and it's good to know that he gave an excellent talk about people, and I'm so interested to actually ask him for his thoughts on this topic, because I was wondering, actually, a couple of months afterwards, why he organized a meeting on the topology of physics in 1974. Gosh, I would have gotten into a conversation with him about compass theory. Thank you for your attention.

1:07:30 I really was impressed by this guy. I don't know, I don't know. I thought he was staying here all the way until the government department. He never mentioned that. He was staying there. But clearly, you know, he was staying there. But also, he was right. Paul, he's supposed to provide for the guys and he doesn't. I showed him last year how to make a problem. He never knew how to do it. He just shoved everything in under the mat. And it looked terrible and he never knew how to do it. And I don't think we're all going to have to do it. There were two things he said which rang a bell. One of them was he was talking about the In the office and the new office in the office you keep in your pocket, something out of Jesus Christ. We're very, very reliable. We're very, very reliable. He said that Lemaitre had tried to solve mathematics in the late 19th century, and later Lemaitre was asked to pay for his lectures, and he said that Hyacinth had not In fact, it can be very complimentary and free to describe his Grants Book of Physics as mediocre, but it was quite clear that his Grants Book of Geology was kind of supposed to be abysmal, which I thought was a good line, but no one talked about it because it was quite clear that it was a witty line.

1:10:00 Physics is a video for this. Theology is a video for that. And how do we do it? How do we do it? You know, if you think about that remark, the suppositions that are allowed in the text are the ones that we're trying to please. Thank you for your attention. When I had done some of these things, talking to these guys, yeah, in our day and age, one of them was a student, and then the other one was a teacher, and the other one was a student, and then the other one was a teacher. And the other one was a student, and then the other one was a teacher, and then the other one was a teacher. And the other one was a student, and then the other one was a teacher, and then the other one was a teacher. It's a discussion group where people post questions and then whoever wants to answer answers. And Nancy knows a lot of things. Yeah, it's so easy for me. It's info that I have and it's given me a job somewhere else. And I contributed to it because once I found... The last part of the program that David Carroll did, he didn't actually have a character, clearly, but it was more weird. Fair enough, he only had one character, but it was really just to kind of just look forward to it, right? There's even more about the question. It's two years ago. There's a village in Hawaii. There's one road that goes into the village. There's a parking lot that you do not enter. You do not enter. And we called the place and they said, go ahead and go inside the building. And there's this road. And it's really the width of a car. And it's a rock. It's a big rock. And you have to go in thirds of the way. And it's not wide enough to pull over.

1:12:30 Thank you for your attention. And then there's this sloping pasture with goats, a green pasture. There's no barrier to the thing. One of the goats jumped in the water. Is that right? Yeah. It's somehow an old barrier. They converted it into a B&B, into a farm and restaurant. And for one, you didn't get one cent. And you need some country plants produced by the father, grass produced by the father, goat cheese produced by the father. At the time, we were probably the only established forms of cleanliness here. We experience every day all around us. All around us. Thank you. But still, I love it. When people hike there, it may be a nice hike, from the village it's about two kilometers, an hour walk, and that's why they come up to the park where it says people don't go any farther than they have already. It's a beautiful walk. It's really amazing. It's one of the best parks in the whole world. What is the name of that gorge? It must be where those cliffs were. That must be that gorge. That one with the really deep river. Of course, except science. Because otherwise, I don't think around that region there's that kind of thing. It's not called science. Well, it's not called science. Well, it's not called science. Well, it's not called science.

1:15:00 Well, it's not called science. Well, it's not called science. Well, it's not called science. So that was the astrology conspiracy. So that was when the problem started before that. Well, if your idea of taking whiplines as a leading expert, then we have to accept whether it is a god or not. So they have very good people, Randall, in their reviews. So, these are reliable reviews, you know, marketing, and also, many Americans, they would know about making the hospital bed, or whether it was used, or the growths, or the advantages of Randall, that you don't get from a, from an academic brochure. No, I don't, I don't know. I don't think he was. Not the original. He really was. And also hotel reviews. What they recommend, what they tell you. I don't know. Stay away from them. Category theory is the original algebra. Well, please let me answer that. When I had my, it was supposed to be about something like that. I had a very minor operation for a benign carcinoma on my forehead. It was a squamous. I don't know. All the Americans are going with that. And the dermatologist said he didn't have it. You don't really need anesthesia for it, but you can have it for a long time without being high strung by nature. Anesthesiologist was this really nice guy, this young guy. And he was putting in the, I don't know how to even give him the right word, a stallion or something, and shouldn't breathe. And he said, you know, well, you're American, you like to live in Paris. And I said, yeah, but I also, I love to travel and... You know, around Europe and Barcelona. And he said, oh, my wife and I live in Venice and we've been to all this. And I said, oh, I have La Baladras. You know, I have the greatest place to stay in Venice that confesses. I'll accept them when we rent the apartment and it's a lot too expensive and it's a lot cheaper than the company. We have a kitchen and an antique furniture. There's a company here and I started talking to him about it. And he said, oh, I gotta write that down. He's supposed to be here. And he got his book. I don't know if he got it, but I told him to look up the website. It's not hard if you get the name, if you get the name right.

1:17:30 You see, we were supposed to say, and without any official clearance, they'd be part of the answer. Oh, well, I mean, it's still not so important. It was way important. I think I tried to engage you in conversation to see when you're using it. I was actually conscious of all of it, which was good because the surgeon was also using it, and he's writing next month. The client came around. He came around. On behalf of his wife. Who was that? His ex-wife. That's right, lady. So he was so satisfied with what he was doing, and he did a great job, and he had put these big slides in there, and you don't really see them, so you know what you're looking for, and have a light on it, and he lined up the cuts and the wrinkles, and it was one of Baez's main points, the Siegel of quantum mechanics. But, of course, that's probably stupid. Well, you're true. Just because you're true. And the same, even worse, when the other people just do something like that. This is traitorous behavior. You could be justified. In this case, it was just about them. You're really hard to believe that you can have such a man in his element. Oh, it's a beautiful picture. You know, when we heard this, we were really pleased. We had the guts to say it to her in her life, but not to wait until she was dead. I mean, that was really, really difficult for me. And my friend, my childhood friend, Calgary, her grandmother got very old and had to leave. And again, non-stop, there were doctors and nurses coming in.