Discussions, incl. FW Lawvere, M Wright (contd.)

0:00 Not the Coastal, because the Coastal, they tend to be always infinite. In that sense, probably one could even prove that. Yeah. In other words, if there is no alphabet on the left side... The definition of alfabetism is quite asymmetrical. Hegel said being, impure being, non-being. The non is pure. The non is the non. So you can clear it. Here it's like this. The alfabetism is just as it was from left to right. In other words, given something which is k-dimensional. What is the lowest level for which it's actually also in the co-skeletons? Well, formally, you could equally well ask for the opposite, which usually doesn't happen. The statement that it doesn't happen is essentially what I was saying about all co-skeletons are infinite. They never get stabilized by any skeleton. In the case of the student, we have both alphabets and code alphabets, and indeed the alphabets function is an unusually expected increase by one in the dimension at this stage rather than something else. So can you comment on what the definition of the problem is? Yeah, in fact these are rather more easy to picture too. Australian terminology, actually. They would, in fact, call them, anyway, the globes, as we show you. We call it ball complexes. So, ball complexes are the analog of simplicial sets.

2:30 In the case of simplicial sets. It's S to the delta-hop. All diagrams and sets are shaped delta, but delta is all the order-preserving math, and you've got to find that order. That's what delta is. Well, instead of that, what am I talking about? So it's S to the... It's a rather, rather simpler category. So we call the general thing ball complexes and so the long category is sort of the generic sequence literally of balls. So you have one dimensional ball or two dimensional ball. These are the objects. But the whole thing is generated just by two maps at every stage. The two maps are intuitively this, you see. A two-ball is just a disc. And a three-ball, do you know what a two-ball is? But notice that its boundary is the union of two two-balls. So from the generic two-ball... With the generic two-ball, there are two maps, known as the northern hemisphere and southern hemisphere. And actually, so these maps themselves should be called hemispheres. The maps themselves show that the objects are walls and that the maps are hemispheres and the composition of the earth is a symbol of the universe. And in that generic example given by the Armada investment, the... The sphere, you can define the sphere, it's the push out, the gluing of these two copies along, in this case of a circle, the equator is a one ball, is a one sphere, so you have a kind of exactness at each stage, you can glue together the two arrows.

5:00 Which will not give you the next object, but the boundary sphere of a ball, which is in fact the push out of the previous two arrows, over an amalgamation which is the previous one to that, which in fact is an intersection. So your intersection goes back and gives you the exact object. The union of the push out goes forward and gives you not that object, but rather its boundary. Now, it's particularly easy to get the, you know, to understand it in the case of the three-spectrum. Well, the thing is reflexive because, say, the three-ball one has a canonical map for the two-ball one, and you collapse it to the equatorial disk, right? The boundary of the equator is the boundary of the equatorial disk, and that holds the whole solid three-spectrum. This is a map coming back, just like the degeneracy in the effects of gas. We have one map coming back, two maps going forward at every stage. And, in fact, it's an internal unity of mathematics. That is, there's no sense of speaking of agilentness between these objects. These are objects in the category that are not categories, you see. But the formal definition of a single map coming back... Two maps going up, with composites equal to the identity, it's exactly what you have, more or less, nothing more, nothing less, at each stage, and then all composites. So that's the model, that's the generic, that's the innate sequence of objects. Right actions of that. These are arbitrary combinations of balls and spheres glued together in all possible ways. This is a typical combinatorial object which allows a geometric visualization because this abstract picture...

7:30 It has an obvious interpretation in the category of topological spaces or whatever, so that induces the actual period on the singular and the realisation, except now it's the singular ball complex associated to an arbitrary topological space, rather than the singular, the triangular complex, which is sufficient. So that model is very simple, but as I say, my students show me. So I'll make a question, and I'll just answer it, but it's not related to other things, but you can see how they want to do it. It's an open problem that you have to speak about. You have to talk about the classical. You have to talk about the classical. I'm not sure about that. Not only that, this example even has co-authored. It's the same number of theoretical options, i.e. n plus 20. It's the same number of theoretical options, i.e. n plus 20. Also, also, if you want to go... Instead of taking the alternating sum on the basis of some derivative, we just do the obvious thing, we sort of think of north minus south. We have two things which take their difference, that's all. No alternating stuff. Now actually, you see, most of the examples factor through the problem of the case, because really, given the simplicial set, it's locked together. Sorry, the simplicial thing about proof now. You glob together the odd parts and you glob together the even parts, and then you take the difference and you just do those two globbings separately, you've got a glob, you've got a globbing, you've got a glob, you've got a glob, you've got a glob, you've got a glob, you've got a glob.

10:00 I think that the old cometary analysis, which was invented in the old cometary analysis, which was a special group of groups, was a linear output of the special sets. It is, in fact, still one today. I'll be very changed on this. The boundary of the square is zero. But that square zero comes from taking the difference of two sides at each stage. So I think probably the unionization... The linearization of all these would be the same. It's a valid model. The linear algebra is still the same. Well, this is certainly a different model. Linearization is still the same. Thank you for your attention. There must be a real space for the stars, as we can capture them. I'm going to keep clocking because we have no results or theorems. No, it's not. I'm all about de-generalization. Yes, it's a practical illustration of what you were saying about search and search. That generalization is not just for... Well, I don't know what those end cabinets are supposed to be, because they claim they have applications in physics, but the applications in physics are not real in physics, it's just this kind of stuff, which has been constructed just for this.

12:30 It's mainly in order to provide, effectively, examples of the kind of stuff which would provide applications. The topological quantum field theorems are happening, or at least in mathematics, because there are theorems about quantum physics, and then in turn the infinity categories, even though they don't have any theorems, they're supposed to apply to... In the field theory, I don't know if I have anything to do, but it's really a two-level thing that we can do. Thank you for your attention. I think he quickly got into another degree in statistics. There are several master's degrees in mathematics, and someone who has a Ph.D. in mathematics could easily obtain a master's degree in those two subjects in one year, and then he has a whole broader bachelor's degree in mathematics. You know, one such guy is a vice president of a travel agency.

15:00 It seems that people in that position don't plow back to the financial and scientific science. It seems they're going over to the very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very. I know, but you have no idea of the realities, but the qualitative nature of the reality is that, you know, you have to carry a team to take care of these women, and your wife has to have a personal trainer, and a personal psychotherapist, and that's the sad thing, that they've come corrupted with this whole thing. Thank you for your attention. We're off the couch, we're only constructing, and sometimes it's referred to as pragmatics, chilling homotopically. It's something that will co-stellarly form, and it has all possibilities, and so in particular the possibility of deforming one property to another, it's also one of the possibilities, which is really there, and so it all, so all the whole thing is added.

17:30 What that point is, it depends on which level of coincidentalization is applied, and so again, they can't associate its basis with things that would be, you know, in certain parts of the field, and try to squeeze this on. Spectral sequence, I'm sorry? Spectral sequence? ...or one of those things that I never... Oh, right, okay, I thought you'd actually come to the words I had in mind. ...one of these very useful gadgets that come to mind. I mean, of course, even the... Witten, Huygens, Kleinschmidt, Kant, Söder, and so on. I'm not sure what they said in that. You know, I'm just wondering how much did I gain with that last step so that I could arrive at this place, which I've only wanted to be, and that's what I'm going to do. All of this is part of a sphere, so it is if you're going to solve some equation and fill in an electromagnetic field or something like that.

20:00 It's just the essence of boundary value problems that sort of detach from particular quantities of steel, but the underlying... The essence of the domain space of the quantities which is still involved, the contrast between balls and spheres, has always been the problem of cohomology. But somehow it's always concealed by these complicated things, and they're treated complexly, and the word alternating sum is always the worst saying. I don't remember quite understanding why the word alternating sum was used. Now that the alternating sum is clarified by the cubical case, because a cube has a front and a back face. See, that's sort of the front and back face of the film. What that has to do with the bounding sphere, the box around the cube. So, you know, in other words, it's possible to do something just in a completely nonlinear way. You take the actual, you know, the space, you take the actual space of all three balls in it, There are two spheres in it, and one of these is included in the other, so you take the quotient space, in other words, you've strained out the essence of those spheres which actually were fillable, but you've thrown them away, and you've maintained them, which are not fillable, that is the holes, that's where the word hole comes from, the actual holes, except usually equivalent to them. The holes that are really similar have been identified, so why isn't this resolved in space when we want? It depends if there's any adding or subtracting. Do you like this hot and cold? Do you like the hot and cold?

22:30 No, the use of that technology is very helpful, yes. I think it's much better than the shriek in the other one. The next question. Of course, it allows for intermediate cases. Yeah. Well, hot and cold are special kinds of agile. What kind of adjuvant is it? This is sort of the opposite of the unicad, the opposite of the unicad. One inclusion, two projections. Yeah. It's the same as the dual equation, or the dual isomorphism. Yeah. So there are a lot of those, like already the... The components and points in the discrete infusion are an example of that, too, so you have these four in a row, you have an example of both types of integration. But, of course, components and points in a cohesive space is precisely not an example of equality. So, what's odd is, to which of the examples of equality, abstractly, I mean the ones that arise that way from cohesion. But are they in themselves? Some of these agile triples are special in that way. The ones which should be called hot and cold is what I'm getting at. Hot and cold should not be a universal term for left and right, but rather a left and right with certain further properties which are typical of those derived from the heathen. I think that tries to illustrate the idea very well. As you say, particularly in reinforcing the point of

25:00 There being the things which are examples which arise from cohesion, not just general examples. This is a separate question, but again, since the history alone is... Was it Eisbel who was the first to coin the labels of adequacy and kindness? Oh yeah, that's right. 1960. Ah, so that's where I got you. K-I-G-Y. I say that the K-I-G-Y... Conception of the general method for geometrically analyzing a category was established in 1960. The K is Khan, Isbell, Isbell, not Isbell, Isbell, sorry, both G is correct in being wise in either way, so, putting, putting these ideas together, you're right, I guess. Well, actually, I should have added Cantor or Le Verre or something, because of this idea that you not only have the adequate, hopefully adequate, but the construction doesn't.

27:30 The property of adequacy is the property of a certain construction to make the construction. You start with a small subcategory, let's say a full subcategory, that's one thing. But on the other hand, you have to dualize that into something. And it's usually described as, well, you take the category that makes sense. Cantor already said, no, that's also inside the given category. So, so really, you should specify that as well, you know, and then the whole discussion is internal to the particular, let's say, coercive category of spaces, you know, you have the, in any space x, you have the actual space of all figures of shape a, that's just the exponential, that's the power a, but now since you have the reflection, the points, Cantorian, ...maps from the whole category, reflection of the whole category, into discrete parts. Discrete parts. Therefore, there is the discrete, so to speak, it's not a set, it's an object, of figures of shape A. And similarly with maps between A's, you get the incidence relations just by exponentiating, then you apply the points function, so to speak, and you get a diagram in the... Discrete objects, diagrams of these things. And so the adequacy then means that the entire original category is fully mapped into the category of all these diagrams, by that procedure. Again, it took me some years to realize this, because I always said that the way Israel plays this game, the category of abstract sex is external. And therefore, it's a higher order condition. Every natural transformation is induced by a math. That's the adequate thing. But the every natural transformation is a quantification external to the original category. It's a higher order condition. So it's, you know, as typical in mathematics, you're relating higher order and lower order things, but you can make it entirely lower order.

30:00 If you just explicitly recognize that there's two sets, in Cantorian style, which is part of the category, and I forgot to mention this in my lecture, but it's a crucial thing, there's a natural condition. So you have a subcategory A and another subcategory that's a U for the Cantorian type objects. Well, there has to be a relationship between these two. And it is simply that for any object S and U, so set makes set, raised to the power any more than an A, All of this is really just equal to that set. In other words, the idea of a figure is that it's connected, you see. On the other hand, you can't move in these discrete things, so the motions reduce to constants. S to the power of A is just equal to all the motions of shape A and S, just equal to S. So that sort of orthogonality and that lay between the two categories is part of the analysis. So you have abstract sets on one side, so to speak, which are going to serve as background for analyzing structures. You have A on the other hand, which philosophically is the abstract general now. Part of the reality becomes the abstract general, and the concrete general is this internally defined category of U-valued functors on it. But now the final axiom is that the reality is faithfully and fully included into the concrete general of background U and abstract general A. And it's a first-order axiom now. It's an elementary condition on this whole study. Specifically, if you take A just to consist of T, the infinitesimal... Then you have the whole series of Euler reals and the proper definition of mu, which now turns out to be the Galois sets instead of the totally abstract sets.

32:30 They're already the same thing because they're tied to the Galois. So that all comes out of Galois areas on the line. In the case of the kind of oriental, the streetcar street, the basis is just the fact that we're getting into that. Galois groups might vanish. Galois groups might vanish. Either the Galois groups vanish in the case of an algebraic or closed waste field, but that's just the usual way of describing it. But you insisted on reality, that it's so real. The real closure, the thing that's real closed to begin with, you have trivial dialogue in that sense. So it has to do somehow with the closure. At least the perception that this U reduces to an abstract sense, because even that's small, so it's a little bit of an approximation to U. In other words, I don't think you can prove that U equals Gödel's L. Just by playing with the choice of eggs, you know, you can just take back all the things that are subject case. You enforce concepts of the ultimate, ultimate, extreme. So, in terms of this explanation, of course, it's an orthogonality, s to the alpha equals s, like any orthogonality, it induces the Galois connection, the abstract sense of the Galois connection.

35:00 So a class of, putatively, discrete objects as opposed to a class of, putatively, connected figures, figure types, one decreasing one, decreasing the other, because you just focus on that one condition only. But then there is the idea that mathematics is based on abstract sets, which Ledeckin and Hausdorff have got to moat at how it perceives views by breathing air. Mathematical objects are representations of an abstract general in a fixed abstract background, which is sort of like Cantorian sets. Where does that come out? Well, of course, if you start talking about structures of these spaces, then you can always criticize them. So you get a vast number of... Categories are sort of defined over the same new, which are not in your original category of music spaces, but may be parts of it, or a bunch of categories on it, or structures of an almost arbitrary kind in it, you see, are in turn actual structures in you, once you've got the adequacy which says that you can view the spaces themselves as structures, quote-unquote, therefore any structures over structures are still structures, and so on. That's great. I mean, that's great. That's how most mathematics goes on. And here, perhaps, really enters the place where Harvey Friedman is right. Mathematics and the foundations of mathematics are two different things together. Because within mathematics, all these different categories tend to be connected by adjoint countries. They're sort of commensurate with one another. They have the same foundation used. Which is reflected in the fact that your function is going both ways, with a more or less constitutive example, or a more structured example. But from the point of view of foundations, you worry about varying U itself in a more serious way, so to speak. So, for example, if U is B, then Cantor's L is a part of that, so there's an occlusion function.

37:30 There certainly is no function going back. All of this is really entering a whole new realm of investigation, actually. Of course, you can view all those. Modeled theoretically, all those different ideas are still modeled in some version of abstract sense. Probably a valid view, but precisely within that view, you have to accept the idea of functors in one direction only. Functors that preserve limits but not, you know, finite limits but not infinite limits and all this stuff, you see. So, it becomes another matter. Whether you can actually get at the hypothesis that D equals L, just in terms of the whole figure of emissions-relation analysis of objects, is dubious. Dubious by this, you know, by the equation between the commensurate pairs and the one-way-compared different foundations. There's this branch of totalist topology in the 1930s or whatever, and people found statements about the plane which are equivalent to the continuum hypothesis. No, not that. No, no, no. They're equivalent to the continuum hypothesis. Well, yeah, you're right, the action of choice, but again, this is sort of, well, that's a separate issue. The action of choice is sort of routinely assumed for constants that's almost as characterizing them, even though it doesn't. And it's routinely known to be, at least among people who look at things, right, it's routinely known to be false.

40:00 Anything if it were more interesting. Cohesion or... Cohesion or whatever. Of course. So, right. So, so, so, continuum hypothesis. No, there are statements that are formed, I don't know, I guess with Kinski. I'll say some nonsense, you see what I mean. There exists a non-divulgable family of lines in the plane which intersect. Every one of which intersects every other one and comes in the same place as all of them. Something like that. Statements of that character, but statements of that character which are clearly, you know, statements within what's considered the limits of ordinary mathematics, quantifying over open and closed and countably uncountable sets in a specific place. For example, in the Johnstone topos, this kind of statement would make sense, but it may or may not be true. It depends on the nature, just in a discrete sense, in the open analysis, because you're talking about... The plane which consists of pairs of Dedekind wheels and blah blah. So it's ultimately reducible. Well, all these statements would be ultimately, any statement about Dedekind style and arithmetized geometry and analysis would ultimately reduce to a statement about discrete sets, but, in the usual point of view, but most such, mild such statements would be insensitive to. But you can find some which are sensitive. So that's what I'm saying. This kind of thing is really about figures of incidence for equations. So maybe there is a way of enforcing U equals L. Not that I care about that. I don't want you to get the idea I care about that.

42:30 I'm not sure about that, but the reason for thinking about it is really in order to convince more people to change their point of view so that they can actually start making advances in geomathematics. So they're not trapped in the view of mathematics. So that's how points are added with a particle. Yeah, this is a different view of life. Well, let's take a look at all the spaces that are on that point of view. Hilbert ignores the existence of organically varying domains. Certainly, I think trying to make people understand the intuitive reasons why the accent of choice leads one astray. Think about real-world gradations are quite good places to start, if you want to explain why there were obstructions between those and mathematics. In other words, if you're varying in real estate, you're well packed. Yeah, and that probably helps. I'm just thinking of a time actually. Yeah, well, I want to go down and see if it's ready. They might even have tried phoning while we've been out. Yeah, we do. Let me get the next one. I just want to run in and use the loo.