Further discussions, incl. FW Lawvere & Colin McLarty (contd.)

0:00 So these basic things, you know, they're so turned off by magicians, you know, they know that they've overthrown magicians today and all that, and that's something better, but therefore they don't talk about the transformations, for example, in the same way, and they don't even consider that there are set functions involved. Clearly all those things like factors of functional... What sort of, are you using a metric on the, on the front of the page to approximate these functions? No, they don't, they don't, uh, I mean, there may be more books about it, but they're, they're, they're sort of... Friends will shy away from all the stuff that they're associated with. What class before Witten was A, again, with that? Oh, with all this stuff, yeah. What class was that? Make a big paragraph. But it surely can't be all measurable functions, because the fundamental theorem of calculus just can't be true. No, it's not. No, they aren't measurable. It has got to do with one way or the other. Well, but I mean... It only talks about finite... Well, but the Lebesgue Interval works for me. And the most obvious reason fundamental theorem fails then is the two functions that agree everywhere except on the setter measure zero, one, and the same thing with an integral, and so you can't help but recover an integral. Riemann integral, I guess... It basically works for continuous models, but then you could say, well, if it's piecewise continuous, what makes a piece of this model?

2:30 But the continuous isn't enough to give it a name. It's not enough to make it a question, yes, yes. But they don't all have a name, they don't all have a name. That would be cool to look at that. Well, I'm glad I walked it up, even at the very 11 o'clock, because it was something I always wanted to study a bit, don't I? T-points. It has sort of two T-points. One is finite, and the other is a variable delta. It seems natural from the point of view of numerical mathematics. Numerical mathematics is a small part of it. Oh, yeah. In other words, in the regions where it's difficult, you can use a higher batch of those parts where it's easy, you can use a higher batch. Oh, yeah. I'm not even clear about that. In a way, if you unravel what this epsilon is variable delta means, it is just the definition of derivative turned around. When you talk about derivatives, you don't have a global one. You're just worried about two things in the universe, and something else. That both explains why fundamental experiments are the worst that were made there, and also suggests that they're concluded for a notion of integration, which in fact is still around here. Would you like to have a theory of expression which made the fundamental experiments as applicable as it can possibly be? Yes, yes. That seems like a good thing to give you. Yes, it does very well. There's a charismatic part about it. You...

5:00 ...seek to generalize... ...to generalize something. Generalized functions, generalized elements, generalized cohomology theory. I was explained all this by this odd character. It's an American who has a permanent position in Florence. Harold Goodman. He actually showed up at our meeting, unannounced. Yes, that's the guy. The guy that we headed up and then the guy I kind of got out of the car and managed to go through. He's not the one that's interested in billiard balls bouncing around the political tables. Oh, I knew him at Harvard, too. He didn't even introduce himself. He just walks up, because I knew him years ago. I remember it very clearly. Colin and I actually had to leave the duck team and then take him around to the other side of the department. I had to tell him some comfortable stories. We did, more or less, because he didn't turn up. Anyway, he worked out part of this himself. He wanted to write, it's funny because he wanted to write a curious monograph about it. He found some of these results that weren't known to people in the literature. Moreover, he was really sure that he could talk to somebody. So at that time, the American Math Monthly still had himself. The MAA still had its headquarters in Buffalo. So he came to me and I took him to the old guy who was in charge. I know he may not have the obvious good idea, but he presented to me that he said the course.

7:30 You learn in this business that there are a lot of crazy people, of all sorts of incredible powers, but a very narrow, very obsessive, deeply boring person, who is just a carousel man, repeating the blinding of the office, who is much more obsessive than I remembered him from the beginning. Well, in that case he must really have been pretty obsessive with his career. Thank you for your attention. He did a great service in the past, mainly he introduced me to this whole business, which I then, you know, I studied a lot on my own about a year ago or something, but I wasn't known about it at all if he had pointed it out, because it was completely absent from any mainstream literature. Yeah, people like that often are very good at pointing out things that are absent from science, really. It was just as instructive as a guy who was getting a lot of instructive on the health perspective. Thank you very much for your time today. Actually, another thing that he liked, he was actually a student at Leeds. He was this guy who combined topology and analysis in a very original way. He had ideas about phenomena and regroups without a unit element.

10:00 Also, generalized determinants, which are a great answer in analysis. So, I mean, Lerner seems to have been a very creative guy. Very interesting. I was talking about your book about this. You were talking about your book. I was trying to push him into the door in a way that he... Well, I was happy to get rid of him then. Ah, it's crazy, wasn't it? No, no, it's fine. He's got this borderline, what I call, green ink. Crazy women who write letters in green. Oh yeah, I've got a person that I'm on. Is that crazy? Well, I didn't think that he was going to be on. Well, I spent 10 minutes trying to convey to him that I knew that my friend made mistakes, that my friend made lots of mistakes, and it's not that he felt he had something else to tell me, he did not make the point that I knew that. It was pretty obvious to me with, you know, with a great respect for your efforts, and to be honest with you, in those 10 seconds he was kind of dying to not take part, disappearing, you know, he was not going to listen to anything you tried to tell him, so that was one reason to try to get him away from the class. Anyway. Thanks for having me, I mean you can see it's really fun. But you read a given line, and there'll be some claim in it, and you think, well, if I think about it for a minute, I'll see that. Well, often you do. If you don't, you have no idea. Is this something trivial that you're missing? Something hard that you don't know? Or something that's not even true? And here Caradoc wanted a cute, scrupulous separation of those possibilities, and that's in one very narrow problem. I'm fine, but when I was going to have that, something pushed me. You saw him at Harvard recently? No, when I was there, when I was visiting. Oh, when you were there. Yeah. That was in 1995. Yeah. And I had just read a paper, and his conference was wrong. And he's telling me, you know, he found a mistake, and I said... Finally, I was standing here in my grave, like, playing in a chair at a room at Harvard, and he said, oh no, there are a lot of them. Exactly. That was exactly the problem. When anybody's that obsessive and has their head down like that, there's no point in trying to make a discussion.

12:30 So that's why I can't get them to see if you get it right. You've got to ask, you know, if it's one of the same old concepts, I can just tell you something. But not one of those. Thank you very much for your attention and we look forward to seeing you again soon. Thank you for watching. If I pay for this on my credit card, can you give me cash? All right. How much cash? A billion. All right. This is a very reasonable restaurant. Thank you for your attention. Thank you for your attention. Finite practitioners.

15:00 Walking down here, I was trying to review how the basic norm minimality results work, and I discovered that I don't know. I can go back to Van Der Ries' book, but I go back to a lot better orientation. So I'm guessing... Oh, surely, yeah. You take, well, Tarski's... You can't even say in that theory that every subprime of a subset of the line is a union of intervals, right? That's not a statement in the theory. Uh, well, yes. You can't, how? I mean, the statements in the theory are about sums and products of numbers. Yeah, but you can take an arbitrary formula in the theory. But you can't talk about it in the theory. It's one three variable, you know. There exists a much simpler one. Yeah, but you can't say that in the theory. I think the idea of theoretical truth based on Sturm's method is that you actually can figure out what the endpoints of these intervals are. It goes back to Descartes even, trying to solve these equations. All of this is a real solution, yeah, between the deficit, you know, compute from the problem, bounds inside of which, yeah, solutions live, yeah, but I think you see that from outside the theory. You use arithmetic, for example. Well, it's not special. I think it's a definable function of one way or another, given a formula incredibly consistent with the other. Whatever that means, it's a definable function of one way or another. It's kind of like whether an axiom scheme is recursive or not.

17:30 But it can't be a theorem scheme that says, for every formula with one variable, there's some number of quantifiers. Well, maybe it could be that. Okay, it probably is true for every formula must be variable. There's some number of quantifiers, such that you form a sentence that says, for these points, an element has this property between these two, or between these two, or between these two. But that's, but you have to pick the number of points suitably. Right, you can't say inside the theory there's some number of points. Oh, but it depends on the degree of the... The proof of Tarski's term is very convoluted. It really goes into all the details of the shturm. But, you know, I suppose there's some sense when you say, you're right, literally, you know, quantified animation. So then I vaguely thought... It is the case that in the models only these subsets exist, the first order models, no subsets exist in the models. I mean, I'm just getting it wrong when I say that. The model has a domain. There are definable subsets in the domain. So the model doesn't have a second order level of subsets, say all the definable ones are there, it's just a first order model. Most of the first order models are there. The model is a functor from the theory into the category itself, where the definable function is the value, the actual value of this function. There's no problem about quantifying over the more general subsets.

20:00 You don't normally quantify over subsets at all. No, but you were suggesting that some analysis involved that. Well, no, I'm saying I realize it doesn't because you can't even do that at all. Right. I think it might be related to a lot of our scientific research and everything, which does quantify the recess now in its hypothesis. It sort of characterizes what quantum physics is in terms of all possible one-dimensional things. All polynomials, really in-couple polynomials, one variable, substituting them to give an abstract question of invariables in order to give another question of invariables in order to give another question of invariables in order to give another question of invariables. Thank you for watching. In terms of individual things, the behavior of individual things is somehow determined by the behavior of the one-dimensional object. So, in space, that might be a lot more general than having the one-dimensional one species, or again, it's not like that.

22:30 That's right. It does mean that if the one-dimensional ones are simple, then the higher-dimensional ones, in some ways, have the same character as one. Right, right. Well, I mean, you see, the one type of result that they concentrate on is coincidability. Well, of course, if the one-dimensional thing is incredibly complicated, then it's not. But if you don't focus on that objective criterion, then there are objective problems. But it does not end on it. It's not. It's not. It's not. It's not. It's not. It's not. It's not. It's not. It's not. He writes in a lot of his papers in the Foundation of Mechanics that he's assuming that the very state spaces and bodies and whatever, topological spaces, that many of Marx had done, the whole reason he needed topology was because of the continuous curves. Categorically, they say, oh, well, let's just say that the curves are the topologies, and so that a warp is merely something that takes continuous curves and continuous curves. A map between higher dimensional spaces. In each of which we have specified a notion of good curve, good curve. If it maps good curves into good pairs, then we should call it a good curve. So that definition of the category of math, which, you know, seemed to be all that it was requiring. That's when Johnston, Pavlovsky, and Popov came out, because it turns out... This doesn't work too well with the usual notion of continuity, but again, this is because the usual notion of continuity is too tied up with a bad incentive to untangle it. That is an unsolved problem, I think. Namely, well, it's another formulation of Duhigg's idea of change of topology, but not tied up with formulas and all that, like van der Waals. Simply to somehow specify a class of endomapples in an integral

25:00 which contains, let's say, all analytic functions. It's contained in the usual continuous information, but which... Well, for which, for which in the topos of canonical science, the line of the order, the order relation is total, because the existence of these pathological, so-called vague functions and so forth, and the usual continuity, destroys the, the fact that it's not internally in all order, it's linearly in all order. Therefore, in order to get the geometric realization as an adjoint, as a morpher like tokos, one had to introduce bad infinity all over again, so the Johnstone tokos is based very solidly on bad infinity and the whole gutter, and in particular you get... You get that the integral is totally ordinary, but that's because it's still an advantage capturing the notion, the general notion of continuous according to the traditional epsilon delta notion of continuous. It does do that, but you see, from this you see that there's something defective about that. It should be tamed. But the meaning of these ramifications is made important. The ramifications of termed topology may be intelligible. And headstock intervals. Well, that would be one of the... You put that together in one title. Headstock interval and... How much was it? It was 69. Just give me 20. Oh, 20. All right. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. The point here is to do some things that they don't usually do.

27:30 The point is, I think, is to get across the huge ramifications of this idea. And obviously it involves consideration of the pressure integrations therein, so again, it's also about the polarity, the consequences of that infinity. The central realm of good functions, spaces, and maths, I mean, just involves so many topics, and it would mean you're an extraordinary genius to kind of all begin in a way that would be totally into mathematical physics. I think actually such a meeting might be considered more valuable than a semester of this class of category, although I don't think good things will come out of that. Anyway, it's something we work on. I really think you've answered the question. Well, you've had exactly what you've answered. I suppose that's the best way to end the seminar. We could jump on the RER and then you'd drive all these. There are a number of different types of mathematics, such as quantum physics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics,

30:00 You know, I don't think even Rumsfeld says Europeans don't know what suffering is. He says they don't understand terrorism. That's what he says.

35:00 Well, of course, that was Bill's point. That's another thing. He's not saying you guys need to suffer. Even Rumsfeld isn't saying that. Some of them, I'm sure, think the French exist only as cartoon characters. The point about terrorism is that it's a secret weapon. It's one of the tactics of the rest of Europe, of South America, and of the United States. And just as this all-purpose tactic of we'll never defeat terrorism until... You can follow that with anything, and it's true. Your conditional will be true. We will never wipe out all terrorism until... You know, I'll put anything in the blank you want. So you can use it to justify anything if you consider that a justification.

37:30 Well, it's pretty easy from here because at this timeline almost all the trains go to Rossi, and it obviously doesn't make quite sure it's not the way to go. Yeah, so you just walk back down here.