Further discussions, incl. FW Lawvere & Colin McLarty — Henstock integral & other topics

0:00 It says basically if you had a cover and you looked at it, and of course it's just, so then it says that you're looking at a cover, so you know all about a top cover, so you know all about the top cover, so you know what it is here, it's a scheme, well yeah, maybe a little where it's an irreducible scheme where the components cross, that might be something like that, but, okay. There's other things you could include in the Purdue Scheme, so you can ignore, you know, the database when you're studying any calculus. So if you want to know what's smooth on the scheme, you don't have to worry about any contextual printout. You got it smooth. So, I think it's time to ramify everything, but that's in French. You might think it might fail to extend. If you want to try new opportunities, let's take a look at this first. This is a rod, wire, and a fork. Sate. Do you want a second? Sure. I like it. Thank you.

2:30 Thank you very much. Let's have some fun. 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 for your attention. You could write this in string theory, and how everything is represented if you want to, the whole of math, but also a lot of things really have to do with string theory, I think. I clearly didn't quite believe what I said there. You didn't quite believe what I said there. I didn't quite believe what I said there. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I didn't. I'm wearing that hat. It's the part that you were putting on a white box earlier. He was saying hi. Maybe. Hello. Hello. Hello. Hello. Hello. The only place you hear about function space is in functional analysis. If I go over it, that is treated in a different way. The only examples are Calibre, relativity, Hilbert space, and all this kind of stuff, which actually gets away.

5:00 So if you hear the word function space, I kind of suspected that's what happened. All of these terms are used to explain the concept of quantum mechanics, and they are used to explain the concept of quantum mechanics, and they are And particularly, this is a point of view I'm not paying attention to, but you said you thought you were a great, what did the preacher say about St. David? He said he was a great preacher. You said you were a great preacher. Oh yeah, a great teacher. A great teacher, is it? A great teacher. Oh, no. It's actually the Kurzweil I'm in, the company's shared campus with me, so it's the Kurzweil we use to prep those papers and throw them up there and use them when we get on to the next class to do this class for me. The basic point is it's a big integral. Physicists were told within 30 years that we're off to a great start. Although difficult, they have to learn this in order to be able to deal with these calendars. So, one of the things I was thinking was, who would correct it? Well, the Riemann Institute. But, you can correct that flaw in a much simpler way. The basic point is that like Riemann, Kurzweil, and Stopp use finite partitions that require re-approximation.

7:30 If you want to integrate a curve, get the area under a curve, divide up the interval into pieces. ...multiply the lengths of the pieces times the heights of the function, and that's a typical approximation. Notice that process itself is a function. Given the sample points and given these lengths, you can just take that linear combination and the Dirac equation. So that's another function. The idea is that the ilk ought to approach the... But now, so the bank's first move was the... Graphs, band, and pedigree say, okay, let's divide the integrals. It's incredibly many features, and so you've got more functions. But you did get some bad features, like one basic idea is that for any function whose integral exists, an indefinite integral exists, and the derivative of that should exist. Fundamentally, which is not true for the Lebesgue interval. Typically, with the Lebesgue interval, you have to assume that the... The absolute value of the function you're working on is also integral, not just the function itself. You make it positive by putting the negative parts up above the axiom. Those just count, but they don't count as integral unless the total area above is a bigger number, typically.

10:00 The condition that you find is always negative. You can divide the integral into a finite number of pieces. Of course, your sample is four, but the criterion for saying that you have to define a directed set, maybe a finer and finer partition somehow or other, and then the idea of the integral should be the limit, What is this directed set? In other words, when do you say that one partition is finer than another one? Well, according to the traditional rendering of the remodel, it just means that you have a number called the mesh, and that the maximum size of all the intervals is less than that one number, that constant number. Well, no, why don't we say that there exists a function, positive function, delta. So we have this partition with the sample points. We say that this is delta phi. If evaluating delta at a given sample point gives a number which bounds the size of O, but this function delta is going to be used for all functions. We consider all possible functions of delta instead of all numbers of delta. No, no, sorry. For all epsilon, there exists a delta. The epsilon, of course, is going to be on the constant, which is the value of the integral, because the values of the combination, that's the usual sort of epsilon. The delta that exists... It's not required to be a constant bound on the mesh, it's just required to be any positive function which bounds the interval in the same direction. The diameter, as I already mentioned, the diameter of the ball is smaller than the delta of it.

12:30 So you quantify over these deltas as functions, and then you prove... But always with finite partitions. It's very important that finite partitions are the key of the old minimum. Well that's actually why I raised it in the context of that moment. Because this integral is obviously closely related in the concept. Improper integrals, right? Improper integrals. Well, not only that, lots of improper integrals. Well, there's a link to that. Improper in the sense, yeah, there's this integral. Like if the function blows up at an independent point, in that case, you can show that even for simple functions, you get more quickly better approximations. All the big integrable functions, hence not integrable, but more functions besides, and you get the general validity of the fundamental theorem. The function which has an integral, indefinite integral, is derivative. It's again one of those things where the analysts love to say, oh no, your intuition is wrong, look at this awful thing. Really? They've been bypassed into it because their vague definition is not the gospel. More basically, this whole business is about you have to consider countable things to define that thing. See, they preached for years, but you have to use, you couldn't get those integrals that you need in physics. Actually, Riemann, he's always, he's always, uh, because you can, it's not the function that always, but I mean, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what,

42:30 The ramifications of tangyphology may be negligible, and just give me 20 minutes, is that okay?

45:00 Sure. Right. The point here is to get affinity across so many topics all together in a way that is almost as strong as classic mathematics. We'll have time next year.