Afternoon talk

0:00 I've heard of some knowledge for years, and none of the approaches I've seen have actually been related to this type of science. In this case, the representative is very useful. Each object like that is represented as a representative rate, at least slightly different, but indifferent, sorry. The second and complete question is the values of the modules of the Bernoulliian group.

2:30 And it would be very useful to have something in others, even for the case of just a variety of rates. The congruences for a rate are not just ideals, but perhaps there is some way of understanding the congruences, an illuminating way, and I haven't been able to. The other day I asked you, we were on the street and you seemed to have a number of ideas. In order to deal with this example, I think you'd understand enough.

5:00 An object there which does a lot of motion. I'm going to put a little bit more than a bare part, but as you need. To make the same mistake that was. Now this is my idea. Am I right? Part of it, definitely, the part that you get absolutely. This object-defined idea of motion, the conception of... Euler, sorry. Euler remarks on theorem, ratio, next to first order, and this too.

7:30 More exactly, the point is that addition and multiplication and even subtraction have a much more definite character than division. Division is really an existential statement about multiplication, right? That there's a ratio is a process which takes quantities into quantities. To say that one quantity is the ratio of another means that there exists such a ratio of a process that takes them all. In other words, it's not determined by the individual points that you're looking at. For the infinitesimal, it's certainly not determined. There's a whole set of things in that. There's a whole set of lambdas that map A to B and A to B. If you've just given two particular A and B, there's no unique lambda.

10:00 To say that one exists is to say that B is a... The lambdas are determined uniquely by what they do to all A's. We can't put the lambda as an operation, it's not determined by lambda, so there's no operation b divided by a given a specific result, it's not determined by a. The same, this problem arises in teaching calculus, you see, difference quotients and stuff. The best way to describe those and not to mention them at all is to talk about corresponding multiplication statements, like delta y is equal to f prime times delta x. There are deltas or differences, but f prime is not equal to delta y divided by delta x, because that, I mean, at some points it is, but at the crucial point it's not, because it's a place where delta x is equal to zero that you evaluate. In other words, the basic contradiction was always there that Bishop Barclay couldn't understand, and so forth. It's precisely, in a way, it's that, what a ratio is, really. F prime, the derivative of some function, is a definite process that takes all the delta-axis into something approximately equal to the corresponding shelf of Y. On all of them, not just one at a time. Anyway, a different form that arises in commutative range theory when we pass to an open subset. If we look at the line, it has small polynomial functions, but if we want to look at the part of the line... Where we exclude, let's say, points plus and minus one, then there's another, a different ring of function. All the rational functions with the given quotient, with the quotient, you know, one over x squared minus one, in that particular. So a commutative ring theory has evolved in a way that's compatible, correctly. You don't adjoin all denominators at once and get rational functions. There's no more space in which that's defined.

12:30 But rational functions with a given finite number of denominators gives a perfectly good ring, and it corresponds to a space where, you know, the open subset of another space where those denominators are, in fact, invertible. So, again, you vary the ring a lot. When you just want to divide by something, you have to, in principle, pass to another ring. But that's the way of it. So one has to have this sort of ideology in mind. I don't know how to correctly interpret Euler's statement, because people have been saying this is nonsense for centuries. Some people are aware that he said this, but of course you didn't mean it or you didn't understand about it because of him. The point is, they don't understand about division. In various branches of mathematics, this has become understandable. If you have a one-dimensional situation, not just a one-dimensional situation, then delta x is a vector. You can't even divide by a vector. But you can apply linear transformation to a vector called delta x and give it another vector, delta y. You want that linear transformation which does a certain thing for all possible delta x, and that's all perfectly sensible for division per se, isn't it? So really we shouldn't teach the students that division is the fourth operation, or, it is the fourth operation, but it's certainly not on the same level of algebraic definiteness. And this is about the experience with non-potent infinitesimals again, is that if you, an operation which is defined for all infinitesimals is well defined.

15:00 Small d being the typical element of capital D. So, certain equations have a small d with universal quantification, then you can cancel the cancellation of the universal quantification. Not just a-x into b-x into some certain x. Because none of these things are invertible. I mean, the other obvious case of the cancellation of a-x is invertible. I think we'll carry one of those together. It does work as a constellation of universal components. The equation has composition. This has ratios and u. Since we're in a Cartesian closed context, it may contain some extra bits, not just those ratios, but some extra bits. It really comes from such a ratio. That's structure.

17:30 When the structure that R has is the result of this, starting with T, which has no structure except a point. Which is also a property because it's unique. Yeah. But then by doing simple Cartesian closed category construction, you arrive at this object R, which has certain structure. It's a monoid because it's a sub-monoid of T to the T. And it's got a point that will decide the unit, namely the zero, constant zero. So it's got two points. It's a monoid with it. Specified constant. A single point, a point at object t comes to a bifoint at object r. Actually it's much more than bifoint, it's got a multiplication. The fact that it has two points is to the extent that t deserves the idea of objectified motion. It really means motion in a given moment. Like now I move from one place to another, perhaps r or something like that. Since R has two specified points, it makes sense to say that if you have a map from R into some other space, X, then you can evaluate that at these two points, and you've got two points of X, so you've got a motion, you've got a motion, and an actual.

20:00 I think in Italian you can distinguish movimento and another word. What's happening right now... You've seen a name for motion? No, no. I can move from Milano to Torino, there's two points, so some word can be moved as much as I want, but on the other hand, if I'm going down the highway at any given instant, I'm moving, so I'm in movement. Movimiento or movimiento? Movimiento is the correct word. For both. I'm philosophically deficient. No, because it's very important to miss this distinction. Drassmann explains it in German. There's a difference between verden, becoming, verden, and enderen, it's change. So you're in a process of becoming. Becoming is devenir. Devenir. But it applies to the general concepts of the year and cambiamento, but the specific application to change of position is variazion. In Spanish there is a difference between movimiento, but that's an adjective. We want to objectify that. It's very important. It's absolutely a fundamental distinction. Between a differential equation describing a law of motion, to integrate from one to the other is a major problem in mathematics. Just to underline that these are two different things.

22:30 Yes, Devenier and Cognomento, those are, in a general sense, for change of state, becoming... I think there's a second word in Italian. You could say displacement. Well, it just means that as a functor, it assigns to every client you present to take out a few numbers for that one. If we're talking about these programming functors, there's less need for double dualization. The definition tells you to come through A to the right of A.

25:00 That's because of what T and Z is. In any reg A, you can take all the pairs. This is just to suggest how small those are, but those pairs, I was finding these conditions, because those are precisely the things that map, these are the ratios that map things, but epsilon equal to anything in square zero, then this comes out to be something that does that, so this process is if you take the value of zero, clearly that's A sub zero, by pullback you're demanding that you read. There are a number of ways to put it. If you do this, you calculate what this definition gives you in the classifier for the theory of rings, it provides you with the generic ring.

27:30 I mean, that's part of the mystique, you see, that you don't have to do this double dualism. Since you know the double dualization is there, you are justified in just calculating in R, symbolically, the generic ring A. You just do the calculation like in high school algebra, except you don't do anything that wouldn't be valid in an arbitrary ring. The interest is, now let's take, consider T had T sub 2, the second order. So those are the elements of our cube zero. What are the maps? What is T sub 2? T sub 2 is the power of T that the object comes up with. All the maps from the first quarter, from the first quarter to the second quarter, again, that object itself is going to be the spectrum of a certain range.

30:00 I believe something good is coming out of it. It's surprising. Of course, you can stop there and to carry on what you just said. This has an underlying structure in my composition and are inherited. There's no reason, there seems to be no reason why it should be commutative. In order to ask for commutative evil, one could simply go ahead and ask for commutative evil. The idea is that all the axioms should be of the form that something is a vertical or that something has a universal property, just as simple commutativity is. At this point, with this type of axiom, it was proposed by saying, the axiom says, if you take this momentoid and force it to be commutative, then this should be an axiom.

32:30 We discussed a different axiom based on the peculiarities of... This gives a commutativity of r and also an attitude of retraction. ...calculations as easy as I was expecting it to be at this level of what was going on in the alternative axiom. You take the quick qualizer and this is a large quantity.

35:00 And that forces the commutativity of the modern structure of r. It has the advantage of not needing to go to the gallery. Glad to see you. It's great. Yeah, yeah. It's really helpful. It's very helpful. Thank you. You're welcome. Goodbye. Nice to see you. Nice to hear your talks and that. Good luck with your thesis and research. And I guess I'll see you sometime. Have a good trip back to Argentina and have a nice time in Birmingham next week. I will. Okay. Ready? All right. The versions of axiomatic equation in these three contexts that are category theory, even product-based algorithm, the t-discrete objects form an exponential idea.

37:30 That's quite easy to prove, and that implies that by zero we set problems. One has to say that the R we define is connected. One way to say it is that one way of saying it is that adding the axiom for t to the t.

40:00 Assuming we establish this retraction, it's just worth mentioning that because it's slightly closer to the hypotheses instructions we're starting with. The nature of R may not have anything to be a byproduct that R is connected, but it's a retractor. Now this is just a methodological non-remarkable. It doesn't work in here, 5-0 of R. It might not be the best example, because you can see there, there is a smaller subtope. That's why it might not be the best example to check. But in the case of reflexive graphs...

42:30 You mentioned T, B, and G, so let's just remark that so is R, and R's sub-varieties are those. Because the chief category is always what you call exponential ideal. Yeah. So T, the T and all the sorts of objects that are, you know, all these curves and surfaces, they're equalizers on R and stuff, but there's a huge amount which is implied just by having T, even though it's, I like to think of the whole thing as being infinitesimally generated, so it's that topos, the smallest subtopos contained, it's generated, but not in the sense of co-limits. But just in the sense of generating a sheet of I.G. alone, I think that's going to be a crucial feature. And of course you're right, I mean the topology certainly contributes to what's going to be connected, because it declares... Try to calculate, well, assume that it's true, whether these holes... These discrete objects are... These are, yes, they're the ones... I mean the notion doesn't depend just because they're closed and whatever topology...

45:00 To be known, having no derivations, but I mean in kind of general sense over any aid also in vector fields and part of it, which I think if you look through my key criteria for being this so-called separable object, in general the rule of multiplication here qualifies.

47:30 If you find that the constant terms multiply as you would expect, and the other ones multiply connected by the Leibniz's rule, it's really where Leibniz's rule comes from, that you make its first-order perturbation of a0 and of another one, a bar 0, resulting in perturbation of the product, its derivation is, and instead why it is.

50:00 It's entirely something happening in the category of rings. Everything here is a homomorphism, because the thing about delta is it just includes the constant part. It's a homomorphism, the present one part. And so this condition is that spec b, the derivative part is zero, and this is just the theorem of an algebra that says that these algebras are just a separable form. The thing is actually a finite product of pieces which are made of local rings, field extensions, as far as innovation is a field. Field extensions with some no-potents are joined. The radical of no-potent elements can be something. Field extensions can be product extensions. So it's just a product of fields. As field extensions, it restricts to the specific thing that I mentioned in these intersections of averaging.

52:30 It's an average of finite. It's interesting to know. What is the atomic one? That's just the bar-atomic. It's the same thing as the bar-atomic topology, because for three finite fields, you have to impose these traces, a trace in the figure of an element of a larger field than the smaller one, because the larger field is a particular vector space over the smaller one, multiplying by an element in your transformation as a trace. So, as you said, the strongest topology is this one.

55:00 I'm very glad to show that it's certainly connected in the usual sense, because every representative can appreciate it. It's 5-0, it's not 1. Let me understand the world you're stating there. Well, what Key then says is that, well, the base is a field. And Bs would satisfy its properties. There are no deltas here. Well, it's actually a finite field instead. No, no, finite products. A finite product.

57:30 The notion of separable object makes sense for any topos, any object. And we know that if all objects are separable, then anything is actually proved as a byproduct. This notion of separable, applied in this context, actually turned out to be true. It's precisely that the right topologies make it good, even though the thing that she's in general definitely not good. But this part becomes good because of course it sounds like some kind of general phenomenon when you adjust the right conditions. Subjects and the totals are not appropriate and you pass a QD reflection.

1:00:00 Each day, I want to calculate this co-evaluator since we're in a bridge of topos, bridge of topos, midway, and so on.

1:07:30 How do you establish that, exactly which co-evaluator works? Probably the T-Discrete Object, you don't need this guy. But that's for general reasons. This is how you calculate the reflection.

1:10:00 Well, suppose I don't know that. You can look directly at Q as a fun survey. I see. Let's check it out. I guess you need more.

1:17:30 It's a good infinitesimal criterion, but it's more than just tedious. It's a good one. It's important necessarily that you can actually achieve it by an R path.

1:30:00 You can achieve it by an R path, good, going from A to B. The concept is somehow that the attempts at doing that are dense. It's an infinitesimal path. It's not, in other words, a homotopy. To define homotopy, I'm saying that you have some kind of integral object. You exist. The two points have to be joined by that object. That's one thing. On the other hand, saying that you have components of a function space is another thing. In topology, there's a difference between connected and arc-wise. Connected does not imply arc-wise. The idea here is sort of something in between. My discussion of second-order differential equations is related to that. A solution of it is merely compatible with it. It's not built out of it, necessarily. In a simple case, it might be built out of it, but not...

1:32:30 You need an exponential map. You apply it with exponential operations, and that will be macroscopic. It is a path through time operations. One doesn't expect that, even from a point of view of continuum, one doesn't expect that sort of thing. It's always defined. But the idea of satisfying... It doesn't actually involve anything about the integral now. It's just a, right, so. By a pointed object, modulo zero to the zero, so to speak.

1:35:00 The zero you already had, and the one that is, either the t really is one plus t, plus t squared, modulo the symmetric, plus t cubed, that's symmetric, and so on. That's simply the freak you do in any of those. This is the one that you have to begin with, and that has the effect of our identifications between terms, and so all these disjoint, this totally disconnected object becomes connected, at least in some sense, certainly T-connected, in any case by the fact that there is such a monoid in the anamorphism of internal monoids. It's essential, you don't need the natural number objects for that, by the way.

1:37:30 So, basically, you see, North Star is a hub with respect to the rational vector. It's very compatible. This is what it means to be a solution. There's no sense in which you build up from it.

1:40:00 Part of that, you often, you know, under various hypotheses, theoretically detect that there's, at most points, you see, you look at it precisely like you see it. Existence of solutions is a difference that tells you that there is a problem. Now, it isn't precisely a map because, well, no, it isn't. See, the space, oh, yeah, exactly, the point, here's the, here's the, or the adjunction map, which takes I, I star, I, lower star, the map going back connects long paths, and the real search, you see, so the naive idea is that this ought to be a nice work, and I mean, to every point, as an initial condition, you need curve. But as we know, in real life, i.e. in mathematics, not all attention matters. For some, x, and for some, c, this is true, but for others, it's not. Even existence is supposed to be failed. But as always, existence just means lifting a load of math, and you have to have a theorem that's very unique, that has a heavy model analysis of this.

1:42:30 On the other hand, you can always solve the equation by changing the space. So the real question about meaning is, when does it stay true? Does it live still in the same space? So analogously, analogously, you've got eye-lower-street. It really is a building-up process. Only this is R-tensile will be T. Consistent pairs with real number, arbitrary time, and a point of x, except that these pairs are identified. So the pair that you get by adding an infinite decimal bit to your time. You can identify with the other one, which is really moving ahead without a big up process going on in the place where you precisely have the appropriate mixture in the streets to build an up process. There is time, but you are reviewing these kind of street steps, vague ideas, you know, when the piano is in this instance and this instance.

1:45:00 Again, the analysis is really the analysis of the production map. Topology, so we could ask if this is dense or whether this is dense or not. Where dense really means every worker could be a strong idea, but it doesn't criteria acceptability, plus that compatibility in a somewhat similar analysis.

1:47:30 This is very precise. This is a compatibility. I was thinking just about something around zero. I was thinking about, because, well, assuming r has an addition, suddenly the map would be p of r, or the point of r. Oh, right, okay, it is. But those, and the question is usually about the mutual. Although they're bigger than the point, nonetheless they don't overlap. Again, somehow, you've got something going for you under the crunches, which then they start to correct.

1:50:00 You've already made the notable wax on the test. Yes. You have great thanks, I'll get it off you on my test, otherwise I was going to ask you to let me take it. Everything that I said there is in my 1967 book. Ah, what did the author say?