Morning discussions incl. FW Lawvere & M Wright / further discussions (contd.)

0:00 There are cases in which it is useful for certain things, where it is completely false, and other examples where it is useful for other things, where it is true. I don't want to say that we should throw away the others, because they are very important. Local topos, for example, can almost never have this property. They must be non-local. So, we could immediately develop this concept of connected space, All this in a funtorial way, therefore, could be raised to the study of types of homotopy, that is, the form of each space, counting, for example, the three-dimensional holes, the seven-dimensional holes and their relationship, etc. etc. The discrete aspect of a space is continually expressed. In another category there are homotopies of spaces. I don't want to interrupt this discussion here, which is a reason why we don't have much time, but on the other hand because I wanted to emphasize another aspect, that perhaps we could go deeper into the subject. That is, as you can see, when I have made it explicit in this way, the whole discourse of connessity, homotipity, etc., etc., depends in a fairly strong way on the other songs that go from the study space to another, to another description.

2:30 The other aspect, because characterizing this depends on the morphisms versus the functions in the sense of Volterra, etc., is an objective aspect of the so-called lambda calculation, almost. The subjective machinery of the lambda calculation tries to describe this aspect of Volterra. So it's not a question of superior types of connectivity. But in any case, let's get back to the answer that was made 30 years ago by Nelligan. He said, 130 years ago, he was in Los Angeles. He said that we can measure some of the approximations, for example, let's say...

5:00 Continuous one-dimensional, a fundamental case, but also two-dimensional, three-dimensional. We can approach things. For example, P. Archimedes already made an approximation of P. So, starting from the fact that the positive or non-negative rationales have a structure of order, Let's not lose for a moment now what the rational things are, because this is another matter. The irrational things, and then the values ​​of truth. In the case of Juliano, in the case of Sprague, there were simply two false ones, but even though he still has many years of experience, we know that by studying the movement of Puccini and so on, the categories of Juliano are not enough. But some often exist the same, not always, but perhaps, but they exist in the object of values ​​of truth. So, the speech of Erichens can be interpreted in these terms. If it were not a continuous law, it could probably be measured in some way, then reasoned. They would have a sub-object of the network, which is already very dubious to accept. The fundamental step is that for this reason there is a morphism from Q to O.

7:30 Let's take one of the preserving maps, the reverse of O. The part that can be defined as the Q system Using these values of the equation. For this fraction, let's say phi, for every x in R of the equation, we get the sum of all the rationales using x. If the rationales are a constant in R, then we can talk about an order part and make x squared, and we get the sum of these. Or in other words, phi of x. The word is often used to refer to the words of truth in this field. Technically, it is used to refer to the fact that omega is an accused category and q is an enriched category, in which this word is always used, which is now called omega. Each group has been in a specific group. This result is applied not to groups, but to data centers. This means that you can interpret each point x knowing in what way these curves, these reactions. Then, with a certain hypothesis, you arrive at the result that this is an objective morphism.

10:00 The points are different, but there is a rationale where you can see the difference, not in different ways, but there is a rationale where you can see the difference. If this would be true, it would mean that it is objective, and then, it is interesting if I understood the story well, at this point Derekin stopped, but with some other he said, In this case, we have defined L. L is nothing more than the part of omega Q such that it has certain properties, and these properties can be identified. Among them, already known by Dedekind, is the famous condition of being a cut of Dedekind, a section of Dedekind. If we put various conditions on these morphisms from Q to omega, If we take the totality of these morphisms, of these properties, we can call this L. So we have reached the definition of L, only in subjective terms, because L is the calculation of a factor, we know where to exclude it, and omega is the value of truth. Certainly, the judgment of truth has a subjective aspect. We have reached all this. We have come to a definition of the continuum in terms of the objective in the term of the subject. So, the term of the continuum seems a little high to me. But, okay, why not? In this way, our concession was born, that is, our theory of reality. Yes, if we are in a category where there are colors of truth, for example, where there is an inclusion of rational and real, we must be careful because having rational does not mean having an inclusion, this is a jump here, but also very important, because all these hypotheses already

12:30 In certain cases, it is possible to make a morphism of this type, that is, there is a partial invariance of a point in the continuum and all the imaginable measurements. But even with all these hypotheses, it does not mean that this is objective. This is another hypothesis. They kill the difference between the infinitesimal and the zero. If there is, where is my... If there is in the network L these infinitesimal approaches of each point, zero, it's fine, but they are killed, in the sense that... in the sense that we already have a property. We don't have to jump to the superior types to understand this. The order is the same. Because every static tentative to measure x has a smaller result than any static tentative to measure y, and also inversely, it does not mean that y is equal to x, because non-static measurements must also be made. And this is more than one, exactly like this, in the theory we are talking about. Here you can't see the difference between a point, which I made in a type of Italian word game, saying that this here is the inclusion of the point, the generic point, in the tip.

15:00 Because in a certain sense, at least in the various languages I've learned, the tip is something that is wrong, The study of differential calculus, precisely for the movement of material bodies in the continuous space, is also followed by fields of There are also logistic fields, which are also continuous. All this discourse can be passed on from the field of continuous spaces, but exactly on the basis that these contradictions between the point and the tip, because in these contradictions between the point and the tip, not only this little thing is born, but everything. I have another definition of L. I write T instead of D, which is a little bit of a reference in a certain sense. T is an infinite space of first order. The meaning of the infinitesimal of first order is expressed through the condition that, that is, the rule of the game, which is this here.

17:30 There are other spaces in their categories, so you have to explain what types of categories you live in and then the very, very, very particular properties of this type of object. But fundamentally there is one point and one point only, that is, there are no other morphisms from one to the other. So, the point of the point is this, which is to be thought in this way, for example, t could be for time, an instant of time in the infinitesimal, but the interesting thing is that if we take the space of morphisms from t to t, it is no longer infinitesimal, in fact this is a model. Very correct for the continuum. Very interesting. In a certain sense, functional analysis is the functional analysis of infinite space dimensions that we know, but more deeply, the algebra of infinite space dimensions is the functional analysis of infinitesimal spaces. An interesting contribution. Why? You can easily see from this image that this is a space where the points are morphisms of T and T, but the morphism of T and T is a line like this. There are many. Another value is to identify, from this point of view, with something objective, not subjective, something objective. Let's say that if the morphism from t to x is a movement in x to infinity, it passes through a certain point of plus.

20:00 The composition gives us another movement. Another form of lambda is a process of respiration or slowdown. It depends on the analysis of the relative magnitudes of lambda. But this is the fact that every movement, x, one can imagine that this movement passes at a speed of half. Or even faster. But how many are these lambda? All the models for these images that we have built are more precisely the same, but you have to be very careful, as I said, with loyalty, dominion, codominion, this thing, because if you don't, you lose. So, being more precise, yes, it is true that lambda... They exist, they are really described, but in space there are not only points, in every space there are figures of various types, elements of various shapes, a fact that Voltaire has already discovered and used a lot. He explicitly says that the elements of a space are not only the points. But the elements are also movements, surfaces, etc. in the same space. This is strongly underlined in almost every categorical work.

22:30 In each object there are points, but the history of the object is not finished with the points, because other elements are important, and this is very particular for those who study it. These are more or less the morphisms from T to T that pass from zero and serve. This automatically has a multiplicative structure. The composition of the category itself gives you, for the endomorphism of each object, a multiplicative structure, and this also passes, in this case, to the submodules. So, this is, in this way, with every t pointed, let's say, it is always automatically a modificative monoid, but in fact there is also the additive structure, the real idea, the additive structure, and the non-negative line of the modificative structure. This is very, very general, so I formulated it in such a way that for definitions, I don't know, for all kinds of things, you can divide in any way the part of the tangent space, in general, m to the t is called the tangent space of m. This is the real sense of the tangent space at zero. The tangential space is above zero, above the identity, but above zero, excuse me.

25:00 The liquid, the humanoid, which is an element called 1, for example here, is the inverse margin of 1 by taking the figure of the point zero. This is the humanoid core, which is called the algebra of there. But if we can calculate with all the models we have, it's true that if we put m equal to r, or even if we put m equal directly to t, rt, so t, rt, rt, that's it, but this core nucleus is still a monoid, but a monoid in a different principle, it's an additive monoid. It is a complete theoretical model, but for me it was interesting to see that in this generality that I said, without necessarily conditioning, you can find all these aspects. I wanted to quote that even the logics in recent years have gone in another direction, leaving a little. I'm thinking about the solution called all-minimality, where there is now a book called Tame Topology, by the author Van Den Vries, where he has pointed out the question of continuity in a very serious and different way.

27:30 But it is not very compatible with what I said here at the end, no one has yet understood enough of all the different aspects, the logic, the uniformity and the categories to put them together to see, but I am sure that they are very compatible. In any case, a very fundamental result is all this topology that Grotendieck also developed. The famous theorem of incompleteness by Goethe, the victory of subjectivism, has been confronted and killed, in the sense that there are many weak theories of the continuum, but where, as a subjective theory, it is decidable, and they give many examples, and I bring here examples. But it seems to me that this aspect was also intended to be included or combined with the efforts of the infinitesimal, that is, the movement, etc., to have an ascension, to continue for our century and all. Is there anything you have already been asked? You answered me but I didn't understand your answer, so I'm asking you again, from the point of view of teaching, how would you think of teaching continuous differential geometry, with what models of real network? How would you do it? You don't consider the real numbers, so let's say we know them, would you make a completely axiomatic setting of this type? I don't have a clear idea. Yes, if I understood correctly, I think that the best way is not to start with models, because models are attempts to demonstrate consistency with certain other systems, or even to verify.

30:00 From a theoretical point of view, the content should be taught directly. We have this intuition of infinitesimality, of direct, of continuous, and so on. Describe precisely, explicitly, our prejudices that arise in this way. Compositions of pensions are fundamental to the development of differential calculus and mechanical mechanics. Of course, this is done from the point of view of security, in the sense that we know, because we have studied various models, that this is good and this is not good. So the formulation is right of the actions, of the prejudices I have mentioned. It depends partly on this experience with models, various models. This experience, however, is not said for the first time. Now it is explicitly said what the reasonable prejudices are. It is not clear. For example, in the book of Koch, this story is more or less followed. Models come later, but the fundamental study is more or less on the same level. Of course, as I said, you have to explicitly make certain concepts. For example, the concept of infinitesimality. Of course, this is the purpose of this course, but it can be generated as a composition of math. Space of functions, functional homomorphism, but functional homomorphism and these things are, in fact, very fundamental concepts that you should teach in school, not to get into the complicated aspects, this is another topic, but to make it explicit is necessary.

32:30 It would hurt the student if you made a concept explicit, so don't do it. Of course, we make the choice of what to do and what not to do, but in principle, yes, it is fundamental, functional, infinitesimal. So, you must not be afraid to explain these, even if it takes... I don't know if these... If we put these... if we take these remarks together with Cox's book and you get some approximation of an answer to your question, I suppose. What is the relationship between the studies on minimality and the things that exist? I didn't think of a relationship in the sense that something implies the other. Conceptual. Conceptual. The objective aspect of minimality is not that they are decidable systems, but that the desire for each object is finite. There are no objects, objects mean networks, but more importantly, this is a very difficult question, because normally, that is, our analysis that is now in school, if you have a simple network that you know, but a function, a morphism of the network itself, a morphism like this, and you take the intersection, the vector of something, You arrive at a space with infinite known components.

35:00 Therefore, to avoid subjective incompetence, one must objectively avoid such things, in particular what they have done. So I think it is really important, even for analysis, Even for the development of the mechanics of the continuum, my favorite goal, the fact that the components with which they are finished, because in fact a large part of the counter-examples, of the bad examples that are found in the courses of analysis, which always make a great difficulty for the visitor. Because if they want to be precise, they have to try to avoid these quantum spaces. Instead, if they don't want to do this kind of things, they have to be non-precise, they are in this trap. This is in fact the fundamental reason for my life, to avoid this trap, to do the physics of the continuum in a rigorous way, but without getting into these stupidities. So, in a general way, it would be, of course, not to expect that, for example, in a topos, every object has finite components, but a set of generators such that each of these, for example, if this is my information, it is simply that they are compatible, that one could build a category with all two of them. And then, certainly, we see that there is something good about Penrose, but to predict which one it is, I can't say anything else about these things. They are both dealing with the problem of consumption, basically. Another question, but I don't know how to answer it correctly, so it's not yet mature, but the question was asked correctly. Anyway, the relationship with the models made with the computer for the study of the mechanics of the computer.

37:30 That is, a physicist who today studies the mechanics of the computer uses the models with the computer. Yes, yes, yes. What is the link between this and, let's say, the implementation? In other words, there are three categories, the semi-simplicial, the topos-vice, the topos-continuum, and the topos-analytical. There are two models of the continuum in the broadest sense, with discrete models, in fact, if you look at them as objects of a category, they are not discrete because they are connected. You don't know them like this, but they are also very different from the other types of mathematics, but you can make a transition in some way. Geometric realization, which in the simplest cases is an additional punctuation, makes two topos. And in the study of this, the method of infinite triangles, this is the principle of realization. As I said, the plans to build a house are discrete in a certain sense, but in another sense, the relationship between them is not completely discrete. As a model, it can be said that you can take a geometric representation, which is not really the kind of dedican thing. In fact, Hermann Weill has already formulated the dedican thing. The same fundamental content, but instead of talking about a net, we talk about space and infinite dimensions everywhere, it's the same kind of thing.

40:00 The aspect is the realization of something. There is a final map also in that context. So the space could be the edge of a chamber where we want to put waste or this kind of thing. Approximation of these types of spaces using diagrams and, in essence, the process of geometric realizations. Maybe there will be additional functions between the models of these discussions. It's too generic, I don't know.