Morning discussions incl. FW Lawvere & M Wright / further discussions

0:00 It is often read that it was a difficulty, the foundations of analysis, it was a difficulty 300 years ago, but recently it has been resolved, in the last 40 years. Thank you for your attention.

2:30 This type of problem is born with differential calculus and the problem of the fundamentals of differential calculus. This is called a problem. The so-called resolution in the last 40 years is mentioned in these circles that I have indicated. And non-standard analysis. Robinson was a good friend, but I think the importance of this analysis is very exaggerated. He himself wrote in 1964, in Tel Aviv, I heard his philosophy, what was called formalism in 1964. I'm going to explain these things a little more, but I want to encourage you to do so. In his first work on non-standards, he quoted Leibniz, saying that in order to explain differential calculus,

5:00 you want to find a real extensibility, like complex numbers, which in some ways become... Already there, one should rest a little, because everyone knows that the Ferrarian extension is not an extension that preserves all logic. It preserves certain fundamental things, but it is important because it exactly changes other things. In my opinion, after all these years of study, we can see that, in fact, instead of a... A much simpler and more direct response to Leibniz was to have another quadratic extension, such as a ring, which is no longer a square, but a zero, because the multiplication of this ring is exactly, that is, each element can be represented in this way, but... The product is simply the product of the central part, constant, plus a certain expression for epsilon, which is a0, b1, plus...

7:30 And if we think that a0 is somehow a variable, a0... On the other hand, there is the term of the derivative, which is exactly the same as the famous Leibniz rule for the derivative of quantity. The fundamental rule, specifically published by Leibniz, is perhaps this one. I think it is not explicitly found in Newton, I have never indicated this, but in any case in Leibniz it is completely explicit, this algebraic rule that is used to move forward with the differential calculation, its fundamental contribution, one could say, to analysis and, in particular, algebra. This has been a much simpler answer to Leibniz's idea of expressing a differential calculation with a complex form extension, i.e. a squared extension, but naturally not the same. Neither is an extension that preserves all the logic of the first order. In reality, the analysis is not stable, especially in the 20th century. These are simple things, but you have to say them. Instead, Valentani's proposal is that instead there should be a type of extension

10:00 that has all the properties of the first volume as the real one, but in some way in contrasts between the two. These are standard concepts. These are the standards that are not there, they are non-standard. In fact, there are rules to make a demonstration and to make a translation of a demonstration using this in the normal language of mathematics. Procedure to translate proofs from one system to the other, that is to say, from the non-standard system in the language of mathematics and algebra. And the positive product of this is that, in a sense, the deductions that must be made at a certain level are simpler. Because there are, fundamentally, in the language of the last 150 years, there are alternations of quantifiers of this type. To define the derivative, to define the integral, complicated logic is used, in a certain sense, complicated in the technical sense, because there are these alternations of quantifiers. But in the non-standard world, certain things can be expressed without using them. There are some multiplicators, but the same content, in a way, can be expressed without using so many multiplicators. So, these principles are all proposed. The use of these so-called dual numbers, already of dual studies and...

12:30 I forgot, but there was an Italian who published, also in Palermo's accounts, what is it called, attempts to do mechanics, this is, I don't know, 100 years ago, 70, 80 years ago, attempts to do mechanics using these so-called non-powerful quantities, but it was not possible. In my opinion, there is a very simple reason, among others, because this proposal has not yet been accepted, and this explains it very well. The concept of category is missing, but the concept of category is one of the fundamental ones. Every application of dominion and co-dominion is simple, but to develop this concept, this foundation for analysis, you have to be very careful about what is co-dominion, what is dominion, and you can do it, but without the category, without this knowledge of what is co-dominion. After a few pages, we come to the conclusion that this must be the power of the other yes and that of the other no, we don't know, but by using the very simple rules of domain, domain, culture, reality, etc., we can resolve these conclusions, but without, it seems, that this expresses, in fact, autobiographically, The geometric vision has always been, at least for the last 100 years, the differential calculations, the concept that in the infinitesimal, a curve becomes straight, and this very intuitive principle has also been denied by philosophers.

15:00 In other words, we use tangential uniqueness more and more every day, and the tangential uniqueness of the things they wanted to express is just a little stronger than the usual justification for tangentialness, because it means that, in fact, it is the curve itself that becomes perfect, not an approximation. And so, if we take the part of the grid where the square root is zero, the concept is that every curve, if we make the jurisdiction of this small part, becomes equal to the grid of events, that is, it means that exactly the intersections of the two are equal. The only thing is that this image is often expressed in these words that we have found in the last 30 years.

17:30 In fact, you can repair everything numerously, but using, as I said, punctuality, Now, there is another aspect, not the most significant one, but the most general one, which is to talk about how to accentuate or fundamentalize language, illustrations, symbols, expressions, language, language classifications, etc. It is a point of view that makes an emphasis on the subjective aspect of science. In other words, on the aspect of the movement of our thoughts, the movement of our imaginations, let's say, without referring directly to the meaning, in this sense, the subjective aspect. This image is not real, but objective. There are many confusions. They all come from a very real objective conclusion. Objective and subjective, in my opinion, are both aspects of thought, strongly opposed aspects, contradictory, both of which are necessary, theoretically necessary, for the group of sciences. And all this is a reflection of reality, of aspects of reality. But it is not the same, it should not be. I have done a lot of mathematics of this type, but in reality there are many things, of course, not only in reality, there is a part of reality, there is thought, especially collective thought, but also the thought of each one, and the approach between collective thought and thought of each one.

20:00 In fact, I think that this report is the most fundamental one to understand today's problems, but instead of going into the definition of thought, remembering that it is thought as something partially collective, thought has two aspects. It is a part of reality, but on the other hand, The function is to reflect the dense reality of these two aspects. It would still be a mistake to identify these two aspects, but if it is not part of reality, how can we accept it? It is a paradox to forget that these two aspects exist, but in particular because thought is part of reality. And because thought reflects everything, in particular, it reflects this context itself. So there is an image of thought in reality, an image that always exists, and an image also of thought in thought, which is part of the image of reality, Reflection is still a reflection of an image. So this would be a theory, that is, the part of the thought that consists, in a large way, of the philosophy of the cosmopolitan. So when I say objective, it means this. It is the aspect of thought that tries to recover reality, to form an image of reality. There is the subjective aspect, where we go to study the process of thought itself.

22:30 So, the principles of this type of science are not, for example, the logic of subjectivity and logic, the formula of administration, etc. etc. etc. On the other hand, there are the concepts. These are concepts, they are attempts to form an image of reality. Of course, there is a very fundamental aspect because the objective would never be like reality in this sense, in a certain aspect, that reality is always particular. The objective is always general, always, so the way in which the objective reflects is complicated, in the sense that we are going to deal with the world by always using general concepts, every concept in principle is general, but the way we talk is always... Now, with this background of my use of the word... The resolution of the non-standard analysis of the problem of continuity is a subjective, which makes sense that in some way the subjective is more fundamental between the two. But it can be seen from my image that it is clear that the objective dominates the subjective, because the content is one or the other. So the subjective is indispensable, but instead it is subordinate to the objective aspect. Humanism, precisely, in its instances, in the field of mathematics, is the philosophy that says that the objective is a kind of dream.

25:00 Piano, for example, is strictly said this way. If we talk in school about a line, it's a lie, because they don't exist, they don't exist. It's a symbol and nothing else. This has been repeated many times by Piano and, in principle, other formalists follow this philosophy. From the point of view of philosophy, which is called Subjective Idealism. Subjective Idealism, in general, says that it is not the thought that is contained and reflects reality, but instead that it is reality that is part of the ultimate thought, and that is so. Or through the thought of one or the thought of none. There are various forms of idealism, but both of them have an image exactly the same. Subjective idealism, therefore, is what says more about the subjective aspect of thought than the fundamental aspect of the world. The world consists of my impressions. This is it. The concept of Berkeley and the concept of Dalai Lama. There are, in fact, other variations, not only of formalism, but also of traditionalism, which is usually done in contrast with formalism, but in a substantial content, in my opinion, they are the same. They are in a slightly different form, perhaps, but they are the same in essence.

27:30 I have mentioned Dalai Lama because I saw yesterday that the American government has paid millions of dollars to develop new systems to teach geometry in schools and this theory to teach geometry in schools It is explicitly based on a pair of words of the Dalai Lama, which is exactly this, which is not the word, not the meaning, but it is explained, which in essence is truth. And what is the result of this in the teaching? Of course, the result is that the teaching is here and I am doing this project. In this project, it is said that it is wrong to teach concepts. Teaching concepts would be like, for example, China's electric car. Science consists of concepts, but this means that science... This is already authoritarianism, authoritarianism, authoritarianism, yes, yes, yes, and it would be an oppression in some way for students to explicitly mention concepts. Instead, you have to show things without explanation, and perhaps later certain concepts are developed, but it is all based on the lack of concepts. The experience of each one, but not only science, not only cognitive experiences that have already been conceived with certain objectives. In fact, there are many variations on this topic today of subjective realism.

30:00 In my opinion, what destroys the teaching of geometry is that Bourbaki was a formalist. So I said that this fantasy is often read today, but it is also often read in other fantasies, that Bourbaki is a formalist. And to understand this, we have to go a little further back. Looking on the internet in the circle of philosophy, we find Hilbert. We know the greatest mathematicians of the last century, Hilbert, right? But for today's philosophy, it seems that the meaning of Hilbert is that Hilbert is a formalist. That's it. A formalist as in France, right? In this case, we know that it is false. But why? We have to look into it. Of course, Hilbert made several statements about formalism in other universities. But why and how? To solve certain specific questions in mathematics, Hilbert proposed a program to iterate the formulas.

32:30 We need to objectify the formulas, look at the formulas as mathematical objects in themselves, as natural numbers, there is another type of negative infinities, but we have to work with these negative infinities, and let's see if we can demonstrate something about this type of system, using various conjectures. The majority of these combinations have been verified, but among the fundamental combinations, few have been proven false. That's it. This program was the contribution of Hilbert to formalism. In his vision of mathematics, Hilbert was a formalist in the sense that These are the formulas for the content of the work. It would be clearly Spanish. In my opinion, it is a possible point of view for certain problems. In fact, instead of denying Hilbert, the concept of objectification of the formula should have been denied, Hilbert was much more serious. In fact, he wrote several things on the dialectical aspect of intuition, and so on. Geometry and imagination. He understood the dialectical aspect. And then we come to Bourbaki. I have made some formalist statements, but I have known several of them, for example my professor in Ireland, because they have said, ah, we are formalists, for completely pragmatic reasons.

35:00 We wanted to avoid any discussion with these philosophies, so we did it in mathematics. Populism, that's it, Norway, we do mathematics. This is the approach. Now, it is an approach perhaps not too serious, you could say, but clearly it is not an approach of the philosophy of the subject. No, instead there were also their geometries, analysts and mathematicians. In other words, both are important, concepts and demonstrations. In addition, in these speeches on how to destroy the teaching of geometry, these things are quoted to prevent someone from reading Bourbaki. Bourbaki has already surpassed, because he was a formalist, and we are of a different type. So it's connected to today, maybe this has been too much in the sense of a philosophical background, but to think, when I arrive, I want to continue, the concept is that one could clarify the objective concept of continuing using geometry, analysis, etc. to develop mathematics, which for us is... Humanity, in general, uses them to navigate concepts and ideas. I had a discussion with a computer scientist a few years ago, I've known him for a long time, he's quite serious, where he proved the materialistic gesture. Materialism is precisely this image of the relationship between reality and thought, more or less.

37:30 He proved that, yes, the real world... But, after we entered into the mathematical discourse, he said, in fact, there are only symbols, there are discrete processes, there is only the discrete. Continuity is a dream that has been surpassed by thousands of years. So I took advantage of this fact and said that I was an archivist, but no, the teaching is precisely, in the first step, exactly a model based on our concept of reality in an infinite way, not in a negative way, but in another kind of infinity, infinitely deep. Even if we make a very, very simple model of this part, there is this aspect of our documental thought, on the one hand, and the content of our thought, which is always something real, in terms of concepts. There is also an interval, which is an objective property, and this interval is connected, in the sense that every morphism must be in a discrete space, like the one we have here. You can factorize between one point and the other, in other words, it's constant. Every morphism is constant. This is the definition of connessor.

40:00 From this example you can already see the contrast, despite the thought that this is the final model. But on the other hand, there is this contrast between a concept of continuity and force. And in fact, this type of thing is often used in architecture, in engineering, in the sense that these finished diagrams serve to How to build a real house, a real house that consists of wood, steel, etc. So, three aspects of what I said. First, to understand the continuum, you must have not only a concept of this object, but also a concept of the category in which this object lives. Both are important. They are connected or are connected in a higher sense, they have aspects of continuity. But on the other hand, there are also many categories of this type in general, with which one had to vote to understand exactly all these things. In particular, the relationship between a topos of a communitarian type, as in simple terms, and a smooth topos, or a topos of continuity, in a more visual sense. Geometric realisation. In fact, those who have certain X powers are still connected to the same property, that is, to a discrete and constant space.

42:30 So, let's say, the X-th category, as it is true, is a measure of the degree of connection. If X is equal to 1, it is part of this category. This is connection in the basic sense. But if X is equal to the circle, for example, this property means that I is simply connected in the sense of topology, in the sense of quantum mechanics. So in this class, X is a measure in a certain sense of I. And one must have a reasonable theory of what it means to have an object connected in a category of the right type to face. All these are concepts that can be expressed in phenomena, but they are not reduced. A reasonable theory, for example, in a category of continuous type, which has many non-phonesic objects, but in a discrete category, all discrete objects, or we can also say of negative infinity, In fact, every restricted space is a particular case between the various continuous spaces. Since zero is a case of quantity, continuity is a case of zero. There is a term called P0. This means to take the set of known components, that is, for each space, for example, P0 equals 1, something that is really known, it is only one component known, but instead the sum of two is equivalent.