Categorical theory-building

0:00 And also I skipped some parts, because in the Zara parts we just repeat a little bit what I already, you know, some standard thinking about category theory, so I skipped that. And OK, in the title I also didn't decide finally. Last thing I thought was something like towards categorical theory building, beyond the formal. So I'm trying to make the case that in category, say, logic category, mathematics, you can do something else than formal logic. But I'll try to be precise also in a part of this paper formal. And what I mean by, you know, something else than formal, it wouldn't be just informal in the sense of something great. It's exactly the thing which I dislike about this notion because there is something, let's say, kind of moral, say, formal is good and rigor, and anything is is somehow not rigor, and so I just try to analyze this differently. Okay, and in the first introduction part I actually tried to analyze this rather specific notion of language, it's a colloquial notion, mathematicians speak like that about said theoretic language, if you call it theoretic language, and when I heard it first, I just thought it's, you know, when they don't want to be serious, as philosophers say about foundation, they wouldn't say. sets erratic foundation because normally mathematicians don't care about foundation or even don't like foundation. And I always thought, and I think it's true in a sense, that they would just say language and they just kind of way not be serious and not responsible. But then I thought that probably there is something a little bit more about this idea of language. Okay, so the term language used in mathematics to refer to a theory which grasps common feature of a large range within all of other mathematical theories and so can serve as a unifying conceptual framework for these later theories. What makes a difference between this specific notion of language and the more traditional

2:30 notion of conceptual scheme or foundations, but it's not the same thing, is the assumption that a given mathematical theory can be formulated in different languages and translated from one language into another. And that, I think, is something really interesting and, in a sense, non-trivial part of this idea of mathematical language. For a simple example, think about Pythagorean theorem. First, as it is formulated in Euclid's elements, and then as it is presented in any modern technical theory. You know, in Euclid's elements, basically, this theorem says that if you make squares, and because there is some English name for this construction, a Russian thing, it's like Pythagorean trousers. And basically it says that you can cut these two things in such a way that you can recompose this big thing, you know. And when even, I even don't speak because sometimes now rather, you know, algebraic things called Pythagorean theorem, but even if we take something like A, B, C, C square, A square, plus B square, even that, actually, it's not so trivial in which sense is the same thing. It's somehow obvious, right? And that's true in a sense. But, of course, if you try to really translate that, you need to hear all the zero real numbers, a lot of things. And then you can ask, Of course, it works as translation only when it can be applied to other examples, not only for Pythagorean theorem, right? But then you can look where it does apply, this translation, where it doesn't, and actually it's rather limited class than it does, so if you try to be precise with what's going on here, you know, it's really, you can say a lot, but okay, that's actually a topic of different paper which I sent by email. So I just leave. What are exact conditions of identity of theories and particular theorems through such translation is an interesting question which I shall not study here. But anticipating what follows, I remark that this question cannot be given a precise answer through any straightforward application of formal methods for the simple reason that the formalization required for it is itself is nothing but a particular case of the translation phenomenon in question and here just i made a note when i i point to this problem

5:00 to proponents of formal methods in philosophy they usually agree that there is indeed a sense in which formalization always remains somewhat informal affair you know air of being paradox but progress not progress at all but then actually i i can name it i spoke if you know ulys mulines who is dean philosophy munich and he has this big project of formalizing physics and then i just and he writes kind of you know burbaki-like structural mathematical formula makes his things rather unreadable but then i i just ask you have this formalism on one hand right And the physical theory is also something mathematical, it's not just, you know. So you do have these, say, two mathematical structures. And the good question is, in which sense one is good representation of the other? And I think it's not good. No physicist would recognize this reconstruction as something. is this theory, but then of course you can say, okay, I just make this particular reconstruction for some specific philosophical purposes, but I think that you still need to be precise, I mean not just leave it informal formalization, but I don't know, make kind of functor or something which shows that there is kind of map from one thing to another which really works, which is not just, yeah. And that wouldn't be part of formalization, because that's what formalization is. You get a kind of functor or whatever, which has some nice properties, let's say. But just something which already was presupposed when people formalize stuff, you know, when they speak about logical form, etc. Okay. The language of set theory has become the principal mathematical language in mathematics of the 20th century. The most immediate effect of using said theoretic language in mathematics is taking the extensional viewpoint on mathematical concepts. In ancient times, mathematicians tend to work with objects rather than just with concepts. In other words, mathematicians tend to hypostatize or reify the concepts that's actually in Plato's Republic. He somehow almost defines what's mathematical, right, because that process is a philosophy just simply about ideas.

7:30 but in mathematics it means kind of two ideas you can play with, and that's something not really philosophical, but probably in math. I think it's a rather fair description. And so what I try to say is that this Satsuri somehow allows to make it more proper than before. This is a refined concept. The concept of natural number is a typical example. In the traditional naive view, supported by school education, until today, a number is an object-like thing one can operate with. However, traditional mathematics deals only with particular numbers, like 1, 2, 3, but involves no object corresponding to the general concept of natural number. So we don't have concept number, but we don't have object number, right? We just have particular numbers. Setsury allows for it in the following way. The general concept of natural number gets represented on whatever if not identified with its extension. So we just can refer to the set of natural number conventionally denoted by N. I think it's really something, you know, a little bit this side, all this story about foundations, whatever, which really makes set series something, say, very convenient. You have this N as a symbol, and you can make an operation with that series, play with, and that really represents all natural numbers, and that's something that wouldn't be possible without at least, say, a naive set series. Wouldn't be such a thing. Of course, people couldn't think about numbers. In this way, before Kantor developed a theory of infinite sets. So that was actually important to that theory, like infinite set, not just set. So the extension of viewpoint made possible by Kantor's theory pushed mathematical hyposteization much further forward. This is, in my view, a major factor explaining the success of the set theoretic language in mathematics of the 20th century. I remind Hilbert's description of set theoretic mathematics as paradise. Okay, now I skip part where I tend to use very basic about category theory, but I did yesterday. So I continue. The use of mathematical theory as a language should be distinguished from its use as foundations. A strong foundational claim of set theory amounts to the following. And here, probably, you might correct me because you might know this better. All basic mathematical constructions can and should be recasted as set theoretical constructions.

10:00 ambiguous, and all mathematical proofs can and should be recasted as set theoretic proof, that is, proof is a formal extrematic set pseudo, like Cernillo-Franco, for example. Most working mathematicians agree that set theoretic language is quite useful, but only very few of them, if any, agree that mathematics in some sense reduces to set theory. Actually, the reason interesting thing, I think that Hintzik, how was he, thought at least about that, that actually how set theory is applied, meaning this kind of structural is Burbaki's mathematics, is actually rather different from this idea of set theoretic foundation in the sense that you have this list of axioms, you can't deduce any theory amounts of them. What people do is rather kind of monotheory, so you have this notion of set, and as I explained put structure, so-called, you define operations, relations, whatever you want, and what you basically do, you have some short list of axioms, say, for vector space or for topological space, and then you do model with sets, and before you have the set theory. So it's rather like you're making models, rather than you just deduce these facts about, say, vector space, whatever, from axioms of set theory. That's quite different business, how these two things relate to each other, actually. It's a question which, for myself, at least remains open, probably, has a really good answer for it. I don't know. Okay? Remark that in order to use set theory's foundation, one needs a notion of abstract set rather than concrete set of points, numbers, etc. When we say n, it's a set of some things called numbers, right? It's not an abstract set. And what Sir Melo did exactly, he just introduced this notion. Actually, what these abstract sets are sets of? Kant's answer is the following. Abstract sets are sets of pure unit, later, Eisen. He called that, so it was just something. which he distinguishes from sets. And Cermelo's answer was abstract sets are sets of sets.

12:30 So you just don't have these two types of things. You have one type of primitive thing and you have this relation, membership and they somehow equal basis. They are not distinguished as different types. Then you put all your action, you forbid for example things. like this by foundation action etc but still they are not distinguished as types okay uh a similar case has been made for category theory in the early 60s by laviere who proposed to think categorically to begin with uh first putting category theory on an extrematic basis independent of set theory and then recasting convenient mathematical concepts terms. I already mentioned that yesterday, right, in his paper, 64, like here, showed how an ocean set can be developed from this basis, from this category of sets, and in the 66th paper, this also put forward an ocean of category of categories, like, there is nothing in mind, it's like sets of sets, it's no more, it's literally, if you start to think They call this funsters, transformation, you somehow have an impression that it's some very high-water absolute concept. But actually the idea was a little bit like this, just think about it simply. If you have something like your primitive construction, right, morphism, and you think about it as falter to begin with, so your objects you call categories to begin with, and that makes another category, so again we don't have some kind of object like numbers, and then sets of numbers, but we just have categories. of categories. He actually, in this paper, he called the category of categories, somehow like a set of all sets, and apparently that didn't, how say, imply that kind of paradoxes, or you can, but then actually he didn't like this idea, rather we can speak about different

15:00 categories of categories, like different sets of sets, okay. in this early paper Ibera starts with first order axiomatic theories of categories similar to but quite independent of axiomatic set theories however in his recent paper 2003 this also takes a different approach and hopes for a notion of foundation understood I quote in a common sense way rather than in the speculative way of the Balzano, Freyne, Piano, Russell tradition, Bill O'Reilly is absolutely militant But on the other hand, what I think is not something, because he doesn't actually propose a clear idea of what would be this different notion of foundation, because he says it's about commonsensical. It's not very radical, I would say. and he speaks about kind of pedagogical, you know, he became old, you know, 70, so for him now foundation is kind of pedagogical concept, you shouldn't teach student writing, something like Euclid, introduce basic, but he doesn't propose any, I would say, philosophically, in my view, this alternative way of foundation and actually it's my ambition to try to make that I'm not sure and this is my motivation of writing this paper so in this paper 2003 Lavir doesn't rely upon the standard formal axiomatic method any longer I refer to that thing, speculative tradition, in the formal experiment, I don't know, about it on that. This change of Laverius' view on category theoretic foundation of mathematics seems to be remarkable. Primaficial, Laverius' choice in favor of the commonsensical notion of foundation rather than speculative one, looks purely pragmatic. A normal worker mathematician who doesn't care much about speculative foundation will approve the need of some commonsensical foundation of his science. Because refusal from speculative foundations and formal axiomatic method associated with it, albeit not from axiomatic method as such, can be also understood from a different viewpoint.

17:30 True formal axiomatic method allows for building not only a series of sets, but also a series of lines and points, like geometries, of paths and holes, meteorologists and all what not. So, Wayne did not use it for developing an axiomatic theory of categories and then tried this theory as a tentative foundation, alternative to known set theoretic foundation. Here is the reason why. That's not a very precise reason, but that's an introduction. There exists an intimate link between set theory and formal axiomatic method, which has no analogies in other cases. To see it reminds the idea of Hilbert Gronlaggen, where formal axiomatic method has been first introduced, probably then, Pasch did something similar, but I'm not talking about this historical thing, probably, it's not virtually the first, it's a principle reference, anyway. Geometrical points and straight lines are thought of as abstract things, as Hilbert put it, of two different types. According to Hilbert, one is free to imagine these things in any way one likes, or not imagine them in any particular way at all. However, abstract and unspecific the notion of thing, and what here might seem, one cannot avoid making certain assumptions about it. So in order to make Hilbert's proposal precise, an approach of general theory of things is needed. As it has been shown by Tarski and others, set theory proves appropriate for that purpose. It has been shown that sets provide the standard Tarski semantics for classical, for stotology, and for theories are axiomatized by the logic. Hence, the idea to use set theory on a pair with formal axiomatic methods and the claim that mathematics is ultimately about sets. Building an axiomatic set theory becomes then rather tricky business, since any such theory involves an infinite regress. In order to build an axiomatic theory of sets, one needs to assume some possibly different notion of set in advance of semantic properties. This and other relevant problems about set theory and logic have been scrutinized by mathematicians and philosophers for the 20th century. I shall not explore this vast issue here, but only remark that given the intricate relationships between set theory and formal semantic method, within the overall construction of foundational mathematics developed in the 20th century, in any of this version, it is very improbable in my view that this construction could allow anything like replacement of sets by categories or something else. To get rid of speculative foundations, one would need to change the method and the notion of foundations.

20:00 The principal aim of this paper is to show that category theory, like set theory, can be indeed provided with its proper method of theory building, which applies quite young category theory itself. I shall call this new method categorical. If the method comes with its specific notion of theory, it also explains it. In the next section, I stress a distinctive feature of formal axiomatic method, synagogues traditional axiomatic method, which controls the notion of interpretation relevant to mathematics. Then I argue that formal axiomatic method doesn't provide an adequate treatment of mathematical interpretation, while categorical method does. I conclude with some general epistemological remark. Actually, again, about this interpretation, a different paper with programmers. yesterday already. But I think it's interesting because you know the Diltay, the guy who introduced this distinction, Naturswissenschaften, Gaiswissenschaften, and he didn't actually, the term is why I couldn't find out, he took actually, this Gaiswissenschaften was a translation of Mill's term Moral Sciences from his logic. There was a German translation very early, just a few years after Mim published this book. In that translation, the guy translated, I can't remember the name, he used this Geistwissenschaften, but then Dilltei somehow, you know, semitized that. And his principal idea of Dilltei was that Geistwissenschaften are about interpretation, or somehow principal method of Geistwissenschaften, interpretation, while So he tried to make this distinction on that basis and the funny thing is that exactly about that time, you know, in the works of, before Gauss, that notion of interpretation became important in mathematics. Actually, in 1865, there was a very important paper of Beltrami, an Italian mathematician, who gave what's now known as a model of Lobachevsky journalism.

22:30 And in the title, it's exactly the title, about interpretation of Lobachevsky journalism. At the same time, this guy also said, okay, interpretation is something, guys, we see a chance of nothing to do with math and science, I don't know, in mathematics, at least in mathematics, in physics, probably just going to come. Okay, I just made an introduction, I just started the second part, which is called Formal Axiomatic Method in a Nutshell, about Huberth Brunelago. Today, a mathematical student can read in various textbooks that formal axiomatic methods invented by Hilbert is nothing but a perfectionate version of the traditional axiomatic methods known since Euclid. True, Hilbert certainly had Euclid elements in mind, writing his Grundlaggen, so his method can be rightly seen as a modification of Euclids. However, I don't think that the description of this modification as perfectioning sheds a lot of light on it. To see clearly what is specific for formal axiomatic method, as distinguished from more traditional versions of axiomatic methods, a historical regression seems to me helpful. Soon after the publication, actually about Euclid, it was a topic of my PhD, but it's just absolutely wrong, I think, plainly wrong, just think that in Euclid you have something like axioms and deductions from axioms, in a sense, then we would speak deduction and logic. absolutely not the case they're not even saying that the kind of imperfect deduction you should operate more just a different project and one second thing we have is postulates you know which for georgians is more important probably the actions and postulates they are even if you you read it in greek they are not propositions at all they are i uh it's something it's actually formulated as an infinitive construction you know just draw a line and if you read good translation it's like his translation he preserved that he doesn't put postulative propositions he said to draw the line from given point to the uniform and so i think a very very natural way to interpret that is kind of primitive operation you know which is taken for granted yeah and then out of you develop, you make constructions. But Hilbert's reconstruction...

25:00 Actually, this is somewhat... Hilbert's reconstruction of... you think this is proposition-like? Exactly, exactly. What I would do to say... Yeah, yeah, yeah. But that's what makes the difference. But Euclid is the same thing, but less perfect somehow, you should make your derivation more precise. It's just a different project. But now I will make again another distinction, because I will say a little bit of this controversy between Hilbert and Frege. because Frege actually had this idea which probably could be good as a shade with Aristotle rather but forget about Aristotle because he had this idea that we should make some actions which kind of first truth to me which are true because we cannot do anything he was still good pupil of Aristotle in the sense that you can't require to prove everything So you should just choose what is more likely than any other, the first truth. And then you develop by precise logical rule all this stuff. And the thing is that that was absolutely not what Hewlett was. And now I'm going to talk about, Frege, he didn't accept absolutely Hewlett. He just told us an intuition. And soon after the publication of Hilbert Groves-Blaggen in 1899, Frege sent Hilbert a letter, the precise date is missing, containing a severe criticism of Hilbert's approach. Freyja has the following traditional understanding of axiomatic method in mind. A given theory starts with axioms, which are truths taken for granted. These non-demonstrable truths are truths about certain objects, what these objects are, depends on the given theory. The theory proceeds with inferences from the axioms made according to rules or winning theories, which must be also assumed. As a matter of course, for any given theory, meanings of terms used in its axioms and further inferences might be univocally fixed once and for all. In search of what he says, I just give Putin's exploitation to make sure that.

27:30 The general epistemological view, dating back to Aristotle, has been recently labelled classical model of science. You know, in Amsterdam there's people, they make these conferences on classical model of science. I think more or less the same. Frege pointed to Hilbert that his Grundlagen falls short of meeting the requirements just mentioned, and in particular, the unikivokale, unikivokale, actually, what Frege says to Hilbert, he says, okay, what are your axioms? he would say okay, this is all actions like that to give points, everything. But what's that point? Then he would outpost now say okay, then you can think how you want. But then they are not true right? These uninterpreted actions, they are not true and since they are not true, you cannot infer from anything So it just breaks, for Frege, the whole idea of logic or science, whatever. This idea that you can think something which you would interpret afterwards. Something, for Frege, absolutely crazy. Then he tries then to come around to make some sense of that, say, okay, this is kind of scheming, but actually he never agrees. Actually, it's interesting, because Hilbert replied once, and I'm going to quote this reply, then Freddie again put him in a big, big letter, and Hilbert just, you know, ignored that. But then the discussion continued very interesting, because in Vienna there was a journal, there were these people, Tomá and Kostel, and they actually, this Kostel, very interesting guy, now fortunately all this is on the internet, so if you go to Vienna you can find all this stuff on the internet. And this Oster, for example, he wrote the whole book on, and basically they defended this Hilbert approach. In a sense, probably we shouldn't attribute the whole idea to Hilbert, that's what I didn't hear, like, because, of course, it was something, something, rather general consensus.

30:00 That came from rather specific things in mathematics. So all his basic approach to mathematics was, in a sense, quite external, because he just didn't understand what was going on in mathematics. Brilliant. Yeah. Well, yeah, I put here, actually I have this in German. Okay? I think the difference is different between, we would today say that Frey had in mind and axioms should be qualified, justified by an intended model. If you have such an intended model, you would like to... But he is basically... He would have the idea of unintended, uninterpreted formal system. But you can say that, but I think it's important that Frey absolutely explicitly rejected the whole idea of interpretation. He said, axiom is nothing to be interpreted in that or not in that, because unless, it's just something not, it seems at all. He was absolutely against this idea of interpretation. He didn't even need the idea of interpretation because all the interpretation by axioms came from something elsewhere. But he was explicitly gained then. Actually, it's a good thing to probably read more, but there is one small book called Hilbert Frege's correspondence translated into English, what I first read, and then there are fragments. But then I think it's a good idea to really read more. Here is his reply. It's actually quite often quoted now. At one point, some systems of things, for example, the system of the Korn. Dear Gesetz, Sean Steinfeld. Sean Steinfeld, yeah.

32:30 Yeah, yeah. And then, very important... Yeah, it's his first answer. And then, very, very important, because it's something that, for me at least, for my reconstruction, because it's something that Hilbert says very explicitly here, but then normally when people who speak to this mother or whatever, they absolutely forget it. Man braucht ja nur eine umkehrbare, eindeutige Transformation anzuwenden, um festzusetzen, dass die Aktionen für die transformierten Dinge, die entsprechend gleich sein sollen. then it's also a different another part of quotation where he speaks about physics okay yeah so he said that okay it's very interesting in my view all this controversy but i try to go further since the point is allowed to be or thought of as a system of love or chimney sweeps beer mug according to another popular saying and all this within one and the same theory frege's notion of semantic method is certainly no longer relevant but let's look for a serious mathematical reason behind hulbert colorful result in the end of the quotation Hilbert refers to the duality principle of projected geometry. You know, you know things, no? Now, the thing is that, okay, you have this axiom in Euclidean geometry, then for two points there is just only line, huh? And you actually have, now just try to replace points by lines and lines by points.

35:00 turns out the way around. And you say for two lines, there is one point. So here you have this duality. And of course, it doesn't work completely in Euclidean geometry, because it has parallelized lines to no point. And projected geometry is exactly when you say you somehow add kind of IT or point, and here you obtain this complete duality. So, and specific, particular reason. It was actually discovered, I don't know, also I can't elaborate, there was a guy called Jed Joné, who worked in Switzerland, but he put it as a duality principle, he just tried to make it a kind of big philosophical principle out of that. I just recommend paper of Ernst Nagel, which probably I mentioned here so-called. He very, very, very... The formation of modern conception of formal logic in the development of geometry. It's of 39, and this is on G-Store. We don't have J-Store at the university, but J-Store is at the railway, the Strasbourg that they pass. So if you search Google by this and Ernst Nagel, you should find it. Otherwise it's kind of untourable. in particular in project geometry one may think of lines as points think of points as lines such a liberal treatment of mathematical object is common in today mathematics okay in the end of 19th century it was not yet common but a number of important examples Actually, I wrote this before reading this paper of Nagy, because Nagy really made complete research of this question there. But basically, I didn't say something really wrong, but I did. Hilbert-Gruhl-Laggen provided justification for this apparently careless conceptual game.

37:30 The problem Hilbert addresses can be formulated as follows. How to construe mathematical concepts which can be occasionally interpreted as another concept. So the whole reason for this interpretation came from geometry, not only from geometry. There was another very important thing, who proposed to think about complex numbers as points of the plane. Something really illuminating. But for us, for our, I don't know, mathematical education is something that we eat, you know, in school, that you can think number, S, point, something, but of course if it was not at all classical, there is nothing like that. it was rather rather new feature which emerged in the 19th century that's why I think this question of I don't know of course it's not humanity in the sense of Schleiermacher or Dillte but at least you you see that that concept interpretations and how it became operational became very important how to formulate a theory in which basic concepts are defined only up to interpretation. Hilbert answers roughly this. One should first conceive of mathematical objects as bare things, as he said, possibly of different types, standing and sending certain relations to each other, and then describe these relations stipulating their formal properties as axioms. Any system of things, his term, Hilbert's term, which hold relations, satisfying the axiom, would be a model we would now call it model of the theory. as Hilbert makes it explicit and that's a crucial point in the quoted passage he thinks about reinterpretations of theories as reversible one-to-one transformations i.e. as isomorphism in the sense which we speak a category theory Hilbert's clearness of an absent or late exposition of formal method allows one to see its limits mutual interpretation of mathematical theories are generally not reversible to a specific case. To build a theory up to isomorphism is not the same thing as to build a theory up to interpretation, for interpretations are generally not isomorphism. Let me now demonstrate this fact using a historical example which was already available to Hilbert. Now probably I'll skip because I don't give a big example about this in Bertrami works,

40:00 the way he interprets. Basically what Berkrami did, he found this surface called pseudo-sphere, which is surface in three-dimensional including space, such that its intrinsic geometry is Lobachirskine, actually not exactly, because it has singularities where you don't have infinite line there. Geometry was a notion introduced by Gauss already in 1927, meaning rather that you have geodesics. Geodesics are the shortest lines between points, right? And think of these geodesics as striped lines in this sense of interpretation. you get your Labacherskin geometry. And actually, when he found that first paper of Labacherskin, he just thought, OK, all this stuff about non-Euclidean geometry's books is what really Labacherskin did. He just described that very particular mathematical geometrical object in Euclidean space, so you just don't need to come to non-Euclidean geometry. Kind of crazy misinterpretation. First, because it was not really a model, it was only a patch of model, and Hilbert really quickly proved that there is no full model of this type. And the first thing, which probably was even more important for him, is that he didn't work for dimension 3. He interpreted the Lobachevsky plane, right? But you cannot, even if you use something like four-dimensional Euclidean space, you don't get anything like Euclidean model of Lobachevsky's space. Actually, what Lobachevsky did was extremely interesting. It's extremely interesting. He just did absolutely different things. okay but then then he read the Riemann's memoir he was just actually it was very interesting because it was the guy who was actually a school teacher you know and he made this communication between Germany and Italy I think very very important you know he translated from German to French and Italian to

42:30 French, always published in this, I would say, Annals of the Quoi Normale Superior, all these people communicate through French, and I think at the time French Academy was absolutely conservative and, you know, they still were looking for proof of fifth postulate until very, very late, I don't know exactly, when all these people already been trying, they still didn't recognize it. And now my argument, what is my argument? That if you have a model like that, right? Is it where we have this one-to-one reversible transformation here? Where is it? I think there is none, actually. What we have, we have embedding, and that's how it's called today, of Lobachevsky, not good embedding, Lobachevsky plane into Euclidean space. And actually, after reading, in French translation, Riemann-Minua, he just found another idea. Okay, look, that is a manifold of constant negative coverage. And that is normal mathematical answer today. So in that same, you might call it the intended model, but in my view it's a little bit something more than that. because it is yeah and then of course you can see that very clearly you say just one manifold is embedded into another one and of course if you try to think about this categorically for the explain yesterday something imagine you think about category of manifolds right it's a very interesting mathematical that i can speak because shut up on that because This is really a very interesting topic, a category of manifolds, and you just say you have this embedding, monomorphism, which is not really monomorphism because of all this problem about singularity, and so on, so you have more standard, you understand what's going on, but thinking that that is not reversible. So, how to think it's reversible? You kind of carve part of Euclidean space, right? And say, this part is reversibly translated into, I don't know, some other, say, arithmetical model.

45:00 But that is wrong in the sense that you cannot carve it out, you know, you can just take something from Euclidean space and taking that part of my model. It's not reversible, it's not isomorphism. And actually, the same applies for these arithmetical models here, because it probably was all this geometry you can do with arithmetics, more or less, somehow you reduce everything to arithmetics, but of course, again, you reduce, say, interpret geometry, say, Euclidean geometry, but just geometry in arithmetics, but it doesn't work the other way around. I mean, you cannot just translate arithmetic into journalism in this way, you know. So, it doesn't work both ways. It's just, you find interpretation, but it's not reversible interpretation. And then, of course, if you somehow, to begin with, you have this notion of formal theory, and then probably it's not a problem, but my argument that I'm going to develop is that unless you don't have reversibility in this kind of class of isomorphic models the whole notion of formal theory doesn't make sense yeah that's I'll try to to say why okay okay let's keep