Category theory, structuralism, hermeneutics (contd.)

0:00 So this view of categoricity is unnecessary and misleading, I would say. And actually, if we really do things categorically, we can't explain it now. It's really clear. In reality, we have good categories of math. and categoristic in old sense would amount to saying we just have like category which reduces to one object with all three of our human beings. Which is kind of absurd. I think it never happens. Yeah. So, this very kind of happened. I think that humor confuses actually two notions of interpretation, of model. First, he thinks of interpretation of a human form of theory as appropriate, intuitive form in some kind of flesh, which can be associated with it. This is a philosophical, psychological, psychological, but not mathematical issue kind of question. Do different people imagine that system in different terms? But second, and more importantly, he thinks about the model of a human form of theory T is a specific construction, maybe there's another theory, T-prime, supply by some random M-prime. And he will not to realize also because of the second kind, like standard, the natural model, so general theory, right? And my claim is that there is no reason actually to report motion only for protein. If you think what they really share it for, just, I'm not going to discuss anything about intuition, but it's clear that it's nothing like a theory, right? can be better because there is a translation that participates in human theories, and interpretation of the theoretical component of the theme concept, as a theory to prime. This revised notion of interpretation of the translation cannot be extended to the case of intuitive funding, just because intuition is nothing like a theory, they cannot translate it completely to intuition, whatever intuition might mean. And second method, actually, how he put this all together? If you have two theories, right, let's take one of them, this target theory, as meta theory. So we can't forget that it is theory. We can think about it later. It's something like intuition.

2:30 And I don't have time, but I just about finished paper. In some cases, you'll be trying to play just with epistemic absurdities. For example, Levitschewski, instead of trying to find a model, in Hilbert's case, of his geometry, he actually, in the, how I say, huge geometry, what he did was found non-standard model of Euclidean plane within hyperbolic space. Which seems to be in this epistemic setting kind of an absurd thing to do. But mathematics is very clear. He just found this map which allowed him to make what they call hyperbolic mathematics. Actually, it's all fine. The scheme is wrong, it's not... Yeah, okay. Let's say, mathematics in different translations, not historical translations, let's say something which is a synchronic aspect of mathematics, are generally non-reversible, not isomotivism. And actually, that's something clear, because if you have isomotivism, idea that they would like hopes and one. And actually this example of duality is very, very specific. It shows what? It shows we might have non-trivial translation of a theory into itself. That's interesting, important part, right? But we can't generalize that example. It's rather special. Yeah. And of course, I would say that Hilbert ever thought that the arithmetic is either work to geometry, in a sense. But what did he do? Talking about the models of geometrical objects, Hilbert, of course, didn't mean to identify geometry with arithmetic. Yeah? But he thought he could kind of carve out a special arithmetical construction from its theory, consider it with appropriate mathematical losses, a self-stating embodiment of geometrical theory. Forget that theory is kind of meta-theory. We have this arithmetic construction, I mean, it is a model of geometrical axon, right? But the argument against, I think we're saying that you cannot forget, I'm saying, mother theory. It doesn't make sense. Just to kind of a science epistemic title meta-theory in many countries does make sense at all.

5:00 And in many cases it's not, I'm saying, that's it right. Here we can locate it. Given all that, probably we can kind of generalize what we have to say. Okay, let's include more pieces, not more and more pieces. So let's see what happens with it. And actually it But it does survive, at least in this kind of extent, it does survive. Because when we have the reversible level, isomorphism, we can think, say, AB, as people put it, after isomorphism, identify it after isomorphism. One account of this kind of thing is a great, great subtraction, right? And that is why, because this existence of isomorphism is equivalence relation. But just existence of the morphism is not equivalence relation. If we just have one reason, there is no sense in which we could say, okay, let's say A, B is the same thing represented by kind of structure, C. We can't do anything similar in this case. All this idea of, say, distinction as a form of theory model, I think it falls, that it doesn't survive. And that's unstructural. The level of structuralism allows for a rigorous definition of the general Rostromorphism. I mean the structure of Rostromorphism, the structure of a man. However, this framework is based on preferences as what we begin with. It was a very notion of structure required by the capital. So, thinking about Rostromorphism is very limited view on what Rostromorphism is, in that sense, misleading as far as we are trying to generalize it. And now this is a little cryptic, but I don't know how to develop it. But there is, I think, an algorithm showing that set theory is kind of nature's framework for this concept. It is not by chance that we have all this history about set theory in exactly that context. And just a hint how it works, that when we are talking about functions, right, or, say, maps between structure and set, but for me it's efficient, it would always go something like this, okay, here is AB, here we have elements, and okay, what is the map between them?

7:30 of something which sends each element of first or some element of the second, right? So, really, that is factor. But, look, on this elementary level, we have this anodic, actually, anodic pile. The idea is because, let's say, there is preference of order here, exactly because there is this condition we have 1 cent to 2 here, right? But if you just look on points, and exactly in cent theory we have this notion of non-ordered pi as kind of basic, as a pair in action, and we define the notion of ordered pi as rare artificial ways, right? But naturally Russell kind of not what people normally do. I think it is profound. So, we may say this category theoretical switch is just, we should take, we take this ordered part of basic and then through conditions we define what wouldn't be not ordered. So it's different few things. And category theory On the contrary, it's a general theorem, and that's a natural framework for this kind of generalization. I'm talking about that, okay, we don't have any other proposal, but we shouldn't think about category theory as a Finnish thing. So, there are stuff to develop, but something would be under the title of category theory, I imagine. Even though people made some proposals, allegories or something, but I think it would be fair to call it category theory. Anyway. Yeah, what about structuralism? Okay, what I call Hilbert saying is not exactly structural, but even in the recent entry in the Britannic album, I think he referred to Hilbert, I think he absolutely appropriate this kind of something basic for mathematical structuralism. And what I say is that this categorical generalization, for the reasons I explain, that really we cannot just improve a little bit of this Hergutson scheme to improve this categorical arrangement.

10:00 So it wouldn't be appropriate to call it structuralism, I would say. Because we don't have this, and the argument, which is a little bit like slogan, that structures are specific categories, but not the answer around. Again, this way to look at a category-specific structure, you say, limited. It's not absolutely impossible. We can do things like that. But what's really interesting, it seems to be an interesting problem, is just make that way around. And now, I only mention, and I can't explain, because there are some problems. Now, okay, I already decided that, but how we should do the right approaches to, not very same, but existing approaches, factorial semantics of Oviya, SkateScript, and Larris-Mann, and which I have a lot of time to explain, to compare them, and I think this comparison is very interesting. And in both cases, I just mentioned, right, categoricity and all that just doesn't make sense. What you look for is kind of good categories of properties of categorical models and given that, even categorical model, you can choose kind of generic or the initial model, initial meaning having exactly one or the other model. Universal model just the other way around and it's in that any model, it doesn't have. Three models are probably most interesting. Nobody Nobody knows how to define them precisely, except as the left is joined to the Paget department. That's not the precise definition, so people wrote on that. Also interesting, this is a quotation that distinction between form of theories and their models is kind of plural. La Vie, for example, he speaks, okay, theory becomes generic models, you know, things like that, which kind of mutilating, but then, what you're talking about, make things precise, but what's going on in my view is exactly that this old setting, which he starts anyway, right, somehow is glorious, we really need to make things precise and new and differently. Some people like Ericsson, sketch theory, he just didn't care about these distinctions.

12:30 And the main thing is that inter-modernization of logic. One important feature of the Silberton approach is logic. This is what I prefer to follow. Weak logisticism, just to be distinguished from, say, Rask's logic. idea to reduce in some sense mathematics and logic. And the big logic would amount just to say, okay, we have this logical framework as basic for everything. Then we have some axons that become mathematics or something else. And I think that approach, first thing, we don't have it in Euclid, for example. And perhaps we wouldn't have it in this new setting. So, start thinking would be a foundational, mathematical kind of, I don't know, mixture of basic geometrical concepts, but interesting that we can kind of recover our logic out of that, out of the infrastructure, it becomes kind of sensitive to concepts, it's not just general logic. I think it's quite similar with geometry in the 19th century. Instead of this idea of geometry as kind of, how say, aqueous scheme or everything, we just construct geometries, which are related to each other. I think with logic we already don't construct them, but unrelated to each other. We have one, say, one logic. But then, okay, but how do we all work together, right? And one way to answer questions is, okay, we have this basic construction can be empirically motivated, say we can talk about quantum physical, some pragmatic performance, and then we can recover logical structure which kind of adapted for even the scheme which goes forward in some particular series. Yeah, just conclusions. So category of serenity for equalization, mathematics, supports this kind of humanitical view is just coming back directly to history, and articonventionalism, I would call him, probably according to Shapiro's, Foundations Without, Foundationalism, I rather agree with, I absolutely don't agree with his project about the sector of the larger, but I say, we can say, foundations of course are important, but in the sense of revision and I thought they would fix it in this kind of a bedrock.

15:00 And having my research, I think we've had some very controversial statements. Yes, dear. One question. The question is, why I don't know why I started thinking it as a joke or if you want it as a sociological opinion. The question is why categorised people when they expose, they have always to say that the others are wrong. And in some sense, why categorised people when they make the topic, they have always to say that the others are brown. It's not enough to say that you are brown. It's a common model of a dog from the category. I would like to understand why. No, no, no. Just part of my performance was criticized. I think it's It's a normal scientific business, just criticize kind of general. Take it, take it, take it. Now, I'm not sure that the problem, I think that is a big problem. The problem is understanding it's a certain theory because it's entirely different from the next. But I'm not sure that it has a general philosophical solution. It seems to me, in fact, that it is the job of a mathematical theory to interpret itself parts of other people, parts of other people, so in the sense of the job of vector theory to say, okay, this is my version of Pythagorean theory. And so it seems to me that, in fact, the fact that this vector theory is the same as Pythagorean one, It's not just something in Western, it's simply something that we say within the theory of the form of the policy. It seems to me that the question of translation is not a general question, which one is a good translation. It's something that can be decided, and it's not historically decided within any theory. The thing is that one of the most important jobs in the theory is to reconstruct within the country the important material of others.

17:30 And the way in which it does is exactly something that is actually said with people. I think it is more a contextual, historical problem than it is in general. I don't know, this is my impression, but I think that we cannot solve the problem in general. Now listen, I basically think it leads a lot to liberty and flexibility because of course you can choose. Actually, I am not quite sure that we should be more accurate about the translations. I think it's often that people just find one translation and they say, okay, this is the same thing. Or if they need to do backwards, you know. And then we should just look a little bit more precisely how it works and we can get out a lot of that. That's one point, and the second point why I think all this formalization strategy, it's really kind of the mistake strategy. And I just try to suggest a different one. So basically I don't see real contradiction. The first question about the political question is not a political question at all. I'm wondering if we can solve the problem in general. I remember the same question in a talk a month ago. I was thinking about that. I'm not sure of that. No, I'm a great one. I'm not sure that the O of a general solution. No, you really don't want to. But that's enough. It is not for me at all, let's say for me at all. No, I agree. I agree at all. Okay. And Jeremy? Now, thank you. I want to take you right back to the start of what I was going to do. and ask you whether you would want to say at all, maybe it's on 5x, just occasionally, that those versions of 5x's theorem do not say the same thing, specifically, but the two modern ones do, because there's an informal version of both those theories to group. an understanding method of formalizing this to you until you get to the print of the state.

20:00 But, I might, would you ever want to say that the Euclidean version, because you have to explain it, actually says something different? Actually, I rather agree with Martin here, saying we just should look for reasonable, probably pragmatically also, identity conditions, right? In which sense we want to have it the same or not the same, right? So rather my point is that there is no right answer to that question. And actually we can really, in a sense, that whole question is not that important, I rather use this kind of puzzle. What's really important is how they translate into each other. After all, I think, I don't know, for at least pedagogical, but more than not pedagogical, I think the importance is more to keep track of, I think this Euclidean theory is a kind of, in a sense, sort of background of any cultural, personal, but you might need to sort of push it, because I think you started with this example, but it's simple, I hope I understood it, and also because it's a generic example, it happens to be simple, much discussed. But it seems to me that the retranslations of the first two sentences into each other are much less problematic than this one, because you have to strip out Euclid's understanding of equal, replace it by another and another. And there is some replacing the one with the other. And there's something actually that is lost, and that's the point of your show. Yeah, maybe she's not surprised about it. But in 300 years, this apparently naïve has actually failed. So, I know this is not the main theme of your presentation because I wrote the category theory end of rethinking these things, but I just want to ask if you would accommodate both the understanding of the patriarchal theorem, these three things say the same, And also, sometimes also, you know, two of them do, but that one... Yeah, because of the historical distance. But my ideal, I would say, probably it's not realistic, probably an interesting one, but if we really find good notions, say, of translation, which I will discuss with coherence and so on and so on,

22:30 and then we can kind of replace it, make it coarse-grained or fine-grained. And then we say, look, we just should, who's going like that? Then it's verified as identity. Then you need to find a way of a person to person. So, it's kind of a knowledge of science. I'm going to respond a little bit after that. You have a definition of translation that needs to determine elements. I didn't know what you meant by all of this. Yeah. something like subsystem or something, that is not the real part of the theory. It would be like elements of sense, right? But I just want to have, say, to a little bit grasp what we are normally doing, right? When we are saying that, say, second book of Euclid translates into algebra, what do we do? We are put this, how to say, on a lower level and look, okay, this coherence is preserved profile and some limited domain. It's how it works. I just described it. We're aiming to be there, I don't think you're hearing. The whole idea that, whatever they are, they could be different things. They could be different things. But what I try to say is that we need more. We need inside them, right? And we need, which is not presented at all, the diagram is outside them. So we have this, some levels outside, some inside, and they would hear us preserve the serpity forms. More questions? All right. Is there no other? Sorry, Michael? Okay, so we should say thank you. Thank you.