Category theory, structuralism, hermeneutics

0:00 No element of what I've said is nowhere to be found. But maybe there's a specific level. And another point which I wanted to make, and I spoke after that, on the question of better settings. Yes, for sure, trying to understand what's going on is finding a better setting to and so on. This is standard, and who would agree? But for instance, when you have a conjecture, you can try to find a good setting so that you can prove the conjecture, that's one thing. But what Erisman did is, no, once you've proven the conjecture, you still need to work again and again and again to find the proper setting. So there's something else than turning loose notions into quick-cut definitions. There's more to it than turning a conjecture into hard mathematical path. There's another quest trying to address that. And another thing is, a standard way of explaining what a better setting is, is to give examples of domain And in this case, it's not what's at stake, there is no domain extension. The proper setting is not a new domain with new objects, ideal or not, which would help solve the riddle. So there are specific elements of mathematical understanding in this practice of whatever we would like to call it. So maybe the last question will show. If you want to make some more of the advantages of the categories that we have assessed from the rest of the world, we will be making a particular one. In that, the rest of the world is a sketch theory. We can set out a category of diagram, which expresses the core of the different group. And then, we have three of these diagrams, and two, the authenticity of interpretation, which the rest of the world's realization. If you realize the sketch of a book into the particular sex, you enter with a book in the other sense of the word. But if you realize this very sketch or book into the category of a couple of different things, you enter with a new book and not a book in the other sense of the word.

2:30 And so, I think it's because of that, and you can say one of the structures, and which means one of the structures, because if you stick to the framework of set theory, another framework, to shift from a group to a linguist, you have to add some new accents, which specify a number of accents of linguists compared to poets. In the framework of sketch theory, you are working with the same sketch that you decide to project or to realize it in different ways. And maybe it gives some flexibility and some insight to what the structure is all. Yes, there are two questions. It's for sure, you can do things in the category setting. It can't do easily with set settings to get communication and so on. But Ericsson worked on sketches of categories and internal categories and so on in the mid-60s, so it can't be the brilliant works. So what led him to this restating between pedagogy theory? Maybe it's what had a minor way. OK, I think we should stop here. I suggest ten minutes for us. I'm sorry, I'm coming now. If you like the text, our third and last talk in the afternoon, or evening, I'm Ray Rodin from right here, and his talk is about category theory and structure. I'm not going to get to the winner of this. Yes, thank you. There is two parts, actually, in my talks. And in the first part, I'm just going to speak what I call Rumi Neutriks of the Saborian Theorem. Just actually it's about, I'm going to compare a few versions from different historical theories of the theorem

5:00 and ask kind of a parting question in which sense, if any, it's the same theorem and actually approach more generally this question, how we can, say, compare mathematical theories very different conceptual frameworks and still we somehow we should probably find it's my motivation here kind of justification of this strong feeling of intuition whatever that we're still talking about one and the same thing in spite of all this conceptual difference and in the second part i'm going to actually it's part of my wider project which i have not developed here. Actually, I'm going to speak about the category of theoretical methods of theory building, the category of theoretical thinking and mathematics in general. And I'm going to develop it as, I want to call it what Renaud just did, as kind of distinguished, almost probably opposite to structuralist way of thinking. And I know that probably it's not very usual way to look at it, but just pointing to one historical thing, in spite of the time that people like Erisman and McLean also, he made general statements about mathematics, which sounds very much like structural, like form and function, and so on. But just consider the fact that what we can probably look as kind of structural, as result of the structural idea, I mean all these four by key volumes, elements of four by key, they didn't include category theory. I think there is a profound reason why, and I'm going to develop on it. And you'll see that actually there is a strong link between these things, because the solution of this puzzle, I'm going to propose that, say, category theoretically, in spirit, how I understand it, And it's not going to be a technical solution where we use category theory to, say, to compare different versions of Pythagorean theory. And this order is probably already clear in some sense reverse, because probably in theoretical order for theoretical, I would keep on explaining more detail in general conceptual framework, and then look at that more particular question kind of application. but I just realized I would never put it in one talk, and moreover the source part, I hope it's just more relevant to the topic of this conference,

7:30 which is theoretical understanding, and I hope that this, how say, the example of the approach, the historical dimension of the program, will make my modern point also clear. So here I just took, I don't know how to say, where there is really a representative selection, but I think in a sense, please, just from three different books. The last one is classes, Euclid 147. And the first one is actually a kind of school course for even not under-graders, but for like a little work from the first time I see that. I don't know how it's popular in the book or not, and people just say, okay, if we have right triangle, as they do this, just cranking with right angle, then if we, you know, glance of sides X and Y and Z, then we have this equality, follows. Okay? And this one is actually kind of Burbankist, where rather we involve account of the same thing. And here Donetsky calls, and they all goes under the title of the Tafakorin. Here I am not cheating. And he says two non-zero vectors x and y are orthogonal if and only if this form was passed. Actually, there is one thing which I think is not actually important because if and only in Euclidean we have, we would say if and only we have that Pythagorean theorem and reverse Pythagorean theorem, which is in Euclidean, I think it's next one, 68, if I'm So, I just take only if part, that kind of terminological solution which probably not very interesting to discuss. So, the problem is, okay, we have these three different propositions also categorically, right? Not arbitrarily, not absolutely arbitrarily. What they share really in common. How can we, how to say, respond to this question? So firstly, before really trying to give responses, just making kind of

10:00 exaggerated effort about each of them, right? We just try to... This is kind of humanistic in very, technical and primitive sense. By the way, Schleimacher, when he just made it somehow popular, before it was kind of not ridiculized, but made something very special, but guilty, Schleimacher really wrote, which I think is a really excellent kind of manual, how to interpret texts. So I'd rather use humanitics in that sense. So, that, actually, textbook of rather low level, right? But if you think from, let's say, higher viewpoints, you would say it requires real numbers, right? Because otherwise you wouldn't speak about letters as numbers. If you need real numbers, it's probably something else. But he speaks about... So, of course, the authors don't provide anything at that level of education, anything, like real numbers, and what they do, they kind of informally, you can read it, you know, introduce axons of metrical space, and they say, okay, just to learn as they measure them, they just satisfy these axons. That's of course, in a sense, not true, right? Because I don't know if we can measure a real number, but what we read from a graduate school is not a real number. So they kind of, and this is actually a side question, which I don't want to discuss, and that not feature of that particular textbook is more than most of them, that we really kind of hide this particular secret about non-crimulability, existence of non-conservable management until, you know, much later on, they kind of treat students to begin work, okay. what these people are doing. And now about the second one. Here just some care needed about reading this formula because, say, this plus, this plus between two real numbers and this minus is between three vectors. So it's not a total concentration. So we should really

12:30 take care to read it. And actually in that book everything is introduced, and I don't remember exactly about real numbers but even they had not developed this in that book it's supposed to give reference to some other textbook where it is developed so we can say it's kind of really rigorous treatment meaning more or less to raise standard rigor and the price of course for this But among other things we kind of almost lose use of geometrical intuition, so it's rather clear that we cannot give this account just in school. Okay? Now back to the pleader. Here actually it's also much more subtle than it might seem, what the pleader means. This is his translation. In right angle to angle, the square on the side subtending the right angle is equal to the squares, he does say to what, one square equals to two squares, but what he means, I'll explain it, but we should ask what he means one square equals to two squares, okay, on the side subtending the right angle. And one interpretation which is a customizer, he speaks about errors here, but that's exactly, that's not true at all. A closer interpretation would be every composability, meaning that basically you can cut two smaller squares in such a way that you can combine back the big one. That's what he means, the big one involves these two. Okay? But that's also kind of interpretation, because if we try to be exact, what if we just read axioms, and these axioms which explain what is equality in this sense, which is used in the statement and the theory, and that's quite fine thing equality in Euclid. One thing to understand that is kind of, you know, axiomatic definition of what equality means.

15:00 And basically, this is the relation which satisfies these things. List added, subtracted in a geometrical case. Of course, this means contamination, we can say in English, yes, something like this. So, subtract it, meaning if I take one two-angle, I can subtract, say, cut off the small two-angle, or add the meaning. I take two-angle, and I say, you can add that back, right? This kind of thing. So, actually, of course, in a plain case, we can say that that's the equivalent of how people go down, that say, if these holes, the two plane figures, are equal in Euclidean sense, then they have the same year, and converses too. But in space, in a three-dimensional case, you know what happened, is there a difference? Because I think it was first, he was probably in his first one, which I heard this legend, that then kind of found the solution, after he published it. I don't know, it's true of that, it's just colloquial. And then Hilbert somehow prevented it from publishing. We'll check, no? Ok, so it must be colloquial. So in three-dimensional cases, it's not the case. Okay, so you see that there is quite non-trivial mathematical issues which involve these simple things. And of course congruence, which describe as the fourth axon, in just particular case, right? Of course congruent figures, they are equal but not congruent as normal. Okay, and this actually is, in a sense, it's kind of a negation of equality. In a sense, all equiline axioms are about equality. And again, side question is, one new axiom, just saying they apply equally for geometrical, for magnitude and numbers. And actually, my term, I know, I said, actually this thing we can also apply to numbers, meaning At least there is possible reading of these axioms, which will involve both for numbers and math.

17:30 And of course, we should mind postulates, because all these operations and subtracting must be doing well through postulates. So, again, we have to stray from this cycle. So, this is rather complicated thing, which is important. Okay, now we have a little bit clarified what those statements stated, and now the generation, how can we compare them, not even saying they are the same, because they are very different in the ground, because I just developed very, very, I don't say, limitably, but we can't really, I don't know, recall all this brick geometry, very, very different ideas from, say, at least if we try this move. precise Donedu account, which is really precise. First account is, say, a pedagogical thing, we don't really have something precise there. But at least in second and third, we do have. And so, how can we compare them? Should we, say, find some thoughts, some general background for them both and saying, okay, that big, I don't know, meter background theory, that would be this, that, and then somehow we would already say, putting some identity conditions there, the same one or the same, this kind of question. And just again, that side thing that's still quite a point I would like to mention. We see there is one interesting phenomenon which I didn't see much discussed in the literature, that we see that, say, mathematical truth, this one I wanted to remember. Apparently, it survived the changes of foundation. So, we will still grab, preliminarily at least, that Pythagorean theory was not invented by Donoghue or whoever, right? But still, like the antiquity. So, we can say somehow it survived all the changes of foundations which happened in history since then, right? But how it's It's not possible. If really a foundation is something which everything else depends upon, right? And if we still use this kind of architectural metaphor, which I would argue is misleading, that thinking about theories, something like building, you remove foundation, everything crashes, then we see mathematics doesn't look like that, and actually science in general, at all.

20:00 So, probably, I understand there is really reason to rethink ideas about foundation and probably I mentioned once again in my talk, but I will not develop this issue further. OK, and now I think there are two strategies to answer this kind of question. One which is unusual, which I try to propose, and next will be what is my proposal, will be the kind of category. And then we say, OK, formalization. Next, formalization. What does it mean? Okay, I am not going into every percentage of what compensation means it will take all the time, but I think somehow extracts what all these things share in common, kind of pure, pure form, right? You just kind of get rid of this kind of ballast, you know, media, And if we believe that it's the same theory, there must be all these things somehow shared in common, right? Let's call it formal structure. And actually we do it. The whole mathematics, that perhaps algorithm provides exactly the ground we are looking for, right? And so we can just then project any which would be called informal, so-called formulation, like Euclides, or this pedagogical one, to that formal, and then we see this project to the same, it would be the same, and so on. So that's, by the way, it's not at all a problem about philosophy. Mathematics, similar strategy, you say, philosophy language, like form of philosophy in the 14th century, somehow works like that. But not like what I see as a really important problem of that approach. And it is this one. Say, what do we get after all? It looks like just one another version of the same thing. And how do you argue that it's actually kind of the best, the standard? Okay, I don't say it's not at all, but I'm just asking the question. And I think the usual argument, which I would say doesn't go through, but I think it's quite usual.

22:30 It was something that is just kind of completely clear. When in the Colosseum language, when people, you know, formalize, talk about events or whatever, they say, how do you really prove that that sentence really formalizes this sentence in formal language, really formalizes thinking in informal language? How do you justify that? Okay, I formalize that. And then probably if I ask someone, this person can say, probably your question cannot be answered, because there are all these informal stuff, right, that we need to be kind of unclear, and now this is this formal thing. So, we just cannot do anything rigorous about this informal. So, if you really want to be rigorous, just forget all this informal stuff and just work with this formal thing. I think that's more or less argument justifying that. And my critical reply is this. The way that F is, you know, this hypothetical form of, say, formulation of Pythagorean, but it's like, some form of it is Donitu. Sorry, do you, are you doing a malaprobism for Deudone? What? No, no, no. Who's Donitu? Donitu. Probably some people know him. Sorry, I don't know the name. It's just, I found the letter here. Oh, okay, sorry, I'm just not familiar with the name. And if we really want to say that that thing we don't think is rigorous while it goes in formal our life, we cannot just do it by this appeal to this formal character exactly because we admittedly cannot justify that it really doesn't show. It's kind of flower in the argument. habits, say, better than others, more rigorous than others, then it must be grounded independently, not by appeal to this formal character. And then, it's a little bit different, man, I

25:00 just think it's kind of historically naive and epistemically wrong, just to assume that that this today's or whatever tomorrow's standard, I refer to formal foundation, but it still probably can be standard today, but I apply this to whatever foundation, we'll survive forever. So just that idea that, okay, we have this foundation and we analyze the rest from that basis, it's kind of a job. I think historical name we really need, this question is not just a puzzle about it, I think we really do need some account, how we can kind of manage historically mathematics, right, because otherwise every time we just, how say, write a table or other and say, okay, all this stuff was informal, now we are formal, it's not serious. Yeah, as Jean Binabou, in a private discussion when we were doing this meeting, probably it's obvious thinking, okay, just imagine, one way, which is, of course, still a possibility, someone finds contradiction with Zepet, and so on. It really wouldn't realistically give anything like a crisis in mathematics. Of course, it would stress some community of people working on the foundation of mathematics. They would find some solution of all of it. But it would not at all impact a larger body of mathematics. It's what he says. Okay, it's kind of a mathematical argument. So, one thing I already told, this architectural metaphor or mathematical science is misleading, and that phenomenon of survival, as we call it, must be really taken seriously and driven rigorously. Ok, what I suggested, it's all critical. And, strategies, is basically this, they just should really study, you see, once one, translation, we just look how we want to translate into another. We've been really attentive, really attentive. And I just try to show what follows, that it's something which is done, of course, people. It was Tandrir who invented this notion of geometrical algebra, right, in Greeks. Yeah, so he proposed this...

27:30 Okay, no matter, but I think it was him, it was two normalian, two brothers, and somehow they translated all texts in more order language, okay, look, it works, but that's not sufficient it all. Of course, it's not a ground saying, okay, that is really what we obtain by translation. And moreover, okay, I just now going to very briefly to address some questions, how to do that. So that would be the first step. And second step, when we do that, when we really study how these translations between theories work, we can already try some identity can be some account of ideas, let's say. In terms of this translation, then I shall show how one way to do that, usually in categories, but perhaps it might be others. Okay, very very briefly, just pausing a problem, not really trying to solve it. What really would come as sound translation? If I say A translate to B, how to justify it? I can say whatever A translate to whatever B. And if this A be really kind of different conceptual schemes, how do I say that? True or not? Works or not? And what we need is kind of coherence, which is of two kinds, I call internal and external. We just say, okay, just think about this, say, standard translation of Pythagorean theory method is in Euclid, right, where we have realist squares, yeah, it's like Pythagorean We say, okay, just take some numbers and lengths instead of sides, and then instead of squares, you take squares, which explains the terminology, of course, or numbers, and so you get the thing. And so we have this kind of element-wide translation, yeah? We say just if you have a strike segment through the real number there, right?

30:00 Not whatever real number, but that's not what it is. It's length, measurement, axles, phonetics, and so on. And then, so we have this kind of element-wise, not really element-wise, but in, say, low-level, I call it element, but not kind of ultimate-level elements, right? And that's what we also want, that operations, which we do later, say, with our geometrical things, and with numbers, they, as people categorically, commuting, which would mean here L, I denoted this operation of putting elements together without pre-sizing. So we translate A to B, belonging to a different thing, say A somehow Here we have two different operations, so putting them together, we have this translation, and then we want this thing for mute. This is the same thing, but for use in the category theorem, it is going that way, and here I just write this thing in the linear way. That's internal because we are talking about elements inside, but we also need kind of external, meaning that what we want, if this way of translation would work only for one theory, you say something stipulated artificially, it can be true, right? We need at least some class of theories, like something we need to put this thing in a wider class of examples like second group and Euclid, and then we say we have this, how to say, lower level, upper level coherence is preserved, we can say, okay, it works somehow. But, of course, if we think more precisely, and I don't have time to correct more of that example, it would probably be a little bit worried, but we see, of course, here we lose actually types, right? We translate that plane square, figure, what is called figure, two number and just a straight line two number, so we can still say different types of things, like one dimension, two dimension, we just lose it in this translation, so we cannot translate it back, it would be reversible.

32:30 And, of course, there's actually a wider domain where it works, it's very, very limited, of course, right? We can't make degree more than three, actually in third degree it doesn't work, it doesn't work, I don't know, for circles. It's rather limited, how to say, part of the theory which allows actually for this translation, which is just one or another. So, it's a rather specific thing, but it's important, it's quite frankly, but we should do it. I don't know, when people normally see that, they're so impressed. It's okay, it's great, it just works. But then you just should see the limit, where it works. And if you see it, okay, it's a very limited limit, not that, I'll say, strong as it may appear. Okay, that's discussion on notion of translation. And now how we can do, just imagine we somehow did our best about translations, and how then we can say something about identity. What kind of translation do we need to say, essentially identity just translates a thing to itself without changing it. OK, I just reproduced what the standard category is, the theoretical notion of identity morphism. in that sense, showing that perhaps something else could be done also in Category Theory. Because actually what is required in Category Theory is that more people call identity. And by the way, as far as we have these identity markets, we don't need anything like objects. No matter what in Category Theory we have objects and markets between objects. But in kind of logic for the mind in Category Theory people never do that. Because this identity matrix in an object, there is no way to distinguish them, say. And what we want that kind of incoming morphism, if we compose this way, A-F-E, which is right, not the wrong. We adjust F, so it's like unit, kind of right unit for incoming morphism and for outer morphism. And what we need is that this works universally, quantifies universally for all incoming A and all R of G, that this thing procures, this diagram.

35:00 Then we say E is identity, but it's a specific identity. Of course, in the contemporary we may have many identities, like many objects, right? Or we can put it this way, each object has its identity, but as I just mentioned, there is no real reason to distinguish objects and its identity. Okay, when we say, okay, category theory is still something, so we bring something, of course, but not much, I would say, because we need some notion of composition, which, as far as I'm talking about, see, translations, interpretations, things natural, we need to be associated, otherwise nothing works. issue I'm not going to discuss, yeah, that I've talked already, yeah, and this notion I'm going to be strong, it's very strong, meaning, and it's context dependent, meaning it all depends what kind of morphine we have outside, so say, if we try to do something like this, like a theorem, it all depends how much we put together, right, what kind of this external, and that's also, I didn't mention this, but I just think it's kind of notion of identity right it just all depends on some other reason which in some sense kind of probably not relevant so if we just have two things we cannot really say it's the same or not the same until we brought all the contacts and the answer depends on which causes we brought Yeah, that's what moved the standard categories of promotion. I think perhaps we, I actually had a paper on identity categories, I discussed with other possibilities, but yeah, perhaps it's just not the last word, and I think differently. And now, just because I'm going now to discuss structure and so on, isomorphism in this context is just, it's not equivalent to total and age. So again, we don't have any of this idea that we have isomorphic structures on this basis is the same. Not at all. So, isomorphism, isomorphism is better, such that these two conditions, how, and they are both wrong, and I think when people think informally about that, they often forget it, they just, okay, we have one way of morphism, that's what morphism is, okay, that's isomorphism, that's absolutely wrong, I think, just, I think, for AAP is sass, finite, little sass,

37:30 and Marxism and function, you see that you have one function there, five functions, it's not the total, it gives a projection. And no, how to say, no reason to expect that thinking by just more complicated things like theories and stuff. So the fact that from theory A we can translate to B and backwards doesn't mean at all that they can capitalize. What we need actually two things, and even if I get one thing, we get again a different thing. If I get that, we get what's called restriction, and here we get section. So, it's quite strong, it's quite strong, and it's used as the notion of identity I just used. Okay, because here we have the thing. What are the objects in the category supposed to be? What are the objects in the category supposed to be? Are they theories, collections of statements? Yeah, pretty good. Different possibilities. I'm just trying, you could try theories, but also it might be done multi-levels, what normal down category is, so if you think theory has category, then you get another category, theories, and so on. I have some formal examples, but not today. Okay, so that's probably just a little, very short resume of this first. But I just assume to oppose this way of thinking to this formalization strategy. Formalization strategy, which in some way probably is too, how to say, coarse-grained to say that, but it seems it's still somehow related to structuralist strategy. Because there is something common there, something common and something non-essential. That's common, not essential, and the rest is not essential. So they just bring that common. And that may work only if we have already all this esomorphic thing, but it's not realistic at all, and we're just in different contexts. Of course, if you have translations where never, almost never, isomorphism, They are just different translations and different theories. Perhaps we can make cases when translated like isomorphic, but that kind of trivial cases, then we can say, okay, it's the same theory. But it's not at all what happened here. Okay, and now I'm trying...

40:00 So I have to use time out until 7. Yeah, 7 to 7 years. Yeah. Okay. So now I finish with this historical example and try briefly to present general evidence without... actually it would be more a critical part because to present positive part I have no time. It would be appropriate to suffer. So that's very I'm sorry, but it's really probably very nicely described, kind of, with vision at least behind what we could, I always think Hilbert's scheme, because I think it assumes really many things which is, became fairly standard, and that's exactly the thing I'm trying to revise. So, replying to Frege's critique about which I am not going to speak, he says, you say that my concepts point between are not only properly fixed, Frege said that unless you fix meaning of your words, you don't have two values, you don't have two values, you don't have axiom, you don't have inference, nothing. So, that's exactly what he would propose. But surely it is self-evident that every series is really framework or scheme of concepts relations to one another, and that basically others can be pursued as one pleases. If I think of my points as a system of other things, and in the group library of 1990, he speaks about the system of things, right? Tom, which later, like the Dickinson, but later it became, as he said, about the system of things. Love or Jimmy Sweeps in different editors, he spoke about the beer mask, and then conceivable all my actions as relations between these things, then we can, I mean, if these four actions are verified with the whole about these things, right, then my theory is, for example, the beer amount goes all over the synthesizer, nothing else. Each and every theory can always be applied to infinitely many systems of basic elements. But here it's still interesting, because I never... In the late account, I think Hildred was really very clear about what he was doing. But mistaken.

42:30 For one, it has to play really mobile and reversible. One-to-one, transformation, and decolates them the axons for the transformed things. It's interesting, she thinks of this, she speaks as time formation, here, but not just corresponding, we correspond with a similar one. This is from the line, for example, the principle to Azatayi, I think he has in my project geometry, and I'll show you a very special example of it. Yeah, so what I want to stress, to begin with, he just has this idea of And then what came to him as a surprise, actually I asked this question on the form list, tried to find out how it happened. All this business about the actual continuity. Already after publishing this Grunewagen in 1990, he discovered that the same system of action might have non-lesomorphic models. And then, of course, with Babylon, the problem was described precisely the rest of this sort of categoricity and so on. So what I can feel back to a scheme is basically that, that formal theory is much less about your models. And as you know, well, say, it doesn't work in many cases, then there are ways to, I would say, some price to make it work. the human solution, from my understanding, say from today's model theory point of view, was just wrong. Because he, probably people who don't need model theory can just correct me, but I think it's just something wrong. Because he assumed as absolute existence of maximal model. And if we think about, say, model theory sets the reticons, that's the semantics, it doesn't make sense, it can impact. It's just wrong. It's never happened. So, the minimum model, in that sense, is a better solution. Okay? And too often, what I would say is Kandelerson, a philosopher of mathematics, not of mathematics, this notion of a tandem model. Just people say, okay, there was that problem, but really, somehow, are filled with the knowledge. So, and of course, it's kind of, say, through the whole

45:00 strategy of this formalization, it's kind of not relying on intuition, and here at some crucial point of all the story, we say, okay, the model behind the pick-up, which is just intuitive. Mathematically, that notion doesn't make any sense, but epistemically, I don't I can justify my claim, and if someone asks me, I don't know what purpose is, but I don't understand. Okay, so now I just give kind of a series of claims with short arguments, intended to show that, say, the schema doesn't work must be replaced, and very short I speak about remaining. So, the idea that he will not, of course, personally him, somehow confuse this notion of Marthism as Isoropism. Actually, I tried to look at this first appearance of notion of homomorphism, morphism more general. what normally people say is in Jordan's treatise. And there he causes, I forgot what I thought, but it's clearly the context in which he just treats as a kind of imperfect isomorphism. And that's exactly what also we hear there. And say different species of these states in this context. Thank you.