Structuralism in maths & physics — what structure is & that structure is

0:00 No, for god's sake, keep it open. Close? In this planet? I was wondering why I know that it's coming into this. It's a real, it's a real, it's a real, no one's in it. It's a feat you will pick on it. That's exactly right. I had, I actually knew somebody who could do that on the first day in Cambridge. So this is the schedule, that is the title you all know by now. Just to explain, the number two is because the first workshop is placed in these last tools, thanks to Stephen French, who has this idea to start the discussion quite much now, Okay, to gather a group for discussing on this topic. And it was so fun and so interestingly that we decided to go on and make a series. And we did it set up, maybe. And everything is completely informal, so just be relaxed. There is nothing official. The reason, this is the reason why there was no schedule up past an hour ago. So I said, wait a minute on the record, more or less. We need a schedule, but we are not strict about the time since we have only four speakers and the whole day we can be just left. And the idea is that the talks are just an occasion for discussing, so if you want to interrupt the speaker a leg, for example, you can do it without any problems. We had a break, we are going to have a break around half past twelve, because the bar downstairs is closing at half past two. So we have, the timing is good for that, and this is the reason why I think we will start again, but we must have finished by that time. Okay, so, and I'm happy with that. Then we can start. Um, do you think I can see some of that?

2:30 Yeah, it's okay. I'll do some of that. Okay, perfect. Okay, perfect. She said, um, she said, um, she said, um, she said, it was such a part of the past year, so I hope she says she has accepted in the whole room, so it will be. Um, next week, that is, um, the name is in Philosophy of Mathematics, a PhD from the University of West Ontario in Philosophy of Maths, and then the organization came and spent a year in philosophy, and tried to drag her into philosophy and physics, and she's come to talk to us today about what structure is and that structure is. Thank you. In typical, informal class, I'm going to change the playlist myself. So it's now Structure and Mathematics and Physics. Okay, so what I'm going to do is outline my project and then state how I see structural realism fitting into a larger agenda of scientific structuralism. And then at the end of my recap, I'm going to sketch a diagram of how I see these issues and then hopefully you'll be able to assist me in some of the problems that I'm having, making sense of both structural realism and scientific structuralism in general. So, just if you want to read alone, you don't want to read it. This is, can you all see that? These are just point forms that I'm going to talk about. So, as I say, philosophers and mathematicians, physicists have all made appeal to the notion of structure as a means of characterizing the subject matter of either mathematics or physics. Floss versus science have used a semantic notion of the structure of scientific theory to give accounts of the nature and the role of physical models, which is known as the semantic theory of theory. However, recent debates in philosophy of math and philosophy of physics have failed to take note of what I call the distinction between set theoretic and algebraic accounts of structure.

5:00 In both areas of inquiry, that is, mathematics and science and physics, resulted in the interpretation of structures as things. That is either metaphysical things existing somewhere out there, or as a theoretical thing. So in philosophy and mathematics, this has engendered discussions of whether structures exist independently of systems that are structured, or whether the structure arises from what's common and consistent. And this is known in mathematics, and it's the anti-Grenner versus Ingres debating this structure. Sorry, I mean, are you talking about in these classes? Is that a club? Yes, the whole thing is about that. I'm just giving a sum of where I'm going to go, what I'm going to do. So, in the course of physics, this has been advised to the structural realism debate. That is, the debate about whether structure of or behind phenomena, rather than objects, is continuous over theory-reaching and has motivated the more metaphors of a bus plane, by a man driving anywhere. That structure is all there is. What has this been addressed to? The recent debates about whether structure exists independently or not. What gives rise to it is whether we should account for structures as things existing out there, or do you set the theoretical counter for the structure? Well, we'll see what's going to happen is that in the background, there would be a lot of the accounts of structural realism that used the semantic view of theory, and they framed that into the structure. No, that doesn't rise to structural realism. It is rise to the debate of that, well, are structures out there, like James would say, or are they somehow in the phenomena? What happens is, if you represent phenomena by n-tuples, for example, by n-tuples, n-tuples, then somehow what you have is that structure is some type of set that are n-tuples. And what I'm going to argue is that that type of representation is a . Are you going to do all places to take the charge?

7:30 It depends what you mean by the structural realist debate. I mean, if you're thinking about Warhol and Ladyman, I mean, historically, what Elaine says there may appear a little But one can think of perhaps earlier versions of structural realism where it does fit that. That's possible. And I'm certainly not characterizing the history of Warhol in that sense. I mean, characterizing here's the sound this is the sound that whole background likewise doesn't fit in It's very interesting, you know, what I'm saying here is that the structural realism in me gives rise to these problems or these assumptions that structure is all there is or that structure is somehow out of bed. So you're taking the first part of the philosophy that it's this type of drama. So what is my objective? It's to inform these debates, that is, both the structural realism debate and the anti-Rabian by distinguishing between set theoretic and algebraic structure. And to show that talking about shared structure of systems, that is both mathematical and physical, you don't have to define structure as a centre or a set of mathematical structure itself.

10:00 So what I would argue there is that what all of these structural counts of science are giving us, or in mathematics, is a way of cashing out what do we mean when we say that two systems have the same structure? And what I'm going to argue moreover is that to give an account of shared structure, we don't have to say that structure itself is set to the record. So my research is focused on using category theory as a language to frame what I call the semantic realism interpretation of a mathematical structure. I'm not going to talk about that today. So my aim today is to investigate structural realism as framed by the semantic view and to shed light on the manner in which shared structure or structural continuity can be used as a methodological tool to argue for structural empiricism. So I'm going to argue that the insights that one finds in structural realism aren't strong enough for scientific realism, but they are helpful for understanding what we want to do with structural empiricism. of the camp of scientific theories. Okay, any questions? So, moving on. So first I'm going to talk about mathematical structuralism. And then what I'm going to do is take this as analogous to scientific structuralism. So I'm going to make some observations about what mathematical structuralism is and what are some of its problems, and then I'm going to say, well, how is this similar to scientific structuralism? So, to situate my objective, I begin with the description of mathematical structuralism. This is typically taken as the philosophical position that the subject matter of mathematics is structures and morphology, that is, the relations between them. So that kinds of mathematical objects are nothing but positions and structures. And the aim of mathematical theory is to describe or characterize these objects up to as one. So, for example, we look at the theory of natural numbers. This aims to describe systems of natural number structure as characterized by a cameraizer. So that its objects may be seen as one of the ordinals, or melanomerals, or any other object that has the same structure.

12:30 This is what structuralism picks up. It's the idea that, look, you can think of one of the ordinals and melanomerals as models. And what structuralism says is that what a structure captures is what all models have in common. So we say that if all systems that exemplify this structure are isomorphic, we say that the natural number structure and its morphology determines its objects based up to isomorphism. That is, it determines what we can say about natural numbers as a kind of object, as a position in the structure. Now, what's important here is that we don't get objects in mathematical structure. We get kinds of objects. to get objects is because when we say, for example, the number two, we don't know whether we're referring to, for example, what word for a new word. So when we say the number two, what we're referring to is a coin. So the philosophical significance of non-analytical It does seem to mean that we certainly do know the best of what we're talking about than we do. I mean, there's a question of how do we characterise and there are a variety of ways of doing that. And so, yes, we don't know what's the right way of doing that, or even there is only a right way of doing that, which is not to say that we don't know what we're talking about when we're talking about natural knowledge. Right. Now, the idea is that, okay, so I've given a set theoretic interpretation for what numbers are. One could also give categorical theoretic interpretation. So I'm not saying that numbers are sets. Right. More than that, what I'm saying is that you're right, but when we talk about the number two, we seem to know what we're talking about, and we don't seem to be talking about who knows the word. This is the point that's also mentioned by Hilbert. Look, we could equally be talking about strokes on the page. That's what we could have in our mind when we think about the number two. The idea is that regardless of whether you think of two strokes on the page or if I'm set for this and I'm looking at normal, the idea is that it doesn't matter because they both have the same structure. So you're right. The idea is that we all do when you say the number two is even. We all know what we have in mind when we say that. The idea is that we know that because anything we put into the slot called two has the same structure.

15:00 I suspect it's really one to take exactly the point of view. It's not by virtually the fact that we're talking about what's going on, too. It's really deep uncertain how I'm talking about the number two in subject terms. So, again, in the country in terms of view, I'm going to ask you, I should leave this to later. No, I think, I mean, I didn't think you're picking up on it, and I'm picking up on it so you can stop, and if I am, is the idea that, suppose, you know, a set theoretician comes in and says, no, what do you mean by two in this order? Is that what you're going to do? Well, I'm saying that, in a way, it's in order of a particular order of concept-exposition. We've certainly done the concepts of natural number before we then do that. I don't think one should be saying that, oh, the reason why a child can know the concepts of natural number is because of some abstract characterisation that the child is not aware of. So, in a sense, I think one ought to be doing justice to share knowledge So what I would say to that is, look, here when I'm getting this philosophical account, what I'm trying to do is talk about the context of justification. That is what allows us to justify that we're all talking about in the same manner. What you're talking about is the context of discovery. Now, the story I'm going to tell about that is that the context of discovery has us acquainted with particular systems. So, for example, is where we start. And in fact, we could put numerals up there and that wouldn't be a fault. So, when we look at how we come to know, for example, the number 2, we start with a system. And then what happens is we realize that this system is like that system. And then we go up and we realize this system is like that system, like that system. This one here is coming down. It's saying, why is that the case? Well, because they all have the same structure. So, yes. Do you think that all structures are isolating?

17:30 you can say this or the future, because this is not satisfied from this test, but that's all the things that said theory itself. Right. So this is what you're supposed to do. Exactly. I mean, one of the things is that, so here what I'm talking about is the notion of saving structure, and I characterize that in terms of isomorphism. One can also talk about what I call sharing structure, and you can talk about that in terms of finalized in terms of embeddability, any type of morphism that's going to carry the structure of one to the structure of the other. So I'm just giving kind of the simplest case at this point. But within this, you can also talk about what I call shared structure, where you don't have that so much. Okay, so the philosophical significance of structuralism is that in And in contrast to Platonist and Nationalist's psychologistic approaches, which focus on the epistemological, ontological, or representational status of objects as independent things as we think, and out there and somehow we're supposed to bring them in, the Structuralist philosopher points his analysis at what we say about kinds of mathematical objects as positions in structures. So they're not out there, they're positions in structures. Seen in this light, the goal of the mathematical structuralist, insofar as he replaces top of objects with top of the position of the structures, can be seen as attempting to move beyond those difficulties with having to account for the independent existence of mathematical objects. So that's the problem, if you adopt the foundation, let's say platenics account of mathematical orthology, is that now we have to pitch down a story about how do we can get to that outcome. How do we know those objects that are unfair? So my objective now is to show that much of the confusion underlying the realism debates of both domains of inquiry relies on talking about structures as things. So here we have an account of positions in structures. And of course the philosopher then asks, well, what's a structure? Where Where is that? So, the significance of this is going to claim that following the Vorbocky tradition too closely, and thereby not appreciating the significance of an algebraic alternative,

20:00 philosophically interpreted mathematical and scientific structuralism has failed us now. So this is the key point, if you're going to take anything away from what I'm saying, that would be it. It would be the idea that, look, we keep asking as philosophers, fine, I'm going to give a structuralist account to mathematics or physics or science in general, but what are structures? They need to be things if I'm going to give a structuralist account. And what I'm saying is, no, they don't need to be things. What's at stake when you say, say a bit more about what's at stake when you ask whether or not they're things? Whether or not they're things. What does it mean to say that they're things, what does it mean to deny that they're things? Okay, so if a structure is supposed to be a thing, then the typical philosopher says, well, when are its identities? And then you might say, well, where is it? Well, it's out there. Well, then if you return to the same problem we have involved in, how do we know it? Do we know it by the systems we look at? Do we have some type of intuitive grasp of it? And so a lot of our evolution of that problem is by saying, well, look, a structure is, and then they give you definition in terms of set theory. What I'm going to show is that if you take that strategy, as a mathematical structuralist, you're probably interested into mathematical foundationalism or set theoretic foundationalism. But as a structural realist, if you do that, right, and so you've now got that structure is a set theoretic entity, and then you argue that, look, theories are continuous because of their structure, or theories are characterized by their structure, and what you've got is that, you know, either nature itself or a knowledge of it is that structure. And that's not supposed to be the point of structural realist. It's not supposed to commit us to one type of structure. I think that the contraposition of structuralist view of

22:30 is problematic in the sense that I might have a structural view of the notion of entity. And in this case, I accept as an entity whatever I can assign for position and structure. If I have this view, where is the problem for debate? What is the point? So I think that only if you have an ideal entity as something which is independently given by each relation is wrong, then the contrast is wrong. Otherwise, this is a problem for, because it's not true that all realists are, say, at least, I mean, so, I think that if you have to do that, the right, the essence, are taken as forms which, as defined structure. Yes, I agree 100% with what you say. I think that, I mean, I'm hoping that after this is done, The idea is to look at Plato and to say that he isn't a plagiarist in the sense that he doesn't exist in a independent. So I agree with that, Eric. And I think what I'm trying to do here is just set up the debate between structuralism and the real world, because that's the debate that's still going on in philosophy of math. And because that debate still goes on, we don't see the value of structuralism as this It has this position that says, look, we have an alternative to different objects as things everywhere, having an identity, and so on. But more importantly, I think what it does is it allows us, so you were talking about this notion of objects, and it allows us to see how mathematics is different from the same science, because the notion of objects is not the same.

25:00 So what I'm saying is to understand or to appreciate what structuralism is about and not get caught up in questions about whether objects existed independently and other structures existed independently, we should separate algebraic and sub-theoretic from structures. So, I do this in mathematics using category theory, as I've said before. I will just say right now, when I get to talking about structure and physics, I'm not arguing a category theory is a background language for the two years. Staying away from our friends. So, we may now distinguish what may be called the Warbucki and the categorical or algebraic notion of a system. I begin, then, with the abstract notion of a system, such as we will see that this is where we find our corresponding notion of what I've created, a cat structure system. That is, the system that has a categorical structure. So in its most general sense, an algebraic system is a schema for what we can say about the shared structure of a mathematical kind. And this is speaking again to the point that was raised about isomorphism. Now here that I'm not talking about the same structure, but I'm talking about shared structure. And that's because when we define what category theoretic structure is, we have objects, we have morphisms, but those morphisms don't have to be isomorphic. Did you want to say I was your brain? Because you put abstract into that. You said... Oh, right. You said I was your brain before you put an abstract. Okay, I meant abstract. Yeah, sorry. So a cat structured system then has objects and arrows as its abstract points, which are structured by the category theoretic axis, because I didn't write those there, but if anyone wants to talk about those, you certainly can hear that. for a kind of cat-structured system, that is, for an abstract kind of mathematical system now viewed or organized or structured as a category, this is the quote of Ali, is anything that satisfies these accents. The objects need not have elements, nor need

27:30 the morphisms need functions. We really do not care what non-categorical this would meant in a sense of category theory, properties the objects and morphisms of the given category have, that is to say we view it abstractly. So that was the sense in which I was using the notion of abstract notion of system. By restricting to the language of objects and morphisms, domains, composition, identity. So the idea here is what Audi is pointing now is that, look, when we talk about categories and how it's thin, so to speak, it's objects and it's not fissure, against the set theoretic foundational lens, those don't have to be given in terms of some type of set theory. This is a huge debate in philosophy of mathematics. So, once we see important differences, on a category theoretic view, not only are there are no objects as sets with structure. This is how Dummett defines an object given by structuralism. Nor are they places with structure. This is how Shapiro defines an object according to mathematical structuralism. There are no abstract structures as either equivalence types of systems with structure or the abstract form of the system highlighting the interrelations among objects. So what I'm pointing at here is, if we use category here as this organizational language, then we don't have to either reify structure, which is what Shapiro does, and we don't have to say, well, structure somehow comes out of what all systems have in common. So what this means is that the Burbaki conception of a system, that is, of a system whose objects and positions in set structures, I use Burbank here in set theoretic as interchangeably, is to be considered as a particular kind of structured system. It is not the archetype for either the concept system or the concept structure. So throughout my talk, when I'm saying that we should distinguish between algebraic accounts of structure and set theoretic, This is what I'm talking about. What I'm saying is, look, structure does not have to be defined in terms of sex. It can be just defined in terms of, look, I have these objects as position of destruction. I want to characterize the sense in which they have the same structure. How am I going to do that?

30:00 Well, I'm going to somehow say that they're related by either isomorphisms or... I have objects which are position and structure. What was that? I see. I mean, you go to the center of the night. Oh, objects as a position in a structure. That's one thing. Okay? For Shapiro, it's places with structure. That is something different. For Shapiro, the way he has it is the following. He gives this analogy between, suppose you have the structure of a baseball game, and you have a position, the left field. Does everybody know baseball? No, I remember doing this at Oxford and there was this guy who knew baseball and he was lost. So we can do it by numbers. You take the American government system, why am I saying American? Canadian. And we have a position in that structure called the Prime Minister. For Shapiro, what the object is, is Prime Minister. For the rest of the structuralist, what the object is, is the things that fill that position, or that place, sorry. So for Shapiro, the object is the place, not the place of the person. And for Shapiro, the prime minister as an object is an object which is a structured object. But that structured object is not referring to the rest of the government. Oh, yeah. It does. It only gets its meaning or its existence, so to speak, from being a position in that structure. So this is what seems to be semantic between a vertical entrepreneurial realm of the government of structure. There's a place in the government of structure. Now, for you, that place is occupied by what the other four prime ministers are saying. No, no, no. See, for Shashira, the idea is, look, here's prime minister.

32:30 Not person who is prime minister. They're not general president. For me, and for Hellman, and I believe for Bransman, the idea is that, no, look, you have this structure called the Canadian governmental system, you have climate, what we say is an object is, as I would say, all the systems that satisfy that structure. I understand what you're saying is that Cretchen, in this case, is not the person who is a schoolboy and who's going to such and such a place. Insofar as Cretchen is invested in the structure, it's government structure. So it is Cretchen characterised as, in terms of that government structure. It is Cretchen characterised by that structure, but what in frontology consists of is not Cretchen and... Surely we have a straightforward ordinary language recognition of this distinction when we talk about the position of Prime Minister as opposed to the place of Exactly It's really the case that Shapiro wants to talk about the Prime Minister as opposed to the position of Prime Minister That's what I understand places that all jets are respected in the place of those functions. But why do you say that Shapiro is in a secular production of the system? Oh, okay. So, right. Well, you think he does. So, for Shapiro, your oncology are these places. Now, the first problem that we've added, and this is an article by Hulbert, which is how are you going to give identity conditions to places? So that's one problem, right? So then I say, okay, but he's somehow defining structures as set structures. And initially, he doesn't seem to do that, right? So he says, look, my notion of structure is like any other notion of structure. I can get it through this notion of equivalence, and any structure equivalence, and it talks about nothing. And then he says, well, look, one thing I could do is I could frame all of this music as that as a accent.

35:00 I think that's chapter 8. And so he goes on and he does it, and then he realizes that we can't get to structural equivalence. And so what he adds is what he calls a coherence accent, which says that if a sentence is well founded, then a structure exists. But that existence claim, I think this is pointed out by Helmin who would take the three line-up structure, that exists, that existential quantifier, makes sense only in the framework that has several. That's the framework he's providing there. Now he does go on and he says look, one could use any background theory one wanted to, one could give Helmin's non-realistic interpretation of math. But again, if you look at Hellman, what Hellman is going to do is he's just adding possibility conditions on top of ZF-C axis. So I think he just uses ZF-C axis. So in some sense, at a mathematical or at a philosophical level, he's being very general. He's just saying, well, I'm just going to talk about objects as positions and structures. I'm not going to get into the nice and easy about what structures are, because all I really need to identify structures is structure equivalence. But when he comes to explain what that is, he falls back on it. Okay, so you're saying that he's committed to sets, sort of the systems. Well, I'm not saying he's committed to that, but unless he can give another notion of structural equivalence, because he needs a notion of structural equivalence, and unless he can give that that doesn't align on the new coherence accident, then I don't see how that's going. Well, it takes cohetics to be primitive notions. Yes. Well, that's his solution to the problem. I'm just going to take it from here and say it's very good. If he accepts that, then his notion of a system is just, and you could call it a model theoretical notion. Yeah. That's a third of it. That's exactly right. And that's a notion of a system. Nanocore structure, as he calls it. I would call it a system. But he's not going to get structure equivalence. That's the problem. We're not going to be able to say we're two structures, not two systems. So we have objects as positions and structures, right? And then we have this next level up where our objects are, in fact, our places in a structure.

37:30 What Shapiro means now is a notion of equivalence of structures. Now you're right, if he takes coherence as primitive, the whole thing works. But I just don't think that we can do it. Okay, we're moving on. Okay, so what I'm giving up here again is that we don't have to give an account of the structure for the second. Everybody can have that. Okay, so we could take our objects to be sets and our morphisms to be functions, and we could present the underlying structure of a more widely consistent. So if we wanted to give an account of set theoretic structure, we could do that using this framework. We just take our objects to be elements or sets and our morphisms to be functions. And the result is a kind of category called set. But this does not mean that objects are sets and morphisms are functions. It means in this kind of category, propositions that talk about objects or morphisms can be interpreted about using that sets or functions. Again, when we talk about structuralism in mathematics, we're always talking about kinds of objects or kinds of structures, isn't it? I don't think we're going to need the example there. So, at any rate, here's the difference McLean captures between the set theoretic and the algebraic category theoretic notion. He says, Bourbeau-Key's concepts define mathematical structures by taking an abstract set and appending to it an additional construct. In category theory, there is no subordination of mathematical structures to sets, and this is the source of supremacy of this theory over Bourbeau-Key. I'm not going to argue for supremacy, but for instance. So, now we talk about science, in case the point of evidence. So, I'm now going to argue that as with the case of structuralism and mathematics, it's following the Borbocky condition, as assumed by a serious event in view of theories, too closely, and not appreciating the significance and use of the algorithmic alternative that interpreted science-making structuralism as failed at the moment. So what am I saying here? All of this began, I'll just give you a little history, because it's important when I ask

40:00 for your help, which I will do. So when this began, I was working in structuralism and math, and then I was at Oxford, and there was these people working in structural realism, and I thought I'd be talking about the same industrial structure. And after some kind of clarification of, well, how do you use the church structure or system or model, the next thing was to write something up, which I thought would be about four pages on the semantic view of theories. Because I thought, well, it's been around for a while and everybody seems to know what it is, so let's investigate that. And I realized that there was no attempt to do a theory that was consistent. So, with Sufi's, what you have is that, you have the question, how can models be used to close the gap between what a physical theory says and what it's supposed to be about? To close the gap, one might say, between language and reality. And he's presenting this in contrast to receive the syntactic theory. And what he says is, look, we can use models to capture what we mean when we say this physical model and this mathematical model have the same structure. Everybody kind of with me so far? So a mathematical theory applies to, or a mathematical model applies to a physical model if they have the same structure. Similarly, be used to represent phenomena if the data model and the theoretical model have the same structure. So what we have with Sufis is we have this account of both application and representation in terms of the models share the same structure or have the same structure. That is for isomorphism. So then what he does is he says, okay, wait, why? Why did they connect up? And his answer to this question is because, well, structures are set-theoretic entities. How does that connect up physical theories with the world? Well, because Sufis assumes both set-theoretic entities and mathematics. That is, he assumes that any physical theory, or any certain mathematical theory, gets its structure from set-theoretic. And he further

42:30 That, that's why mathematics explains, because physical theories, and we'll get to, I have a quote about this, because physical theories, and he mentions, I think it's thermodynamics, mechanics, and then he goes on to talk about even psychological theories, He says, are on all fours with set theory. That is, the notion of structure that one finds in physical theories is set theoretical. And that's why mathematical theories connect out to physical theories. But how does that make sense? So the next person I looked at was Suki. And what Suki does is he says, well, look, let's distinguish between a model and the expression of the model. Let's cut out where Sufis is using set theory. And let's say the model is, or the theory, sorry, is not a collection of a set theoretic model for it's model given in a set theory, but the theory is the model. Notice how I said the theory is the model. Because what Sufis assumes is that, or Sufi, sorry, is that there is one intended model. Now, for those of you who are familiar with mathematics, you know that talking about every intended model is a problem. You can't pick out intended models to follow through that. So the next person I looked at was Mark Fossin, and what he does is he says, well, look, we don't need intended models because we don't need them to go home. We don't have to talk about truth. All we need is empirical adequacy. So he, like Sophe, doesn't lie into this set-theoretic connection. But what he's saying is, well, look, all I need is empirical adequacy. And now, he can't give an account of shared structure in terms of set-theoretic way Sophe says. So what he does is he uses the notion of embeddability to capture the notion of two structures, while the theoretical models, kind of data models, that's one of the questions we have, have the same structure. Okay, so there's the kind of story that I kind of found. And then I looked at French's work to see...

45:00 Colors, yeah. You know what it is, but that's what I'm talking about. And I see, well, where are we now with respect to these natural structures? And secondarily, how does this fit in with the agenda of structural realism? And what I realized is the partial structure of those things. The idea that we look at partial structures of what we call partial models, and we look at the various relations between those, right? relations of partial exomorphism, relations of homomorphism. And then we can characterize in the same way that Suthi said, the idea that these structures have the same structure. And so a mathematical theory applies to, or a mathematical model applies to a physical model, they have the same structure. A theoretical model applies to a data model because they have the same structure. So, what we have is this account, but where Rockfrost and Sophie gave up on the idea of set structure, this gets reintroduced with French. That is what's making that connection between our mathematical, our theoretical, and our data models is that they all share that structure in virtue of being defined set theoretically. I think it's the last bit that doesn't follow from the previous bit. It's one thing to use partial structures as a representation of two theories or models and their inter-theoretic relations. It's another thing to say that they have those inter-theoretic relations in virtue of instantiating the partial structure in question. So it's me hoping to Steve to say, well, what I'm doing is giving you a way of representing the theory of relations, but I'm not saying that they have those relations because really they're set to the record of structures. I'm just saying I can use set theory to represent the structure of the question. Okay. So two responses. One, then that agenda is importantly different from Sufi's. So that's just a historic moment. Because for Supri, I really do think he was a retro manager. So if you're going to make this a point about representation, right?

47:30 So now, and to get to my picture, so what you have is you have this picture of how mathematical models are here, and you have, say, theoretical models, and then you have, say, data models. You can have models in between when you talk about hierarchies. And so the idea is that you have partial structures and those are related by either partial isomorphisms or homomorphisms depending on what interaction or representation you look like. So all of that supposes at this level the level of what you just pointed out absolutely correctly of representation. So that's absolutely fine. It's when you combine that with structural realism that you draw Here's why. Because structural realism doesn't constrain itself to that representation. It then moves from, look, I've got shared structure here, or if one wanted to give an account of theory change, so one would, you know, say have the models of Fresno's theory of light and Maxwell's theory of light, one could account for their shared structure. So you don't have to go just this way, you can go horizontal as well. Again, all of that is about representation. The question then is, how do I move from that to a claim about, look, the continuity of structure speaks to something in the world. That's what I'm not understanding. So I think structural realism gets us to that level of representation, but how is it real? I mean, this is what we've asked about. I mean, you're not supposed to be able to use structural realism from left as a way of representing. So it's not supposed to get you to structural realism. The thought is that these two issues are somewhat off-off. No, I wasn't saying what a... Okay, sorry. What I was arguing was that that, that representational, self-theoretic, more well-meaning thing, partial structures approach, in combination, I'm not saying that it's really the result, I'm saying if you add those two together, then what you've got is either the claim that

50:00 if you're going to move from the level of representation to the level of ontology, and I'm not saying metaphysical ontology or physical, let's really look at that. So we're going to say I'm going to move from this representation level to the level of ontology. How do I make that? Is it sets all the way there? Can I just make a minor comment? Well, that's to be a little bit careful about... I'm getting a bit lost about what we meant by representation. Going back to your... I noticed a quick skim through the history. One issue that James and I and Lacoste and I were concerned with is the issue of the reification of theory in the following sense. If you read time, one can interpret, let's say the receiver you're saying, what the theory is, is a partially interpretive, axiomitized logical and mathematical system. And then Suckey's comes along following tasking and says, what a theory is set-theoretic structure. And then you have people like Baz and then James and I and Acosta saying, wait a minute, we have to be a little bit careful here in identifying what a theory is with a particular way of representing And many of the motivations for the semantic approach can be seen as for representing the practice of science in a particular fruitful way. And then, cowards as we are, we just back off and you can say, well, what is a theory? If you say that a theory is a set-turistic structure, you get into all kinds of problems and people just start beating you up. So you can back off and say, look, theories are the kinds of things that scientists fool around with. And what we're doing as philosophers of science is coming up with a particular formal representation that will enable us to say certain things about scientific practice. And it may well be the case that a category-theoretic representation is more fruitful for certain purposes than a sector-specific one. As I've mentioned before, people like Da Costa have written little bits about it. So that's one issue. What's the best way of representing, if you like, scientific practice? the second issue is the representation of reality

52:30 but that's another issue so having chosen the semantic approach as the best representation of scientific practice as James says there are other things one has to do to get on to saying right and now these aspects of theories represented theoretically represent stuff out there and that's a different set of issues Okay, yes. Now, I think that's, we're both in line here. So, because what I'm going to say is that the way I read some of the things you said, and certainly the way I read James, was that you were, in fact, moving from representation of the practice to representation of the reality. and that's it was that move that I saw as problematic so that's why I'm going to say structural realism is great as a method or tool because it speaks to scientific facts but it's not great at telling us what's the stuff in the world so that's well I mean in defense of James and I think in the paper what he says is that the semantic approach is the best Partly in the paper, James is arguing against a particular sort of Worrell's adoption of essentially the receive view, roundly sentence, and all the mess that that entails. And I think making the perfectly reasonable, and making the way of astonishingly, to me, reasonable point, that if you're interested in representing structure, then pick a representation of scientific practice that brings that to the front, there it is, rather than partially interpreted, you can still perhaps do it, but then one runs into problems, but I didn't think James was actually saying what you were saying so what I'm going to argue is that you're right, that the best way is the semantic view of theories but you shouldn't read the semantic view of theories except the you should read it out of right, that is you shouldn't set structure, but we can open this to what kind of structure it has. And so that's my only question. But my big question that remains for me is, even if you do that, how do you

55:00 move from, here's a theory, it represents, you know, why this is a model in the sense that it has the same structure as the data, whatever that means, that's another question. why is it we can go from that to playing about reality playing about what objects of the world are like sorry to take it can you sort of elaborate because one response this is just me being naive one response is why we can do that we have a theory it captures the data we represent that capture we say partial isomorphism whatever And it does it very well, however we judge well, so we might have to tack on some form of confirmation theory, but we can do that lazy and that's fine. And then, as realists, one might want to say, then run the No Miracles icon. It can't be a miracle this is happening. We reject social constructivism, of course, in all its forms. So that gives us, that warrants the inference that this is hooking on to the world in some way, cash out the usual business intuition. Isn't that enough? At the very end, I mean, I'm going, I'll just get back to this in a minute, but at the very end, I'm going, but the reason why it can't serve that, it can't latch on to reality in that way is that what mathematical structuralism tells you, and looking your way down, is that, look, you can characterize these objects and whether or not the problem is related to life or whatever, of their structure, but in terms of their plane. Remember, mathematical structuralism is about planes of objects. So here at this level of representation, we've got kinds of objects, and at the ontological level, so to speak, I just want objects. Well, we should let you get there. But some people are going to say, from the quantum physics point of view, you're never going to get it. So, Eleanor might say, stick with kinds, that's pretty much all you're going to get, unless you want to you know, you can get a sense of object if you do the kinds of moves that

57:30 Eleanor and sorry? Yeah, yeah, yeah, yeah, have run, but you're never going to get objects in the sense, I'll just we should let you get to this. One begins to worry that in your demand for objects you're asking for something that some people are going to say you're not going to get it. You're going to get kinds. You can get a certain notion of object out if we follow Mackie's mark. A structuralist notion of object. It is an algebraic structuralist notion. We can get into that. We can leave that for that. I just asked a question. I mentioned just getting clear on I want to be a bit more careful about that because one of the things that I've wanted to do with the whole framework of partial structures and partial isomorphism is not tied explicitly to realism because I think Ottavio, if Ottavio were here, he'd pop up and say, no, no, no I've developed a whole structural empiricism in these terms, and he might say, when it comes to continuity, if you're thinking of Kuhnian problems. He's going to say, I get over the Kuhnian problems by appealing to shared empirical structure, and that's all I need as an empiricist. And actually, French runs into real problems on that, he thinks. So I don't want to say that partial structures gives you a realist solution to the Kuhnian problem. But it does allow you to talk of continuity, yeah. Would you hope that it's a piece of the worst issue to the American society? Sure, sure, yeah. I mean, your paper years ago is a great inspiration for that, right? Yeah, sure. It's classic, classic. No, it's one of them too. We disagree with that one. Hugs all around. I would like to make a naive comment as a physicist, if they don't succeed to follow all the others. your technical language. But I think it could be dangerous to extend this concept structurally

1:00:00 from mathematics to physics, because here in a sense you are trying to cut off completely reality, whichever it means. And you replace with equivalence classes an abstract space This category is the best, is the top of it because of the equivalence class. Now, going to physics, the concept of equivalence class is completely outside the practice in laboratories in physics. Let us take an example of the gravitational field. At the mathematical level, if you accept Einstein's theory as a representative of what we call gravity, you can say that the gravitational field is the portion of human structures of a different field. And this is the first step to arrive at a category of definition of gravitational field. But in practice this has nothing to do with laboratories. A gravitational field for a laboratory is a set of records which are named that way. And then the only problem is how to connect the data of a laboratory to the data of another laboratory. Now, the number of accessible laboratories is infinitely less than the number of diphomorphism. Most of the theoretical laboratories which can be reached by diphomorphism are totally outside any possibility of the nine times. So, I see dangerous to a struggle. Mathematics, I have no opinion. Their things are much more complex in a sense. I put it here to definitions of this type because I think you lose the essence of physics. Yeah, I mean, I think that's what in some sense this conversation was speaking to. I mean, my problem is structural, really, is not going to go away that hurt to me. If you're going to stop that structural empiricism, then I do see the value for these continuity and structural arguments to argue against . I don't think we have a problem.

1:02:30 But the problem is this problem, namely, OK, so we've imported this notion of mathematical structure, and it's used for characterizing all that but we have this kind of perhaps naive view that physics is supposed to be about the stuff in the world, not the positions and the structures. So my question is how do we make that next step? How do we connect that, you know, now our data models, for example, or the structure of our data model to the phenomenon. I still don't understand. One would say the structure of the data model gets captured by the theoretical model, which gives us more, of course, it has to. And it does that enough times that it's suitably confirmed. And then we run the No Miracles on it. Ottavion says, you're mad, you're mad, no, no, no, just stick with it. And we say, but we're weird. run the No Miracles argument, we run the No Miracles argument, and then what we say is, and we have to be sort of fluffy, muffy, really a bit about this, but we say, look, aspects of that theoretical structure, mirror, map, look on to the world, that's what the world is like, and that's how we make that step, we say the world, and then opinions may differ, and then we may just all fracture away, Warren's over there, James is over here, but we may say the world is structured in that way or that theoretical structure represents the world structure not just the structure of the world but the world as structure and that would be a more sort of ontic way but the moon really is the no miracle I would have thought would just be the no miracles are and others about that fast and would you know would be with a failure saying no, no, you don't need to do that. Kuhn and others may have other options. Does it put the motivation into a very miracle's argument? Sorry? Does it put the motivation into a very miracle's argument? Then there's the issue of what else you have to deliver on it. Well, I'm not sure. This is why I'm asking Elaine what she wants. Why is it self-righteous? It's partly what you... If you take the no-miracle's argument to be an argument, and we all know the problems of cashing it out

1:05:00 as an inference to the best explanation in going from the age models to the age models, then the world I thought that bit, that's what it covered I don't see what else can cover that to be honest, if you're a realist so whether you take a no-niple's argument to just be a more formal expression of realist gut intuition or whether you take it to be a form of inference to the best explanation that's defensible in some way I thought that was the step that was made And then the argument is, and here's, I thought, where Elaine was interested. If you've represented a theory in terms, in set theoretic terms, then it may be that you, and then you've made that in a miraculous jump, then you're saying the world somehow has a set of objects over which they define families or relations. And if you're James, you may find yourself in trouble with that view. If you're John Worrell, you may not. And then I thought you were saying, sort of Catholic, Category theory comes riding in, and if you, one were to say, represent theories in terms of category theory, that would give you a representation of structure in the world that may not have the ontological commitments that the likes of James is nervous about. Now that's exactly what I'm saying. That's exactly what I'm saying. But that still, to me, doesn't... Okay, so I'll just wrap this up and then... Sorry. No, that's exactly the point. So anyways, all of this really just says that, look, the characterization I gave of mathematical structuralism, one could characterize physical theories in the same way. So one might say that the subject matter of science is structured systems and morphology, that is how they relate to each other, so that objects are nothing but positions of structural systems and scientific theories describe such objects and such systems and why I'm not saying we're sharing the structure. That's what scientific structuralism gets into it. What I see is I see the semantic view of theories as a version of scientific structure. So, semantic view of theories... I don't know, I don't know what I'm talking about. Okay, let's do this one. Just says, we'll just skip down. The semantic view of theories says we're going to take scientific theories as a collection or a hierarchy of models of family structures and family partial structures, where models are taken as non-linguistic. That is, models are used

1:07:30 to represent the shared structure of a physical system, and the constituents of models are supposed to represent the objects with which the theory is supposed to be about. So this is a characterization of the semantic view of theories under the umbrella so to speak because it's a scientific structure. And again, when you interpret this, you know, set theoretically, as I've shown like Sufis and so on in French too, right? Or algebraically. What's the difference? The difference is, on the Borbaqui interpretation, you have theories or collections and models of methodical structures where objects have the same kind of structure, but they have that same kind of structure in virtue of them having a set structure. What I'm trying to argue here is not in favor of formal category theories of background language, but rather just that we should algebraically interpret structure really. That is, we shouldn't commit ourselves to having the same kind of structure because they have a kind of set structure. So we should just say they have the same kind of structure, whatever that kind is a kind of structure. And in mathematics, we have lots of kinds of structures. We have set structure, we have group structure, we have field structure, we have silver-based structure. I mean, we have lots of them. And so let's just keep it neutral as to what kind of structure gets together. And still, we have access to the continuity of structure argument that we need for, I don't know, structural realism So, what is structural realism? Again, this is how we characterize it. So, it's supposed to break the impact between the numeric by shifting our focus from ontological continuity to structural continuity. So, we can be realist because we have structural continuity over period change. So, yes. See the reason of the strong contrapposition between the category approach and the centroidical approach. Because, after all, you said, this is of course I, that the centroidical approach can be be reconstructed in the framework of the more general Kantar approach.

1:10:00 This is true also for the problem of a semantic representation of the critical sciences. It seems to me that all the work, for instance, of Tsukis and so on, can be easily saved and to be constructed in the moral manner of framework of the category of law. So I don't see the easel of the strong. No, I don't know. Yeah, no, you're absolutely right. Yeah, the only, I don't know, for example, I know that Von Frosten explicitly and Sufie in the footnotes, he is not committed to set theory. Von Frosten says explicitly, look, the typical way that one does this and this is in the scientific at the beginning is why you say set theory, I'm not committed to that, so he's over. Stucke's I'm not convinced, because I do think that he is a set theoretic I do think that he thinks that structure in mathematics is set structure. It's no more. Okay, and that would be absolutely false. that that gives us by Stephen, who this partial structures approach defines a structure as a set theory. Now, if you're, you know, that to me, you know, if you're willing to give that up, then I don't have a problem about representing theories the way you do. I mean, I think what partial structures approach does is it shows us that shared structure shared structure doesn't have the same structure so we have other options besides that I mean can you sell us on the category theory or the algebraic approach a little bit specifically when you say going down your points 1, 2, 3, 4, 5 it's about how A mathematical theory applies physical theory this is something that James and Otavio and I many years ago had discussions with about group theory gets applied to quantum mechanics. How do you represent that from the model theory perspective? And that's what led us to do. Well, what you want is a picture in which you relate somehow

1:12:30 mathematical structures to, for want of a better word, physical structures in a way that allows you, in a sense, to draw down more mathematical structure as needed. So you sort of hook into this and you find, ooh, I get all of that too if I want it. And that's where we have the idea of generalizing the notion of a partial isomorphism of a partial homomorphism, and Otago in particular has developed that. Now, we were kind of pleased with it, and we thought, yeah, it fits, it makes sense. Is the algebraic approach going to give us anything different or anything better? Because if I can do everything with set theory and partial homomorphism for the purpose of philosophy of science, why would we need to go to the algebraic approach? Not that I'm not sympathetic. I'm actually very sympathetic and always have been to a category theory view. But, you know, in a sense, this is sort of the weary answer. Good Lord, do I have to learn a more category theory to do what I can do anyway with good old-fashioned set theory and powerful, light, and morbid? I think, you know, the answer... And that's why, right at the beginning, That is not what I'm saying. What I'm saying is, if you want to give your partial structures approach, then, and I don't know how you're going to do this, because if it really ends up the rhetoric, the way you find what you're doing and the relations and so forth and so on, is what you want to be able to put in there somehow is, look, when I tell you that this is going to capture my kind of object as a position in this structure, I'm representing this set theoretically, but any other kind you would do, group, right? And that's your example of group theory and quantum effect, right? Is the example that you want to capture, right? So the way that you can paint that picture using partial structures, that's only going to go through, that is continuity of structure, of group structure and constant rules, if a group is a set, and if the quantum phenomena more than being group structure is set structure.

1:15:00 Because when you represent your domain, that's set the record. closed. That's not what's doing your work for you, the fact that it's self-traumatic. What's doing your work is groups. So let's not force... What's important about your approach is exactly what you said, namely that you have partialized morphisms, homeomorphisms, you have a couple of homomorphisms and so on. Those are the relations that are allowing you to set the same structure. And that's what's doing your work for you. Now, Any mathematical theory, whether it's group theory, category theory, or set theory, has a notion of partial isomorphism, isomorphism, and homomorphism. So what's important about the approach that you brought to us is those relationships. Because they capture shared structure, same structure, they account for surplus structure in the way you want them to. So one of the things, I mean, this is just a slogan from category theory, but I think we should bring it together. And what it says is, let us not look at the objects, but let's begin with the morphisms. And that's, I guess, what I'm suggesting to you. Let's begin with those relations and then we can say, well, what kinds of things satisfy those? So let's remain neutral about whether those kinds are set structures. So what I guess I'm saying is not let's bring in category theory, go over in category theory, but find a way that when you're representing this partial structure as a quote, that you've captured the relations between your objects, but you haven't said those objects have to be set. That's all. What you mean by it is that you are not committed to ontology. What you mean? That's exactly what I mean. What I mean is, I've captured what it means to say that two systems have the same structure or shared structure, but I haven't told you that those systems have to be set structure. You have the same problem. I have the same problem in the sense of...

1:17:30 In what sense? Yes, that's my question now to these structure rulers. And given that I have that now, how do I know that the mathematicians that get picked up talking about shared structure to match on to natural kinds, right? How do I do that? I mean, you know, this is a problem that's similar in some sense to the problem with the syntactic approach, right, with the, I think it's still out there, but he talks about Russell, like how do we know that the structure of our percepts matches the structure of it for himself? And I see a kind of analogous problem here, which is, so I've got my representation, I've got you know this represents and this applies of a shared structure because of the next work that seems much less well I think still is there referring to the Newman problem, presumably the Newman problem doesn't even arrive no he's going to refer to the Newman problem in this he's just talking about a super obedience principle that's the term he uses that we must assume that not only the structure of our percepts matches the structure of things themselves, but vice-versa. It seems that you're talking about Russell, and maybe it's a point where I am. That's just a no miracles answer. Again, if the structure of our percepts was radically a force with the structure of the world, then we'd come unstuck all the time. So that's my question. So what's the side in that? Is that predictability? Is that confirmation? I mean, what... Well, usually, I mean... Imperial success? I mean, it's not going to be enough to have... I mean, the thing that's usually said is normal predictive success, right? Okay. But I think you were saying that you want, in one of your views, you want a world without a predictive success account, but rather a unified. Did I? Yeah. So it is predictive success. I think you can give all to it.

1:20:00 But then how do you have any hold on the distinction between an effective theory and a fundamental one? I mean, you have to say it's to do with as well, surely. Otherwise it would have to... Yeah, but it would be some combination of... One could add other things. Predictive success, unification and exploratory power. It was great predictive success in the area of condensed matter physics. You wouldn't want to say that that's a candidate for a fundamental theory. So, anyways, I'm just going to skip to a few minutes. Sorry. Alright, so, is how structuralism is a basis for ontological or physiological structural realism, then we're led to the conclusion that nature itself or knowledge of it is non-set structure. We don't want that. If, however, we accept an algebraically interpreted scientific structuralism, because we don't compare ourselves to cognitive set structure, then we're left to conclude that science and it talks about crimes a lot.