Identity & diversity of objects in structure

0:00 That's the crucial point, isn't it, that still seems to me to be got across to James, and I don't know about Hannes. Well, that there isn't some, as you put, inverted commas, set theoretical account of a more primitive or intuitive notion of structure. That's the point. That's simply the key point, which hasn't been got across to them. Otherwise, they wouldn't constantly be saying to you, oh, but we just don't want this set theoretical surrogate. I know. It's just math. The whole thing is math. And that's because they believe set theory is Zermelo-Franco, first order Zermelo-Franco. They think it's a mathematical theory on a par with all the others. Now, what do I... what's your... what do you want to go to? What do I want to go to? I want to go to my mail, which is www.yahoo.co.uk. Is that it? Yeah. And then I have to enter my password. There you go. Yeah. No, it's down there. You have to click on mail. And then I have to enter my password, which I'll do. Well, I can give it to you. You can give it to you. Sure. Was there anything you needed? i'm gonna yeah i'm gonna well no i just leave don't try to shut it down i'll just leave it as it is just leave it as it is let's go out of my yeah okay and um okay i can't seem to find my watch i must have taken it off downstairs when i was what is the time christ is that the time yeah 10 40 bloody hell christ i must have slept longer than i thought well well yes we're all the same oh by the way um on the subject of ribald um tommy's and the little ribald songs song sung by tommy's during the war um all right you're obviously not getting the hint are you um you know the one that's the take take off the marseilles no no i know i don't i know i know well it's the one that i would have been

2:30 very loudly this morning. A Frenchman went to the lavatory to enjoy a bloody good shit. He took his coat and his trousers off so he could revel in it. But when he reached for the paper, he found there was none to be had. He found there was none to be had. the point is you haven't got any bloody paper okay well you should have told me last night i could have got some bulk paper couldn't i yes i had to you know i had to tear bits out of exercise books and generally improvise oh dear i mean there's it's down there there's there's a big well yes but when you've done your business it's a bit awkward as a good way to get up in Where were you downstairs? No, I was in the bog straight across the road from where I was sleeping. No, it's no big deal. But I'm just saying if you've got some bog paper, it would be a good idea to... Yeah, that was all I was trying to let you know. Shit, don't tell me he's had a problem. Thank you. So it struck us that, I think, we've done a lot of activity in both the philosophy of mathematics and in philosophy of physics, but no one had every different way to bring these two together. I think structuralism, issues that come up, and so that's what we're trying to do to really get some perspectives on structuralism about two different areas, see what one can, and one from the other, et cetera.

5:00 And another reason that we came up with this idea was simply that we realized that sometimes we have a lot of people interested in structuralism and related issues here in Barista. So not just James and me, obviously, but there's Thomas Lightgate who's interested in this, Alexander Byrd who's working in beneficence of science, sort of comes close to this, John Mabry, Richard Pettigrew. I don't think it's much of an exaggeration to say that we have the largest number of structuralists that we're put in any department anywhere. So, first of all, could it do something with that? Once again, that's so many other people in this field are in the UK anyway. Most of them are not part of it. They happen to be here now. I'm Stuart Shapiro. So, it just seemed like a good idea. Doing this, I hope we'll have a good discussion, and that's pretty much what I wanted to say by way of introductory, so I think I should just stand for a speaker, James, and then see if you have any further introductory remarks or sort of... My paper's going to be, we'll sort of serve as an introduction because I'll touch upon all the issues which the other people gathered here will talk about in more detail. So, I want to begin by acknowledging and thanking the people that have improved my thinking on these matters. and especially Hannes Lightgate, some of what I'm saying will be taken directly from the paper the two of us wrote together, which is hopefully forthcoming. But also Stephen French is here, Simon Saunders, Oliver Cooley, so it will become apparent, I think, what I owe to them. Okay, so let's start with a quote from Jonathan Lowe. Jonathan Lowe So, this is sort of convincing me that I'm not arguing against a strong man, which is always reassuring.

7:30 Lowe says this, he says, two different individuals cannot both individuate or help to individuate each other. This is because individuation in the metaphysical sense is a determination relation. As such, individuation is an explanatory relation. I have no idea what that sentence means, but anyway. In the metaphysical sense of explanatory, whatever that means, it goes on, certainly it seems that any satisfactory ontology will have to include self-immediating evidence. The only question being which entities have this space, the space-like points, benefits, tropes, and individual substances, all being among the possible candidates. So, Lowe here describes to what John Stachel, in a paper in the volume dedicated to Henry Putnam from 2005, calls Intrinsic Individuality. So, anti-represent structuralism about mathematics, as made famous by Stuart Shapiro, and anti-structural rivers about physics, as made a bit less famous by Stephen Frank and myself, having common the idea that relational structure is ontologically more fundamental than individual objects. So, I used to be inclined to advocate eliminativism about individual objects, in particular about individual quantum particles and about individual space-time points. And Simon Saunders revived the weak form of the identity of the discernibles from the work of Quine most recently. And that convinced me that the defender of anti-structural realism particles and space-time points are not individuals, but ought instead to emphasize that insofar as they are individuals, it's the relations among them that account for this. Oliver Pooh's sophisticated substandardism similarly allows that space-time point to be individuated relationally and hence not independently of the metric field. So thanks to Simon and Oliver, I started thinking, okay, we can forget about saying there are no individuals and just argue against, there are individuals, but argue against in terms of individuality, basically. Account for individuality by relational structure. So this shift of emphasis, I think

10:00 is not undermined the essential point of ontic structural realism. Both the anti-rep structuralist and the ontic structural realist, for both of these people, the individuality of objects is dependent on the relational structure of which they are parts. individuals need not have self-individuated candidates nor be individuated independently of the other individuals in the relational structure to which they are part, i.e. culture alone and Stachel calls this contextual individuality okay, fine, but then more recently, Hannes-Meyker convinced me that at least in the case of mathematics there may be nothing to account for the facts about the identity and diversity of objects in a structure. Yet this suggests that the notion of individuality is primitive rather than being dependent on relational structure. The notion of primitive individuality is dubious for two reasons. Unfortunately, Alexander's not here. The first reason is because it's associated with the metaphysical doctrine of hexiety, and all metaphysical doctrines are dubious. And secondly, it's associated excitism, the doctrine that there may be worlds that differ solely in virtue of the permutation of individuals. Now, it's that doctrine that causes problems of the interpretation of physics, namely, seems in conflict of the permutation invariance of constant states and the diffiomorphism invariance of generalistic physics. It's precisely the need to avoid those problems and seeing that doctrine as causing those problems, therefore rejecting that doctrine, that got me into being a structuralist in the first place. So now what's going on, right? I started out rejecting primitive identity and I seem to have ended up being persuaded of it. So I just want to, what I'm going to do is tell a story of, it's going to end up not being the same kind of primitive individuality that I rejected originally, thankfully. So I'm

12:30 going to set out a number of positions where my take on the relative ontological status of objects and relations, I'll briefly explain how relations may be used to ground the identity and diversity of objects and structure, I'll illustrate the issues that stay with some examples of graph theory, and then I'll suggest what I think the key issues are concerning mathematical and physical structures, probably, but there's no clock in here anymore, is it? They move blackboards. Never mind, the Vice-Chancellor's blue in the Senate building is gold-flaked. Right, but haven't we got a clock? Someone should actually lend me a watch, so I can make sure. Okay, so that's what I want to do. So, here are some views about, well, first of all, here's what I kind of think is a standard metaphysical framework. There are individuals in space-time whose existence is independent of each other. Facts about the identity and diversity of these individuals are determined independently of their relations to each other. each has some properties that are intrinsic to it it's also often assumed that PII principle-related discernibles there's some properties that distinguish each thing from every other thing and the identity of individuality of objects can account for impure and qualitative terms both quantum mechanics and relativity theory We've been cited as teaching us that the nature of space-time and matter creates profound challenges for a metaphysics of self-subsistent individuals. These considerations do not compare us to abandon such a metaphysics. The identity of individuality of quantum particles could be grounded in each having a primitive business, the same could be true as space-time. What we can establish is that physics tells us that certain aspects of such a world would be unknowable. I just want to make contact with something that's being much discussed in Metamooks at the moment, which is the idea of Ramsey and Kantian humility. Frank Jackson, Ray Langdon, David Lewis advocate use similar to epistemic structural realism. Jackson refers to Kantian physicalism, Langdon to Kantian humility, Lewis to Ramsey humility.

15:00 They argue that science only reveals the causal relational properties of physical objects, Jackson says, we know next to nothing about the intrinsic nature of the world. We know of its causal kind of relational nature. Langton argues that science only reveals the extrinsic properties of physical objects. Both then argue that the intrinsic nature of the world is adversely inaccessible. But then, what these metaphysicians have seen in Peter Unger argues that unknown to the world is purely structural, but qualia are the unknowable, non-structural components of reality. these metaphysicians argue for is that we can't dispense the traditional conception of individuals, even if we have to admit that their intrinsic natures are unknowable. Now, part of what motivates me is the desire to keep one's metaphysics closely tied to what one can know, so to reject unknowable idle wheels. I'm just trying to establish that lots of people think there must be intrinsic natures to objects lest anyone think that I am attacking a straw hat and here's Russell famously making this point it's in here to suppose there could be individuals who don't possess an intrinsic property but whose individuality is conferred by their relations to other individuals he says it's impossible that your author should be as any event suggests nothing but the terms of such relations of progression. If they ask me anything at all, they must be intrinsically something that's different from other entities, the points from incidents or colours from sounds. Okay. The definition always indicates some class of entities have a genuine nature of their own without distinct individuals in the first place. There's nothing to stand in the relations that are supposed to confer individuality on the relator. I regard this as a kind of And, of course, Ben Asserat also famed the influence of mathematics in this respect. So I regard this as basically a metaphysical article of faith. And it's not one that I adhere to. So what might we think, then, about the relative ontological status of objects and relations?

17:30 Well, here's Spatuel again, distinguishing between several views. Firstly, there are only relations and no the latter. Secondly, there are relations in which the things are primary and their relations are secondary. Thirdly, there are relations in which the relation is primary while the things are secondary. And finally, there are things such that any relation between them is only apparent. Right? So, liveness, not very popular in modern philosophy. Usually, opt-in-structural realism is considered as either one or three. And then people quickly point out that one is incoherent on following grounds. They say there can't be relations without relata. that this is a projection that Steve and I got a long time ago in personal communication from Michael Redhead, Satis Silos, Morganti, and Anjan Chakravarti, who since he came all work from Canada, I thought I would get a quote from him into my paper, said, one cannot intelligibly subscribe to the reality of relations unless one is also committed to the fact that some things are related. In other words, the question is, how can you have a structure particular, for example, how can we talk about a group without talking about the elements of a group? Even many of those seem pathetic to French language onto structural realism and objective, but they cannot make sense of the idea of relations without relata. For example, Michael Esfeld, Holger Beyer, John Stachel. Okay. So, in the best sense, I think that we can make sense of, I mean, it seems to be just straightforwardly refuted, by the I mean, his, if he wanted to do one of those, he'd go like this, he'd say, look, A is true, the Earth has been the Moon, and that's true, A is certain, there's a certain relation between two objects, the Earth and the Moon, so of course there are relata. The best sense that can be made of the idea of relations without relata is the idea of, the tonic view, I think, that all there are are the forms and all the appearances are illusory. Okay, but that's not what Steve and I have in mind. And John Statchell really, I think,

20:00 has a go at saying what we do have in mind is that look, what one can be read as, the view that there are only relations that no relapse can be read as, is asserting that while there are relapse, that they can be analysed themselves into further relational structure. One means, therefore, that it's relations all the way Okay, so indeed, French and Laban emphasising the claim that relata are constructed as abstractions from relations doesn't imply there are no relata, rather the opposite. Now, Stachel denies that there's any reason to think that this holds. Simon Saunders, though, seems to agree with Steve and I that there is some reason to suppose on the basis of modern physics is relations all the way down. Okay, so I don't want to get into speculating about, I mean, what one would need to do, and Steve and I have gestured towards this, would be to tell some kind of story about how, why whenever we think about relational structure, we have to end up sort of hypostasizing individuals in order to throw our cognition to get a grip. But that's a story that we haven't really got very far with, apart from a nice quote from Michael A. He says something similar, the gross matter which is furnished us by our sensations, but a crutch for our infirmity. So the thought is that somehow we use objects, thinking about the world in terms of objects is a consequence of our cognition rather than reflecting how reality is ultimately structured. In any case, there's another view that Stachel doesn't consider. Michael Espel is clear in his rejection of one and he says relations obviously require a lot and he denies that these things must have intrinsic properties over and above the relations in which they stand. So then he's with us against Lowe and the other metaphositions. And Edsfeld, like Oliver, Pooley seems to hold a view that Satchel doesn't enumerate. Five, there are things in relations that neither is private or secondary. And Oliver puts this in, he calls this a no-priority view.

22:30 Okay. Now, it's not clear to me, my watch, says, not shock resistant, water resistant. would be okay. Now, let's just forget about disagreements among structuralists. I mean, we have to get into the nitty-gritty. I mean, actually, I think there's some sort of reason why we could press Oliver that, you know, five is unstable and so on. But I don't think that's really to the point. I mean, what's important is that none of the people, none of the structuralists affirm two or four, two or four being the view that there are no relations or that relations are secondary and object primary. So none of them affirm the existence of self-subsistent individual things. Saunders clearly agrees with Esfeld that fermions are not self-subsistent because they're individuals only given the relations that obtain among them. There's nothing to ground an individuality other than those relations. In Einstein's terms, sorry, I missed out this nice quote from Einstein. Stachel, when Stachel's talking about the first view, he has this quote from Einstein. Einstein says, in a letter to Max Born, the idea of independently existing objects comes from everyday thinking. and it's a necessary presupposition of physics. Okay. So, basically, there's a bit of a consensus emerging amongst philosophers of physics that we should abandon the metaphysics that posits fundamental self-subsistent individuals. This metaphysics is also abandoned in philosophy of math by inter-rep structuralists. What further similarities and dissimilarities there are remains to be seen. So now on to grounding identity and diversity in properties and relations. Fraser. Unfortunately, Fraser can't be here for very good reasons. So, as John would say, he had a chance, he's the ghost of the feast. Because it's a real shame he's not here.

25:00 Fraser asked this question in a reply to Simon Thornton to some extent me what constitutes the numerical oh this is a reply to me I think what constitutes the numerical diversity of mathematical objects objects in general we could deposit XI to the property of primitive self-identity to answer this question intelligible, there are worries about it. I mean, one worry you can have about anxieties, it leads to a kind of regress. Because we explain the individuality of some objects in terms of possessing this property of primitive identity being that thing. But then you could ask, well, what makes the property of being, for example, that thing, distinct from the property, what makes that anxiety distinct from the excise that this other thing possesses? And you might think you've got to posit a second order anxiety for the property, and so on. And Chris Day had a paper about this years ago. So you might worry a lot about anxiety, whether it really does any explanatory work, and so on, but leaving all that aside, even granting that it works and everything, many philosophers are just not inclined to go that way and they prefer the idea that objects are bundles of properties, this seems to offer empiricists a way of accounting for individuality whilst quantifying only over properties that are in the scope of natural science, or could be in the scope of mathematics. So it looks like if the identity and individuality of objects is to be explained by the empirically accessible properties that they possess, it it was saying that the principle of identity discernible was restricted so that only qualities and not identity modern properties or its scope must be true. Okay, but now, okay, so now we're getting up to date. Much of the debate about this principle has been focused on examples of structures that admit of non-trivial automorphisms. Contrugal automorphism is a map from a structure to itself that isn't the identity map, and which leaves the structure intact, doesn't change the structure.

27:30 So, the standard philosophical example of such a structure is that the mathematical actitudinaries of identical spheres that are a mile apart in empty space, permutation of spheres results in the same world. And it's here that the latest chapter of the story begins, but PII seems to be violated by quantum particles and by mathematical objects such as the square root of minus one. So the single state of two photons admits that the non-tribular automorphism that permutes the two particles. The complex plane admits that the non-tribular automorphism maps every complex number to its complex conjugal. In such cases, the structure that results is invariant and the individuals in the structure appear not to satisfy PII. So, along comes Simon. So this is the motivation Stephen French and I have for saying, they're no satisfied PII, we don't want to go the metaphysical route of positive primitive identity, so let's just get away from thinking in terms of individuals altogether. But then along assignment and challenges the claim that PII is false, the third law is in entangled states by reintroducing Quine's distinction between three formulations of PII, namely absolute relatively weak. So some people are living and breathing this stuff, but not everybody is probably completely familiar with this, obviously we go through it very quickly. Two objects are absolutely discernible if there exists a formula in one free variable which is true of one object and not the other. For example, ordinary physical objects are absolutely discernible because they occupy different positions in space and time. I and one are absolutely discernible because one is the square of itself and I isn't. Two objects are relatively discernible just in case it's a formula in two-three variables that applies to me in one order only. Moments in time seem to be relatively discernible since they always satisfy the earlier animation in one order only. Think of the points in one-dimensional for a mathematical example. And finally, two objects are only weakly discernible if they're not absolutely or relatively discernible, and if nonetheless there is a two-place irreflexive

30:00 relation that they satisfy. Black's two spheres are weakly discernible. So do infernal to the sin that's statement, I minus I. All these examples have a car that the entities stand so you're irreflexing the symmetric relation to each other. So I agree with Simon that the weak notion of individuality advocated by it is coherent. It will be questioned by him to deny the sufficiency of weak discernibility, nearly because stronger forms of discernibility available. Note, however, that while Soros' view indicates an ontology of individuals, it's a structuralist one because the individuals are not assumed to be ontology prior to the next relations in which they stand. Okay. So, now, Fraser brought up an argument from Russell. I'm going to go through this Fraser brought up an argument from Russell which is supposed to show relations can't account for numerical diversity. It's an argument against a bundle theory. The rough idea of the argument is since objects are merely bundles of universals and universals are multiply instantiable then objects being nothing but universals must inherit the properties of the universals therefore objects inherit the property of being multiply instantiable but multiply instantiable means being instantiated in different places therefore the same object could exist in two places at once but that's a contradiction nearly be universals. That's roughly the argument. Now, my argument against that is just that it seems to characterise the bundle theory in a way that bundle theories are going to accept. There must be some difference between a set or class of universals and an object on pain of being committed, on pain of the bundle theories being committed to the view

32:30 that all possible objects exist, since all sets of universals exist. The idea of an object as a bundle of universals is intended to contrast with the idea of a substratum or bare particular instantiating universals. It does not follow that a bundle of universals is repeatable, like universals themselves. A bundle of universals in this sense is a bundle of universal instances, hence the first sight of Breyer's argument appears to rest in a conflation of a universal or bundle of universals, with an instance of a universal, or a bundle of instances of universals. Well, anyway, we could talk about that, but that just gets us into the metaphysics of particulars and universals, and that might be one that you're just inclined to reject anyway. So I'd rather just move on Charles Parsons in opposition to Karen it's a kind of endorsement of a saunders type view Parsons opposes others who Karen and others who demand metaphysical accounts of objects within terms of exiles and self-indulgent and elements of intrinsic natures structure of this view coheres to the thin conception of what an object is. So the most general concept of object arises from formal logic, and we are speaking of objects and we use the apparatus of singular terms of entity and quantification. This thin conception is attributed to its principal representatives of Freire, Carl, and App, and Quine. Okay, so we just go for a thin conception of object. Now, the next part of my talk is going to suggest that the right conception of what an object is may be even thinner than quiet and sawn as thick. So, Labour in 2005, McBride, ultimately, Tim Barton, who replies to, talks about these issues, Simon, Catherine, all tacitly seem to assume that bona fide individual objects are subject to criteria of identity, but the latter are construed somehow accounting for the identity of the individuals in question. but Brian explicitly talks whether or not relations can account for diversity, whether or not, it's all this

35:00 dot, dot, dot. Well Simon doesn't say this explicitly. I think the difference is between, I mean I think Simon's Simon I think wants a formal account of identity emerging from predication but he can say for himself what he thinks in questions and so on I mean, nothing's going to hang on whether these people really think this. They're certainly buttoned Harvey's against Shapiro by fashioning the existence of primitive identity facts. However, Stuart Shapiro, Jeff Ketland, and Oliver Pooley seem to be the only people who have explicitly raised the prospect that maybe we should just reject the need to supply a general criterion of identity for places and structures altogether and have a notion of primitive identity. So, how does light game, is that right on them? Yeah. How does light game end? They don't seem to talk to each other in a lot. They're certainly going, do you think that really? That's why I'm saying that. So, how does light consider cases from graph theory that violate even weak DIR? Whether or not there are corresponding cases in physics is an open question. So, the upshot of the examples I'll discuss, the upshot is as follows. One, the identity of the difference in places and structures is not to be accounted for by anything other than the structure itself. Two, mathematical practice provides evidence as to how mathematicians themselves conceive the places and structures. we include that identity and difference positions and structure are relations as if we ought to be viewed as integral components of the structure in the same way as, for example, the success of relations and integral components of the structure of natural numbers. That's what we claim. So, now I've got to the part of the story where we're back to primitive identity. So, first, I'm going to take too long. I'm going to give out another, hopefully it might be much more than another ten minutes. You're going to give your graph theory Yeah, I'm going to give my right here as well. I let John guess while I was going to talk about it.

37:30 It would be more interesting, don't I? John always says the same thing anyway. That's really unfair. That's okay, I'll be unfair if you were. Is that thing about who you laughs last? Okay, so what I'm going to do now is get to the reason why we might want to reintroduce primitive identity and then say something about why that kind of primitive identity isn't the kind of primitive identity that I was running away from in the first place. So, graphs Graphs are very I mean, simple graphs are about the simpler mathematical structures that you have to deal with and I'm only going to be talking about really simple examples so that's handy They've got two kinds of entities They have nodes and they have edges So here's a node, here's another node here's an edge could be directed or they could not be the graphs themselves could be labelled or unlabelled in an unlabelled graph different nodes are indistinguishable if considered in isolation that's why structuralists ought to be interested in them labelled graphs or unlabelled graphs linguistical, numerical labels to their nodes, okay. Now, what's going to be important is the notion of a symmetric or an asymmetric graph. A symmetric graph is a graph that admits of a non-trivial automorphism, and an asymmetric one is one that doesn't. So, relative discernibility would be exemplified by a directed graph with two nodes. So, these two objects would be relatively discernible. Weak discernibility is exemplified by, forget about that, just through this.

40:00 Weak discernibility is exemplified by the following one day, I've got the version where the pictures don't come out right. So these are supposed to actually join the notes. It's not a smiling frog, it's a duck. So this is the graph-theoretic counterpart of Black's two-spheres universe, or two fermions of the secret state, or I and mine's I. Now, obviously, if it's a non-driven automorphism, the two nodes are distinct. This can be concluded from the fact that they stand in a reflexive relation. So that's all fine, but we can apply a standard graph theoretic operation to that graph, which is the operation of removing an edge. And if we do that, we get the following graph. Now, that graph is not a terribly interesting mathematical structure. On the other hand, it's a perfectly legitimate one from the perspective of graph theory, and now we've got the problem that it violates WPII. There's no relations. A4 to your right, there are no irreflexive relations. And so, we can't look to relations and satisfaction of WPII to ground the identity and individuality or the facts about the identity and diversity of the objects in the graph. Okay. I think I'm going to sort of speed up because I've been quite a long time. So I have loads, we have a lot to say about this, but the thought is that if you take seriously these kind of examples, then you're not going to be able to rely on weak PII in the context of mathematics. Now that might lead you to reintroduce the notion of primitive identity and just say, look, well, in a sense, it's relations that account for identity and diversity, but the relations in question might be nothing other than the relation of identity. There aren't any other relations. Is there something you want me PowerPoint? No. Okay, so, at one time, okay, so that's, yeah, okay. Now, at one time, I

42:30 thought that the question of whether PII was satisfied was the same as the question as to whether the facts about the identity and diversity of individuals in a structure were accounted for by the relations among them, the relations other than the identity relations. But, this is wrong. And an example from graph theory makes this clear. The following graph I'm going to give you in a moment is is undirected. I'm sorry, the following graph I'm going to give you in a moment is asymmetric. Each node satisfies the structure description that no node does. Let the node be described by a list of numbers, one for each node it's related to, and where the number is assigned to a node is the number of nodes to which it's related. I'll certainly come off this when I put the graph up. So, the first graph, the The first asymmetric, unlabeled, undirected graph, other than the one-node graph, has the first ones have order 6. And here's the graph I'm thinking of. The bucket is over there. I'm going to use my flat. One, two, three, four, five, six. I'm going to use the package online. I'm going to use my flat. I'm going to use the bucket. and I'll just go through it okay, so this one is related to this one has the description three attached to it because it's related to one node and that node is itself oh that's right, sorry that node is itself related to three nodes maybe that node is related to one, two, three then this one one node that's related to only one other, and it's related to one that's related to

45:00 three others, namely this one, and it's related to one that's related to... Ah, that's fine, that's fine. James, you're standing in front of it. Yeah. It would be really annoying if I can't remember what. You don't have to go through the labels, you just have to observe that the graph is, in fact, does have no automorphism, because each node has got a difference. Yeah, that's right. That's all you need. Yeah, that's right, because this one is related to 1, 2, like that one. But this one here that it's related to is related to 4, not to 3. So that makes this and this different. So the point is that each node in the graph is absolutely discernible from each of the others. Yet clearly in this example, it's facts about the relations between individual nodes that account for the facts about the identity diversity of the individual nodes insofar as relations account for those facts. So we can use a monadic predicate to pick out each point uniquely. So the question of whether things satisfy or violate PII is not the same as the question as to whether or not it's possible to regard their identity as deriving from the relations that they enter into with other things. That's what this example shows. So, um, I'll finish by just, um, let me go for an hour or so, I'll just stop. Well, we started by taking it to a... Did we? Okay, good. So, um, now, so...

47:30 Thank you.