Structuralism & dependence / Unknown speaker: response

0:00 I thought I was considered and supported, but your comments to Stuart makes me doubt that again, so what I had in mind, what you did at the end was, like I said, there's an ambiguity, or there's two types of what we mean by identity fact, right, so all those facts about whether things are identical or distinct, where you've got one or two things, so you get a pair of objects with it, you know, what are the facts about identity here, and then the other, what I thought you were doing was distinguishing those sorts of identity facts, which you might think about the facts about how many things there are, No, that wasn't the distinction I had in mind. I just had in mind the distinction between, I mean, really, it's a bit confusing when you talk about contextual or primitive, I'm sorry, intrinsic individuality, contextual individuality, and then primitive identity, contextual identity. I mean, there's two things going on. I mean, one is just thinking about a thing and it's what it is for it to be that thing. And then there's sort of another way of asking what is often the same question is, in virtue of what is it true that these things are diverse, are distinct. Right, so if you've got an answer to the first of those, what makes me do you, that would explain why we're different. Right. But you could answer to the second one, so we just don't understand things, right? There has to be messages of stuff that you don't like. Yeah, that's right. so the route was to go for the facts about identity and diversity of objects in a structure can be primitive but that doesn't confer primitive intrinsic individuality on the things such that when you take them out of the structure it becomes meaningful to ask whether they're the same or different without so if you take if you have a two node graph and you take away one node and now form the one node graph that you get by taking away one and then start with the same graph you started with and take away somehow the other one there's no fact to the matter those graphs are identical what you get is identical

2:30 I think we all take the one thing we want to get from this weekend is how mathematical structuralism and structural realism might learn from each other but someone might say that the kind of concerns that Stuart has with securing the identity of numbers for example, they're not necessarily the same kind of concerns that you have with dealing with, you know, fermions or whatever. And so, therefore, it's not entirely clear what the graph theory example is doing. It makes a certain point, but someone might say, look, I just don't feel the need to secure the identity of, you know, fermions in the single state. Simon has shown how we can do it if we want to but some of us think we don't want to or we only don't need to so it's not quite clear what the even the ungrounded contextual notion is giving us even with the way you know, Kevin's helped articulate it in that sense in the, you can say, in the quantum and that kind of relates it kind of relates to this sort of the spat between a friend of mathematicians here, which is, yes, of course, if you're going to, you know, you should always define a mathematical concept, and if you define them set theoretically, then you're committed to set theory in some sense, and you've got a notion of identity, right? So what? How does that apply to anti-structural realism? It only applies if you think the structures that you're saying are out there in the world are set theoretic, but some of would say, no, they're not. We're using set theory to represent them, right? Now, John might say, yes, but that set theory is committed to a notion of identity, so therefore you are. Well, then, there's various maneuvers you can make. One is to move away from Ian Paisley and go to some other Northern Irish politician, stop being a fundamentalist and go non-standard. There are non-standard forms of set theory which are sorry for identity, which may do not what it works. The other is to say, look, I'm just using this as a representative device. It has no ontological So the issue, the fundamental issue is exactly what graph theory and that kind of debate is actually showing us when it comes to ontic structural reasons. Yeah. Well, I don't know the answer to that question in full. The reason why I think it's relevant is because I think it's interesting if it's established that in the mathematical case, you need a notion of primitive structural identity.

5:00 because, I mean, Catherine's absolutely right I mean, I was taking a sideswipe of metaphysics but Catherine's right that this is about what I was doing was just establishing what was kind of intelligible and was very much at the abstract level so, now, then it's an open question whether that coherent notion finds application in physics as well as in mathematics now, however, right people like Oliver have effectively, I think, applied that very same notion in the context of physics to space-time points and said within the structure of the metric then these points have their identity but take away the metric, they don't it makes no sense to talk about their identity so it makes no sense to talk about them being the same points simply because there are people who are wanting to use this kind of primitive notion of identity, and I mean it just struck me as a striking analogy because Oliver's very clear that he's not going the route of the old-fashioned substantivalist, what he calls the hexatistic substantivalist, so there's this crucial difference between his notion of primitive identity and theirs and it seems to me it's exactly the same difference between the notion of primitive identity that Hannes and I deployed identity which has been used in the context of well, which could be used in the context of philosophy of maths to say, you know, there's a fact about whether 3 is the same as Julius Caesar but we don't know which, or whatever so I still think it's an open question, I did have some stuff at the end of my paper about the relationship between the physical and mathematical cases but I didn't really get onto it, I mean there's a philosopher called Diepert who uses graph theory to talk about the world he argues that We have to have, if the world is represented by a graph, it's got to be an asymmetric one. So he seems to think in an empirical domain, you cannot have a world that admits of a non-trivial automorphism. Carnap, Hannah has pointed out, in our paper, Carnap thought something similar.

7:30 He gave a similar example at the end of the Aufbau and thought that every object would have to have a unique structure description. me that once we've established that there's an option available that's intelligible that now becomes an open question so you could say well, go the weaker notion of individuality that we go for and you don't have to adopt Simon's conclusion that bosons aren't individuals partly this is just terminological, right, I mean it seems to me when I was reading Oliver's paper where he carefully goes through what you and I said paper we wrote a few years ago, and I was feeling really embarrassed reading it, because it seems to be so confused in light of how much more advanced my thinking is now, ha ha. And yet, I sort of look, but then I read these metaphysicians, and they all lump individuals with intrinsically self-individuating things, right? So, insofar as you make that assimilation, all the stuff we say about there not being any individuals is intelligible, and you can reconstruct why we said it. that there's a notion of individual that you can generate for yourself that's much thinner. And Simon's got a pretty thin one, and then there's a possibility of an even thinner one. Now, who cares whether you'd end up saying that bosons or fermions are individuals or not? I mean, we're all much clearer, I think, about what the logic of the situation is and the structural relations in one theory as opposed to another. to call the things individuals or not, I don't think is so important. All right, we've got two more people on the list that running out of time, but let's do those two questions and then just... Okay, I'll be brief. James, I think that what you presented is a really lovely articulation of things that you were saying earlier and maybe things that Stephen was saying earlier, so that's really nice, I think. So the idea that there are relations are in LADA, and the relations are primary, and in LADA are secondary, so the relations don't exist anymore, but they can access relations. So here's where I take it pushes some people towards, one, to conceive of identity in terms of something intrinsic. A thing, a subatomic particle, can stand in, there are lots of different nexuses that we can talk about. There are lots

10:00 of different structures, right? They interact gravitationally, they interact electromagnetically, and all sorts of things, right? Now the question is, how do we know, in virtual what, do we identify the nodes, the relevant nodes, in all of these different structures with one another? If there's something intrinsic, so for example, I say that it's a bundle of dispositions to stand in these various structures, you have a means of identifying the nodes in these different structures with one another in the way that physicists do. They talk about electrons in all of these different cases. So the challenge here is, it's something It's something analogous to the problem of operationalism in the logical positive space, right? You say that electrons are widespread in a cloud chamber here, but in a different experiment, you say there's something else. How do you make sense of the way that businesses talk about these things as the same thing, since they have different semantics in different cases? Here I'm presenting an analogous challenge to you, which is to say that there are lots of different structures that these things can. How do we know to identify the relevant nodes in those structures from one another, unless you feel it's something for change? I don't quite see why it would have to be something intrinsic, I mean if the things really stand in lots of different relations then you can identify them in virtue of them standing in those you know, standing in those different relations, I mean Well only if they're doing so standing in all the relations of which they're capable simultaneously and that doesn't happen So, when we talk about the cases in quantum mechanics, we are abstracting, I mean, most, probably it wouldn't be the case that most particles are going to be absolutely discernible in reality, because they're going to stand in a whole bunch of contingent relations to loads of other things, and it's quite likely that you're not going to have the pure case of the singlet state. so I think this is a problem in principle the thought is given that it's possible for two electrons to stand in all the same relations as each other how can we individuate them and that's where the weak PII story comes in so I don't I'm not sure the problem arises in that probably in practice whenever you're talking about actual particles, they're, they're absolutely discernible, but, um, maybe I'm missing the

12:30 point of your question, right? Let's, let's, at what point did the final question are, and I've just got a very brief question, it's to you, John, and from what I understood, you, you weren't undermining the significance of the example that James was presented. at the same time, you're insisting what some notion of primitive identity must be consumed in mathematics. Yeah, in mathematics. That's critical. That's crucial. I just wonder, with this notion of identity that's been consumed in mathematics, would you admit any constraint or restriction of the sort that James passed mainly that if it is are grounded in anything? Is it contextual? Do you agree with that, or would you deny that any such? I think what's happened here is kind of interesting. The point is, the whole structure as opposed to mathematics is one of the aims of it is to get rid of metaphysical questions in mathematics. in a sense, when I say we just you want to be able to talk about sets of things that aren't sets, but in those cases like maybe bosons or whatever. In those cases, you think you're given these things from the outside, and the guy that gives them to you tells you that these are bona fide distinguishable things. So these questions about identity that you guys are talking about are much, that involve physics and metaphysics and so on, they're much deeper than the identity than you can ask in mathematics, if you like. so maybe what this tells you is that mathematical examples are not going to be helpful because they assume the very things you're worried about maybe we should wrap up there again and we have scheduled a 20 minute coffee break so let's then stick with that and we can meet at a quarter to a three Lots of the issues that I went through quickly are going to be brought up by being in much more depth and in particular Hoystein-Linbo, who's from Bristol is going to talk in much more depth about what I refer to quickly

15:00 as the relative ontological priority of structure and objects and so on so he's talking about structuralism and the notion of dependence and then categorical I'll have one of mine. It's a bit chewed, I'm afraid, but have it. All right, yeah, so I'm sort of like what James said, that there has been a lot of talk about dependency relations in connection with structuralism and recent research. A lot of people making strong claims, claims for the effect that always the case that mathematical and optics depend on the structure of the source. Now are the case that mathematical and optics depend on the structure of the source. So what I'll do here is to sketch some of these claims that have been made about dependency in connection to structuralism and argue that this is actually a good way to address questions concerning structuralism. So get away from some of these earlier points of characterizing structuralism and it's actually quite fruitful to talk about dependency. Well, myself, I want to occupy some sort of compromise view. So I'll say that there are counterexamples to both of these extreme positions. So the structuralist, like, represented by stupidity, I think, saying that all mathematical objects depend on the structures to which they belong, and then other people, which should be represented by Fraser McBride, but sadly not, and saying the diametrical opposites. So that is never the case, actually, that sort of thing. So I'll try to suggest that there are actually counter-examples to both of these extreme claims. And then finally, if there is time to try to say something about what this problematic, crucial notion of dependency is, that is in YouTube, I thought, yeah.

17:30 So first of all, what's the sort of structuralism talking about, well I'm talking about non-elimitary mathematical structurals. So let me just say quickly what I mean by that. This is going to be really ground to most of you, and I'll go over it quickly. So this is mathematical structurals. Structural is about mathematics. I'm not going to have anything to say about structuralism, but physics, maybe some of what I say will be relevant to middle-spot. Everything I say is officially just about mathematics. And mathematical structuralism is an investigation of abstract structures, however exactly you spell that out. So this is a local form of structuralism, about mathematics specifically, in contrast to global forms of structuralism. quite on the interview of Radford since it's one sort of global structuralism and polysemantic and there may be others of a more metaphysical kind and obviously it also contrasts with structuralism about other local areas like structuralism about business. So that's the mathematical bit. What about the non-eliminative bit? Non-eliminative here is just my maximally mutual word for something that a lot of people have their own favorite labels for. So non-eliminative here is what Stewart refers to as anti-rem, what Michael Brestig refers to as pattern structuralism, I forget what there are other labels around it. So here the contrast is just with eliminative structuralism which denies that there are abstract mathematical structures in the relevant sense and that there are in particular mathematical objects whose entire nature is exhausted by being positioned in these abstract structures. So this eliminative view can be developed in a bunch of different ways, a deductivist And there's modal structuralism of Hellman's sort that would belong in that canon. And semioretic structuralism, which is pretty much what John was talking about, I think with some slum cabbage, perhaps, those would try to eliminate purely, purely

20:00 structurally mathematical objects. So when I'm talking about non-eliminative, I have in mind the view that there actually are mathematical objects, which is fundamentally all positions in mathematical structures or mathematical patterns. And henceforth, when I say structuralism, I just mean non-limited mathematical structuralism. Right? Okay, so what distinguishes structuralism, let's find, from Platonism? Since we're now saying that there are mathematical objects, it's true to say so. So this is a brand of Platonism, it sort of qualifies as Platonism by the standard definitions of that term. So what distinguishes it from traditional Platonism is that this is a new philosophy of mathematics, a new addition to the traditional isms about mathematics. And if you read the literature on this, and go through it carefully, and look at what various structuralists say about what distinguishes the review from traditional Platonism, pretty much all of what people say is summed up into claims, so what they will call the incompleteness claim, and a dependent claim, and sort of we go through those for a turn. So according to the incompletence claim, logical objects are incomplete in the sense that they have no interim majors, no non-structural properties. So there are questions you may ask about them, about which they are simply wrong answers. So the incompleteness is a little bit like incompleteness of fictional objects, which is a curious idea. Certainly Parsons and Resnick make, I believe Stuart makes it, but it doesn't matter. But at least that's the incompleteness idea. And certainly what everyone does in this game is for Resnick. So let me do that as well. So this Resnick quote that people are very popular, and that's pretty much everyone to determine this, the following passage from Michael Rezman.

22:30 So in mathematics, Rezman claims, we do not have objects within the internal composition arranged in structures. We only have structures. The objects of mathematics are structuralist coordinates or positions in structures. And as positions in structures, they have no identity and no features outside of the structure. Not a claim, the organization of the entity displayed. Then the second claim which is supposed to distinguish structuralism from additional paganism is the dependence claim. And here the idea is that mathematical objects from one structure depend on one another and on the structure. And this plays a prominent role in the steward's characterization of structuralism, perhaps more so than in Corses and Resnick. And a nice passage here is the following. So number two is no more and no less than the second position in the map of number structures. Number six is the sixth position. Neither of them has any dependence on the structure and with their positions. And as positions in this structure, neither number is independent of the other. So numbers depend upon one another and upon the structure that they follow up to. So, given my title and everything, I'm going to be focusing on the independent claim. But let me say a little bit about the completeness claim anyway. What I'm going to say here isn't supposed to be conclusive or anything. I will just say enough about the completeness claim to make it clear that this claim is actually quite problematic. And it would be a good idea for that reason to hold on to the independence claim. As far as the organization on what is roughly the next time. So that's a good reason to care about the independence claim. I'm not just embarking on this as an exercise of all sort of analyzing a couple of thorough remarks. These are important remarks, interesting remarks, made by a lot of people. So, the incompleteness claim, then, if you look at that as well, and in my paper, I've got a bunch of quotes here that I'm not going to input from you, but if you look at all of these instances of people committing themselves to retrieving the incompleteness claim, you

25:00 will see that there are two strains of that claim. So there's one that says that mathematical objects have no non-structural properties. I'll call that N-S in a little bit. And then there's one that says that mathematical objects have no interruptive proposition, interesting strings, and intrinsic properties that people also talk about. So I'll talk about that as I think of things. And for example what the relation is between these two may be a little bit unclear, and I'm not going to really settle that. So there's a question, is every non-structural property also a privacy property in my sphere or so? But regardless, what we do know is that these two notions are intentionally different, sort of, completely different, or separate. So what do people mean when they say that non-structural properties? What is essentially non-structural property? I think the best analysis of this comes from a student here's word, so following him here. So let's talk about a system of objects. So this will be some sort of loose-in-form, a collection of objects. So a system is a collection of objects with certain relations for the two students, so there's a domain team, and relations on this journey. And from the fact that there are uppercase variables here, i.e. a.k.a. if you are supposed to be the basis of the variables. And then the structure is then characterized as the abstract form of the system, highlighting the interrelationships among the objects and ignoring any features of them that do not affect how they relate to other objects in the system. Now this is a little bit problematic perhaps because it's in epistemological language but I think that's not quite the intention. So this is just some sort of abstraction from the particular features of each system. So that you arrive at the structure which is commented and shared by all of these systems. So some people refer to this as Dedicant abstraction, which I think is sort of a good phrase, actually. Since this is what Dedicant wanted to do for that, to write that. Disconception of the . And then finally, one can say that a structural property is just a property that can be like that.

27:30 So that gives you a pretty good understanding of what is structure A and property X. And then the NS in completeness claim would be that the only properties of mathematical objects would be structural properties in that set. So why do I claim that that is problematic as I did just a couple of minutes ago? I mean, firstly, I don't think it does a whole lot to distinguish mathematical objects from other kinds of objects. I'm just saying that there's a range of objects which have zero kinds of objects. There's another range of objects which have zero kinds of objects. And then there are no objects of one kind, halfi-optics of another kind, as versa. No identities falling between objects from these two different kinds. But all of that is completely symmetric. So look at this question again. Is it identical to the natural number 3 or not? If there is no answer to that, why should that be buoyed entirely on the natural number 3? So far the situation is symmetric. It's not clear why mathematical objects would be more incomplete than physical objects. But perhaps the more worrisome problem about the NS incompleteness is that it just simply doesn't seem to be perfect. There seem to be all kinds of properties had by... What's the matter? Do you think you want to solve it? Is a structure a system of the same kind of which it is a structure? I'll deal with that one. I'll make a note. I don't know how to deal with it. You're going to have to remind me about that. But the more serious word then is that there just seem to be all kinds of counter examples to this claim that mathematical algorithms are incomplete in that sense. A number, for instance, could be my favorite number. It could be a number of the people in the audience here. And perhaps more interestingly, they could have other properties, like abstractness,

30:00 a big national number. But abstractness is a good case. So it seems that it really is true to say that actual numbers aren't abstract. And that's quite an important claim about them. understood that, there's something quickly, basically, that you might have got the boundary technique. But that slackness would not be structural property. So as I said, I'm not going to try to clinch that case, but this is enough to be a bit worried and looked to the other characterization of structuralism. If you read a recent work of structuralism, again, So perhaps, there is a pretty good backing off from many of these earlier commitments to incompletes and in particular, then this strain of it, which I call NSNL. It's really interesting to see how far to what extent people do want to back off on NSNL. The other strain just quickly. So would it be better to say that yes, mathematical objects were incomplete, but the incompleteness consisted of having no intrinsic properties. Would that provide a good contrast, a good distinction between mathematical objects according to the structuralness and mathematical objects according to the traditional opinions? Well, there are two ways of developing the notion of intrinsicness. So, one review that intrinsic means something like shearing by all. In an earlier talk, I may be somewhat And I found you a lot of progress as a result of that. So I'm not going to say that, but this is unpromising. It's my modified plan. And what is a duplicate of a mathematical object? I have no idea what that would be. Besides, if you want there to be no properties that are shared by old people, it's not mathematical objects. What about abstractness? You have to get away from that. So that means that there would have to be duplicates of mathematical objects that my understanding of what you do to this is supposed to be just a problem. So this is actually basic. The other way of spelling out intrinsic would be to say something like intrinsic means unchanged even when the rest of the universe is disregarded or removed. The public still remains like still happy.

32:30 I think this pretty much just collapses to the dependence claim, because if the dependence claim didn't hold, then it should make sense to consider the mathematical object in isolation from the rest of the objects, and then it would have all of its properties in relation to the incompleteness rate. So the only way to get the desired result here would be to say that, look, it doesn't make sense to consider the mathematical object in isolation. We can only describe the properties to the mathematical object when it's considered as part of the structure that we want. But then we're backing in something like this. All right. So let me then go on and show how people have. So far, I've just said that the dependency claim is really important. I care about it. What the baculomat structuralism perhaps ought to be focused on. Now let me characterize quickly these two what I refer to as extreme use, which say that all mathematical objects depend on the structures in the particular way or not. So beginning with the former, there are two claims here, objects dependent upon objects, ODO on the handout. So each object depends on every other such object. Each mathematical object depends on the structure, so objects depend on the structure. And both claims are then endorsed in this passage from Stuart that I read really well. So the structure is prior to the mathematical object, which contains just as any covenization. Now this is actually a new one. Alright, so the structure is prior to the mathematical object which contains just as any organization is prior to the objects that constituted. So here in the talk it's about priority, but it's pretty close to dependency. And if you flip back to page 2 and the quote I gave there, then you find a very clear endorsement of both these claims.

35:00 objects depend on objects and objects depend on other structures. How do you find this in the branch of people? So there's a quote from Resnick here, mathematical logics have their identities determined by their relationships to other positions in the structures to which they belong. James, for that matter, made a lot of claims that sounded very much like this in his talk. So these are some interesting claims. And taking at this value, at least, they are speaking things about absolutely all non-negotiables, not just standing in these defensive relations. So I'll refer to that as an upwards dependency. It's going to be somewhat a little smart. But the idea is going to be that if you look at ordinary physical objects, collections of physical objects, there, the collection depends upon each of its constituents. downwards dependency which is maybe not a universal rule when it comes to the physical domain but it is often seen to be and the other thing is meant to show that or say that in mathematics the division is the complete opposite of it So you always have some sort of upwards dependency. So the object in any mathematical structure depends on this upwards or on the structure of the type of object. What about the other extreme view? Here there are two people I have in mind, Jeffrey Hellman and Preece McBride. But perhaps before going to the interview, let me just give a slightly simpler version of the related view. So here's the thought. What is it for one object to depend upon another? It's for the second object to, in some sense, be prior to the first object. Now, one object is going to be prior to, you can have kind of a circle of relations on priorities. Some of these priorities are going to be absolutely prior. And perhaps even stronger, you shouldn't have this priority relationship to be well-bounded. So you can try to motivate that sort of view.

37:30 And this is a sort of reasoning that the opposite can't engage in. You could say WF, on the other hand, that the dependency relation must be well-bounded. And that's certainly a border against the plane that obviously depends on the object. But even if this were right, and I'm in no way imposing it, I don't think it would be particularly problematic to the structuralist, since they'd still have the other plane, that object depends on the structure, maybe that is the more important of these factors. And that would ensure that object depends on the structure, would ensure pretty much what the structure is here, so that you can't have one object from the mathematical structure and not other ones. If you get one, you get all, that would follow. So we've got one structure, and the structure consists of all the positions. And it would do the job of distinguishing the structure of the traditional structure. So I don't think that would be very bad. But let me then state Fraser and Jeffrey Goldman's view. So I think that I guess there's some time I'm going to go over this pretty quickly and not just the case that isn't here. I'll go over this reasonably quickly. So here, what's going on seems to be that they take the well-founded this claim that the dependency relation has to be well-founded, or amotivational, which I just came through, and combine that with the view that there's going to be some sort of universal kind of downwards in that institution. It's always the case that you have sort of anything that you have often in a physical way. And then you would get the result that even claim that structures would be . And that is indeed the view of what people So I think I'm just going to leave it at that, actually, and leave it on. We could come back to that in case of discussion.

40:00 All right, so dependency matters. And there are these two extremities through the effect that always depends in the mathematical realm of this upward direction, or that there is never, ever that kind of opportunity. So I'd like to have a compromise view according to which both of these experiences go before. So in some cases, there is a little bit of a challenge, even in mathematics I think that is the case. This is actually a new view, if you, I think something like this is already present in Kahn, so Kahn talks about two kinds of totalities, so there's a totem synthetica, synthesized totality, where he's talking about what the examples would be, a collection of people, for instance. So that is a totality that is made up of prior constituents and the totality which would then depend upon these constituents. And then the other kind of totality that Kant is talking about is a totium analytical, where his example is, not surprisingly, space. space, the entire space is actually a prior to the subspaces that you find within space. You've got the entire thing and then you can carve out subspaces, planes, space. So I'm not going to base my point on concept, but on some slightly simple aspects. So the counter-example, I think, works against the structuralist claim that all mathematics can explain the opposite tendency and the process. So much like a concept example actually. So think here about the iterative conception of sets. So first you've got the element, then out of the element you form your set. The elements have to be present, available in some sense, prior to the information on the set.

42:30 So it very much seems here that you were saying that the set depends upon its elements. And there would be no similar element of, similar dependence of the set upon the example, since the elements are afterwards supposed to be given prior to the set. So this is giving a certain metaphysical, or taking in a certain metaphysical seriousness, these metaphors that are finding the individual conception of the set. But it seems to be part of what's going on. And that seems to be a very good thing, in a way, since if you could integrate sets in that one-way manner, you'd have serious difficulties characterizing even very, very simple sets. So take the empty set, or take my singleton, or something like that. But it seems like we know perfectly well what these sets are. Although we don't really have a particularly good understanding of the entire structure of what we have set. So why is that? Well, I think they're iterated from the bottom of that. They depend just on, in the first case, nothing. And in the second case, maybe. I think it's easy. Right. So that's my claim. Okay, what could one do, what sort of responses are possible to this kind of example? One thing we could do is to say that the notion of dependency hasn't been articulated. It should be problematic, be skeptical of this, so we can't do that. I'll address that point. Another thing one could do would be to say that let's see how there's some other notion of dependency on which these claims do not follow. But if the interesting conceptual concept is strong enough in the picture, we get from it has enough plausibility to any analysis of dependency that doesn't give any true verdict on that. It's somewhat unattractive analysis of dependency. But that's only, that is not the challenge of people to come up with. Sorry, what's the counterexample two? It's supposed to become, for example, to these two structuralist claims that in every case a mathematical object depends on all the other objects from that structure.

45:00 So this set, for instance, is the case of and accessible particle, et cetera. Another response would be to switch to the structure of this concept. I think this could actually be done. But after one account, for example, so if I could repeat my example to fly, no damage would be done by some other exception, Structuralist conception, you might want to be able to do that. Federalist response has a structuralist we need to direct the challenge of my and I don't want to be able to use it to a personal sense of the sort that I'm revealing myself. Now, we'll see maybe some of you want to make that response. I don't mind having a practical myself. itself. It has been attempted, by the way, I think, in a paper by Charles Morrison to the Czech Center. Other responses, you could say that structuralists shouldn't have all be counterpiced in terms of the independence of the way, but that's given up. All of it could be sort of the compromise side. What about then the other extreme represented by Brace and Bride and every element that you never have this sort of structuralist style of dependency, never ever. Just look at examples. There are lots of them in the literature, some of them we talked about in the previous lecture, but here's mine, two element proof, two elements alpha and beta. Everything, every fact about that structure is given the multiplication table at the handout. nothing more about it. I think it seems that there's nothing more to alpha than its playing a role in its new structure. So alpha is what it is, and beta is what it is in virtue of studying these structural relations. That's all there is. So they do depend upon the model, it depends on the structure, take away the structure and what that is. So how could people like Fraser and every company respond to that?

47:30 Again, they could say that giving analysis is dependency, I'll get to that. Again, they could provide their own alternative analysis. That has to be done, but that is an open challenge. Perhaps it's more promising if you say that we don't believe in objects like Alpen data. I suspect that is what they would say. And here I just refer to everyone else, for example, Norway, Stoensburg, or whatever, to the fact that these are bona fide as objects. I have nothing to add to that discussion. Or one could, finally, go for this compromised theme that some apply problems offer its dependencies, some offer its dependencies. Alright, how much, when did I still turn it off? That's why I won't understand it. So let me say a little bit about how I think this crucial notion of dependency can be understood. I think, by the way, that a lot of what I've said so far could probably stand relatively independently of what I could not want to say, but it would be supported by what I'm going to want to say. But there are alternatives to, at any rate, so the standard analysis of dependency is the obvious modal one, so it depends on why it's possible for x to exist without y existing as well. That's what all works in mathematics and every method. Mathematical objects seem to exist necessarily, So one is definition of method number two, which depend on the empty set, and vice versa, and it just wouldn't do that. So we need to get away from that. And then there were a couple of proposals, non-modal notions, or non-modal analyses of dependency in the literature. Keith Fein has his, John Furlough has his, Newsy Martin, which I think is a better policy I'm going to be talking about. So this is dependency for identity.

50:00 And then when talking about identity, you'll have to say something about individeration. People talk about individeration in both the semantic sense physical sense here. So how the identity of an object is grounded. Then the grounding of identity would be done by means of identity. These are two different kinds, so there's the one-level kind, two-level kind. I'm going to be talking about the two-level kind, which you can find on the handout here, which is very, very nicely exemplified by directions. So how are directions individuated? Well, they're specified by means of line to this direction. And then you have a function that you can be a specified line to the direction that you've been specified. such that only two directions specified by two lines are identical, even if two lines are parallel. So the two-level criteria of identity is what I'll be using for instance. I think that is more informative and that we've got enough of these two-level criteria to do the job. Then, I think sets can be iterated in a two-level way, just take a bunch of things represented by a plural variable, maybe a set, or maybe a form of a set, perhaps, restrict or something, what's that do? But some other things, when do these things focus a set as those things? Well, every object, every one of the first things is one of the second things, and then finally I can get to my definition. So say that one object is, an object y is strongly involved in the integration of non-object x. if y is actually a constituent of the individuation of x. So it may be this specification that you use to specify what x is, whether or not it's relation that you have, so then these specifications do.

52:30 And say that y is weakly involved, rather if it's possibly strongly involved, or at least whatever resources you're drawing on to individuate x And then accordingly you get two meanings of dependencies. So that strongly depends on Y. If every individuation of X evolves, strongly evolves, Y. So, think about the individuation of a set, for instance. To specify that set, you always need to involve the other. So, that would be a strong dependency of the set upon the element. Weak dependency defined in terms of every individuation of the one object weakly involved in the set of objects. And then, on this analysis, you get the source of dependence that we're talking about in my two proportionate counterexamples. So you get then that a set does, in fact, strongly depend on its outland. And the outlands do not interdependent by the set, not only uniquely. So this would be the kind of example to the extreme structuralist claims that upwards dependency brings throughout mathematics. And then I'm not going to go through that business on the brain structures, but go back to the group example. then you do get that every way of specifying what alpha is. We'll have to go via the entire structure. So alpha does then really depend, at least, on the structure to which belongs. So you get what they want in that case. And that's also that the integration of alpha gives you So we have that sideways dependency as well, but obviously we have one and one and the same structure. So I think that all this analysis of dependency you do get .

55:00 So I hope it comes with you. Thank you. But let's hope it's neither. I guess most of the time was doing two things in the lead of the paper. So one was trying to give a characterization of structuralism, non-limitative structuralism. So he made an argument that if that's in terms of dependence, and then what he was going through at the end there, there's a more detailed explication of what is what dependence amounts to. That's one thing he's done. And the other thing he's done is taking that characterization. He's argued that certain kind of half-mechanical objects you, it says, whilst everyone's doing. I'm not really going to comment on the second part of the paper, so I'll leave that to you guys. I'll just say a few things about the first part of the paper, where you see counterizing structuralism. And even here, it's not that I want to deal with criticism, it's more I'm kind of taking this page just to raise some questions that to me seem interesting about structuralism. So, as I said, the question is, how does or whether the challenge is to distinguish structuralism and platonism about mathematics? And that's exactly what you do, like modern new views, you want to say how it's different from the old view. But then it seems to me, well, haven't we got a characterisation already? The structuralists say that mathematical objects are the positions in the structures. And I thought, well, don't the Platonist say that a matter of an object satisfies a structure, or instantiates a structure, or something, nor bargain actually being a position in a structure. And so what I want to do is just explore a little bit of this distinction between being a place in a structure and occupying a place in a structure. I guess one reason, most I might say, is why we need to go beyond that distinction between being a position in a structure and occupying a position, is that that's hard to kind of directly test, right?

57:30 So, one of actually just spelling out the notion of dependence and so on is that we can actually apply it to different kinds of mathematical objects to see how it works out. But definitely, that's another reason that structures might not all hang too much on the idea of something being a place in a structure as opposed to occupying a place. Is that a place in a structure is a kind of obscure, for me, seems like an obscure kind of thing. It sounds like a kind of structure can be reified. So I think I can understand what structure is. abstract form or universal that can be instantiated by different collections, different normalities. But what's a place in a structure? Something like a gap or a hole is what would make the structure unsaturated if it wasn't filled. I can sort of understand what that is, I can understand what it's supposed to be, you know, a property or a concept of something to be unsaturated. But I'm not quite so sure what I'm going to talk about if I try and talk about the gaps, talk about positions. so that's really just a kind of question for a sort of clarification what are these positions supposed to be and how do they relate to the structures themselves and one thing that occurs to me is maybe, well, if you think of the structure as the property of the whole collection or the plurality maybe the position in the structure can be something like a property that a thing would have to have in order to be one of the things that is amongst the collection that is the whole structure But Einstein has to be the objection that he won't try and think of places and structures as kind of constituent property or sort of parts of structures. We're going to get these identically discernible problems. So that's, I think we've got that clear, but if so, that seems really not to get that clear. So one question is, what are these places? What are positions? Another question, I guess, to raise is how, sort of a, if a mathematical object is a position in a structure, how does it get how does the mathematical get constantiate relations again if you're thinking as the structure as a big universal or formal something like a property you think of the things that slotted into the gaps that would constantiate relations and expand in the relations but now it makes sense of the idea that it's the place in the structure or nothing that fills the place that somehow has the properties I guess this is why, I guess you don't hear structures about physics talking in this way. People think, well, to have a structuralist view of physics, they're not thinking of objects as places and structures.

1:00:00 They're thinking of objects as things whose nature is given by the fact that they fill the places of the structure, something like that. It's not that the objects are supposed to be places in the structure themselves. out. Just briefly, I mean, it seems exactly right, but as we try and spell out the completeness notion, especially intrinsically completeness, we do fall back into things about dependence or essence or whatever. I mean, as you said, I mean, I think the notion of intrinsic, extrinsic have been built up to deal with contingent existence, and once we get into the necessary existence, you end up thinking about something essential in nature or within the nature or something like dependence but that seems exactly right but i wonder whether the structuralists if they did have qualms about identifying mathematical objects with positions of structures they have the kind of forms i'm having anyway um but i wonder whether there's a related notion perhaps close to what the structure is about physics and all about um a mathematical object being something that fills the place of the structure often being replacement structure. But somehow that's all it does, because we have a lot of the idea behind the completeness talk. You've got the kind of thin objects that they sensate, they instantiate structures, but that's the most important thing about them. Okay, I'm going to wind up there so more informed people can join the end of the discussion. I want to have one comment about what Einstein says in section 7, analyzing the notion of repentance. So again, I'm basically with it on the verge of the paper. I mean, if you want to get animals to do this thing's the best way of doing it. It's spelled out dependence, spelled out dependence in terms of identity, one thing depending on one another. I just want to draw to your attention what were the fundamental specification you're doing. It's really crucial to this, the idea that one thing depends upon another, or the question of whether one comes in the terms of one thing is another kind, all in terms of what the fundamental specification is of what the supposed thing is of an object. So if you want to argue with me about whether one type of thing depends on another, you've got to argue about what the fundamental specification of all things is.