Identity & diversity of objects in structure (contd.) / John Mayberry: response / discussion incl. Hannes Leitgeb

0:00 ...old news, because they turn structuralism and traditional Platonism into traditional Platonism, which excites us all primitive business. However, the notion of primitive identity that I've just introduced in this context is not that the traditional defender of primitive business or hexiety. So, I would speak of primitive structural identity or primitive contextual identity for the following reason. Okay, so let me put up my tree diagram, which is sort of where I got to. So, suppose you think, you ask the question, the title of my paper about what accounts for facts about identity and diversity of objects in a structure. The first question you can ask is, are those facts grounded or ungrounded? Okay, so suppose you think that they are grounded, then they could be grounded in hexieties. Or they could be grounded in qualitative stuff, right, properties and relations. If they're grounded in qualitative stuff, then it could be that they satisfy the absolute version of PII, the relative version, or the weak version. Now, for each of these options, is the individuality in question intrinsic or contextual? Well, if it's anxiety, it's intrinsic. If it's weak, it must be contextual. If it's weak, PII. If it's relative, it looks like it's going to be contextual as well. However, you can have satisfaction of absolute discernibility, either through intrinsic individuality, i.e. the thing has an intrinsic nature that's different from the intrinsic nature of every other thing. Or through contextual individuality via satisfaction of absolute discernibility like in the case of the graph that I just had, where it's only in relation to other things.

2:30 There's no intrinsic properties available to make the thing the individual it is. But nonetheless, it still satisfies absolute discernibility. So that could be intrinsic or contextual. So, what about if it's ungrounded? Well, it seems to me that if individuality is ungrounded, there's only two questions left to ask. Only one question left to ask. Is it intrinsic or is it contextual I want to say that the primitive identity that we the primitive notion of primitive identity factor that we endorse on the basis of these examples from graph theory ungrounded contextual identity rather than hexiety and rather than intrinsic identity. Now, why is that? Consider the three-node edgeless unlabeled graph and the operation of removing a node. So here's the three-node edgeless unlabeled graph. Take away any node, the same graph results. Similarly, take away a node, put back in a node, there's no fact of the matter about whether the node that you put back was the same one that you took away. So, although there are primitive identity facts in this structure, they're not facts which get out the structure, they're contextual. They're facts about the identity of the objects that are structure-relative. So for that reason, and this is why, this relates to this idea about, in the physics case, about the, not, about, not, about not having hexitism, not having the view that commuting individuals gives you a distinct possible world. I mean, sorry, that a world that differs only in respect of the permutation of individuals distinct. If that was the case

5:00 then the individuals would have to have intrinsic individuality so I'd like to make, I mean I'm running out of time, I'm going to stop, but I think the claim that I want to end on really and I had loads of stuff about all sorts of other things but I'm running out of time. James, on time you've had exactly 46 minutes. Have I? Yeah. I promise you, I've timed you digitally, you've had 46 minutes and 49 seconds. Right, so what we want is we don't want the primitive identity that we're invoking. We mean We don't want it to give us hexitism. We don't want that to be the case. Why not? Because the original motivation was to avoid ending up declaring that worlds of different only in respect for the fermentation of individuals were distinct. basically because of naturalism because physics doesn't respect the difference between such worlds and it's uncomfortable for philosophers interpreting physics make distinctions that aren't recognised in the physics my claim then is what's the difference if a particle if an object is then what could that mean other than that independently of context it carries this individuality with it and what else what does that imply if not that if you commute the individuals so they're in different situations but the world overall looks the same well if they're intrinsically the individuals that they are then as it were the world remembers their individuality There's a fact of the matter about difference between these two worlds. Now, I just want to say that the examples from graph theory don't motivate that. Why? Because if we start with a two-node graph and take away one of the nodes,

7:30 we get a one-node graph, we get exactly the same graph whichever one we take away. we don't want to allow there to be facts of the matter about the identity of the individual nodes outside of the context of the structure now one other point I was going to make by the way is that this business of how when you take away nodes and put them back you don't have a fact of the matter about which one was taken away which was put back I think it would be helpful to think about about a while ago, about diachronic identity for quantum particles in terms of, you know, I think graph theory is a useful framework to think about lots of these issues. Anyway, so how I want to finish is by saying there is a primitive notion of identity that's available to us if we make that contextual identity rather than intrinsic identity then we're not pushed in hexitism and we avoid the conflict with physics and from the mathematical point of view we can say that we haven't just got old fashioned hexitism by having this kind of primitive identity. What we've got is primitive identity in a structure, not primitive identity of individuals to core. And it was precisely the people who argued against structuralism like the Nazareth, precisely being there must be intrinsic identity. They want it to be the case that if you forget about all the relations and the numbers, that the number three is different from the number five somehow. It has some intrinsic nature. And of course they say because it's not, it's not really an object. So the claim is the notion of contextual contextual identity, primitive contextual identity can be used to give us an even thinner notion of object than we got from Simon and from Quine. It's a notion that doesn't imply ex-scientism, but it's still intelligible. It still allows us to say that there are individuals in the structure. Okay, I'll stop there. Thanks.

10:00 I find myself in a, I'm not sure exactly where I stand, but respectively in very simple company. James has described me as a structuralist, and of course I am a structuralist by virtue of my profession as a mathematician, because all mathematicians are structuralists. Structuralism, in the sense I'm going to describe it, is the central method of modern mathematics. But I'm going to... I see myself here in the role of a missionary addressing an audience of cannibals, or perhaps a cannibal addressing an audience of missionaries. I'm not sure which one is the case. Maybe a better comparison is with Pope Benedict I'm defending mathematical orthodoxy against heritage, I don't like to think of it that way, maybe you don't think of it that way. So let me, I'm just, I'm going to, roughly speaking, give you the first year and a half of the Bristol Pure Mathematics course, the basis of it in about 10 minutes. I won't give you any details because that would take too much time. So I'll just give you an outline. Did you walk off with the marker? Yeah. For a reason, right? Did you bring your arm? No, of course not. OK, so I'm going to put it in there. I'm going to organize my remarks, and I'm going to come around to considering James' example. That's the simple thing I want to do. But let me organize my remarks around a series of questions. So the first question is, what is a mathematical structure? And I'm going to give you the answer that mathematicians would think is the correct one.

12:30 structure was invented in the 20s by him and the greatest of all women mathematicians. One of the greatest mathematicians of the 20th century. And the idea, her idea, was roughly this, that a structure consists of an underlying set. set. Underlying set, let's call it S, and the morphology. Well, the morphology can take any number of forms. In fact, it's not useful to give a precise mathematical definition of what a structure is. Bourbaki did so in his volume on set theory, and it's turned out to be absolutely useful. So in giving you the sight of what a structure is, I'm not trying to define something precisely, I'm just trying to give you a rough idea. Okay, so maybe an example or two an example or two will to do. So a group, so groups, let's give two examples. A group is an underlined set, let's call it G, and the morphology consists of a single binary operation on G. And it's subject to the restrictions imposed by what are sometimes called the axioms for a group. So this binary the associative, and it has an event of an identity element of inverses and so on. You can write all that down. The morphology in this case just consists of this single binary operation. Okay, another example would be a topological space. And that would be an underlying set, let's say, S. And over here, we have a family, what's called a T, of subsets of S, subject to certain axiomatically

15:00 random conditions. In this case, that collection, that family of subsets is called topology, and it consists of the set of open sets of the space. A field consists of an underlying set, let's say, F. And over here, the morphology is more complicated here because we've got two binary operations. These things This maps F plus F, two binary operations plus and times, and two distinguished elements, 0 and 1. So there are four things that constitute them. And let's give a final example is a graph. And a graph is an underlying set V of vertices Relation, let's call it E for edges, or edge connected, re-read it, X, E, Y, to mean X is edge-connected. Okay. Now, the point is that when I give such a structure, I'm implicitly, all these structures implicitly fall into certain structure types. For example, the type of this structure is this of a single binary operation. But in order to specify the type exactly, And in order to say what you really say what you're talking about, you have to say what you mean by a morphism of a structure type. So morphism is the key idea that Nertha was the first to actually understand the importance of. realize that the key idea to getting a structuralist,

17:30 what we would now call a structuralist, picture of the mathematical world is this idea of morphism. What's that? Let's suppose we've got two structures. Let's say, let's call this one S1 and this one S2. okay, so I'm talking very abstractly Okay, so that's the underlying set, and that's the morphology. That's the underlying set, and that's the morphology. So what do we mean by a morphism here? Well, we mean a function or a map from the underlying set of the first to the underlying set of the second, which respects or, yeah, respects the morphologies. Okay, so what does respect mean? Well, I told you that these general notions, I'm talking very generally, and I'm not going of a general definition of what this means, but I'll point out, for example, what it means in groups. It means that m of the product x star y star 1y is the product of the, the image of the product is the product of the images in x star 2 and 1. So I'm thinking of myself as having G1 star 1. My first one, then G2 star 2 is my second one. OK, so respecting the morphology in this case just means this, that if I take the image of the group out of the two elements in the first group, I get the element of the second group, which is the second group product of the two images. Well, in the case of topological spaces, the condition is that the inverse image of an element of an open set in the second one is an open set in the first.

20:00 So it's a quite different kind of thing. Okay. Okay. So what we've got now is the, we've got the notion of a structure, of structures and morphisms. Okay. Notice, both structures and morphisms are, roughly speaking, set theoretical options. They certainly use set theoretical notions in their definition. I mean, a morphism, for example, is a function, among other things. It's got a certain problem. OK, so the second question that comes up here, then, is this. Is all this a, quote, set theoretical version of this stuff? No, it's not. There's no known notion of group other than the one I indicated here. One often hears philosophers talk about, or even mathematicians talk about, set theoretical definitions as opposed to other kinds of definitions, or set theoretical treatments of subjects. That's all really missing the point. these simple definitions I give you are taught in the first three weeks of the course at Bristol. It's set theoretical in the sense that we talk about sets and ordered pairs and functions and relations and structures of these kinds. It's a kind of really minimal, it's the minimal collection of facts and procedures in set theory that go into the in every mathematician's toolkit. It's nothing to do with, let me exaggerate a bit, it's nothing to do with the formal theory Zermelo-Frankel. That's not quite true. The formal theory Zermelo-Frankel is sort of modeled on this naive stuff. But saying that you're using the notion of set in a foundational or fundamental way here

22:30 saying that you're proceeding in a manner that all mathematicians now proceed. So let's get rid of that tanr. We're not talking about some kind of special set theoretical version of this stuff. There's nothing to being a group except to have an underlying set and a binary operation on it in the way I describe it. Now there's one other point I want to emphasize. before I can address the example that James gave. And that is the idea of isomorphisms and automorphisms. Okay, what do we mean by an isomorphism? Suppose we've got two structures, S1 and S2. Now, let's think of, and we've got a, we've got a morphism between them. Let's say M is an isomorphism. That's what we've got to define. Okay, so what do I mean by M being a morphism from S1 to S2? I mean just what I said earlier, but in particular it has to mean it's a function from S1 to S2. where these are the universes, or the underlying sets of those two structures. OK, so there are two conditions that have to be satisfied. First of all, M is an isomorphism if and only if M is a bijection biject is a bijection M inverse is a morphism what does that mean? what does it mean to be a bijection? well, it means it satisfies two conditions that it's one to one that is to say that if fx, sorry, mx, equals my, then x equals y, OK?

25:00 It's on to, that is to say, if y is in s2, 2, then there exists x in S1, y equals m of x. Now all this is quite straightforward, but now notice this important point. The definition of what we mean by a bijection, and hence what we mean by isomorphism, depends crucially on using these identities. what it means for a function to be one to one means that its values at different points at different arguments are different you can't even say what one to one means without using these set theoretical notions of function and so on and certainly without using the notion of identity identity. What does onto mean? Well, again, you have to use the notion of identity in order to explain it. So the optic lesson here is there's no question of being able to get rid of identity if you're going to talk about automorphisms, permutations, all these things. They depend on it. So you have to already know what you mean by identity, as it were, in order to talk about automorphisms and isomorphisms. Let me quote Ian Paisley at this point. Never, never, never use technical ideas in mathematics without getting precise instructions. Never, never, never use technical ideas in mathematics metaphorically. okay so now let's get on to the let's get on the business of graphs okay so now let's get back to the question of what graphs are okay now there are two

27:30 the first thing we have to if we're going to talk about graph theory the first thing we have to do Okay, so what's a graph? Well, there are two ways of going about this. One is, it's a relational structure. Let's say it that way, such that R is irreflexive. That is to say, X doesn't R, X. And cement. That is the second. X, R, Y implies Y, R, X. And of course when I say X, R, Y, what I really mean is that the ordered pair X, Y is a member of the set of ordered pairs, which is the relation. an alternative way of doing it is to have a set of vertices again here this is the morphology it's provided by that single set where S is a set subset of two-element subsets of V. So let me give you a concrete example of this, a concrete example of the graph. So I'm going to take two objects. I'm going to take H, which is Hannes, and J, which is James. So I've got two objects, and now I'm going to consider the following graph. V is just equal to the set whose members are Hannes and James. And R is the following relation. It contains the ordered pair Hannes followed by James, and it contains the ordered pair James followed by Hannes. OK, now that's an example of a graph, right?

30:00 Now for those of you who are so sunk in metaphysical confusion that you think it may be impossible to individuate the two objects, Hannes and James, if you're capable of distinguishing your arse from your elbow, I can give you an even simpler example. The vertices consist of your arse and your elbows, and the relations consist of these ordered pairs, arse followed by elbow, elbow followed by arse. I mean, this is absurd, but the point is, it doesn't matter what particular things you take as the elements of your set. You could take the yellow tower in Nelson's column. about those two things being objects, well, they'd pick another pair of objects. The point is, what's going on here is that you really do have to have a determinate notion of identity and distinctness that applies to the things that you're taking as an example. Okay. Now, this definition of graph doesn't look like James's. because he was drawing sort of dumbbells on the board. what we're looking at here So, let me address the question, what is the role of these diagrams in graphs? So, this graph could be represented like this, A, E. Let's go back to the more depthless example of Pontus and James. I can label them to indicate which I had, which was which. But that's not the real function of this diagram. The function of the diagram is not to define a particular graph. It's to describe the isomorphism class of the particular isomorphism class of this graph.

32:30 So this diagram is not a graph. It's a diagram. And what it designates is, it designates either, you can think of it either as the class of graphs isomorphic to this, or you could think of it as designated some particular but unspecified member of that isomorphism class. Now that's important. That's important in the context of what James was talking about because this diagram is sort of, it is in some sense abstract, right? And it's not so clear what the dots are standing for. as you sort of stare the point is when you specify what the dots are standing for you've got no problem about it now the idea that there could be any kind of structure of whatever type that's built up of things that only exist as it were in the structure is just incoherent it's certainly it's certainly incoherent of M. Nertes' definition of mathematical structure, which mathematicians use. Now, my feeling on these things is that we should, if we want to talk about structuralism and the various issues that arise out of it, we should start with the mathematician's own idea of structure and see how far that will take us. and the crucial point is that when you certain steps in this process of analysis you've got to be exactly correct about exactly correct about your definition well I think I've talked on long enough because we want to have a discussion after this I'll remember all right good applause applause applause Since we started 10 minutes late, why don't we add 10 minutes to the discussion time so we can go until 2.20 or so? Sounds good. Let's see, Hannes, Stuart, lots of people.

35:00 Hannes, why don't you begin and then I'll jot down the other... All right, so as James said, we were somewhat inspired by mathematical practice, right? And John Mew also gave a lesson in mathematical practice. So the question that arises, I mean, someone's cheating here, right? They can both be right. But in a sense, I think there's something to be said in favor of what James said and in favor of what you said as far as mathematical practice is concerned. So let me try to separate things here a little to get a clearer picture of what's going on. So first of all, I think when philosophers and mathematics say they have a look at mathematical practice, that's of course a good thing, right? These are the data, I think. It depends on which mathematical practitioners. Yes, I'm coming to that point, right? But of course, as in science, I mean, in science, the discovery was in philosophy of science that there are no real data in the sense there's not something that we can be wrong about. And similarly, a philosopher of mathematics should think of mathematical practice in that way. So they look at mathematical practice, but they shouldn't take everything seriously that mathematicians say. It's finding a reflective equilibrium, where on the one hand you look at mathematical practice and take up things that are done there, but on the other hand you systematize things theoretically, you come up with a philosophical theory, and that's a back-and-forth process. But I think it's fair to say that at the end, you have two philosophical theories that went through this process and found some sort of equilibrium by looking at mathematical practice. The one theory is closer to mathematical practice than the other one. Then this is a prima facie evidence for the former philosophical theory, the one that's closer, the one that's closer. That's, I think, the general structure we thought of. The second problem is, what is mathematical practice? I mean, what are we looking at? And what we were inspired by was how graph theorists actually reason and how they actually talk when you talk to them about graphs, okay? And how they talk to each other about graphs. So their manner of talking and their manner of thinking, if you want, okay? And there, they use notions like unlabeled graphs all the time. And you can show that in graph theory books. It's unproblematic, okay? And they would say things like, if you ask them, okay, what's this node in this unlabeled graph, I mean, if I took that node out of the graph, right, what would that be?

37:30 And then they would say, well, if you took out the left node or the right node, I mean, it's the same thing. That's not the point. Don't you understand what I mean by unlabeled graph? So they would come back with this question, okay? So they have this manner of talking in that way. And that is what we were sensitive to and what we looked at. And people in graph theory talked like that before set theory came along. Now, you made this clever move in your comment where you said, but I'm not really talking in terms of set theory. It's not serious set theory. It's not that I made use of the axiom of choice at some point, right? Yeah. But still, it was set theoretic language that you used. Yeah, but that's what you use in everybody. Yeah, well, so far I've talked about unlabeled graphs and nodes and things like that. Okay? Okay, I didn't make use much, I didn't use a lot of set theory and terminology here. But now the second part of mathematical practice comes in. If, at the same time, the same mathematicians, these graph theorists, they've studied mathematics, and they have learned to make use of the language of set theory. And that's very often the level where the mathematicians end, anyway. They do not make use of heavy set theory, but they use it as a sort of language, right? Very often they do not even know the accents of set theory, and that's not what they need. No, but they do it. But Ramsey's theorem in graph theory is a deep theorem, and it makes deep use of set-theoretical numbers. I agree that there are parts in graph theory, particularly those parts where you deal with infinite graphs, where set theory, the heavy one, comes into the picture. But standard graph theory certainly does not. You don't think Ramsey theory is standard graph theory? Yeah. I mean, the finite Ramsey theory is a standard topic in graph theory. Yeah, but, John, even you would, I think, agree that for most parts of graph theory, it's more on a combinatorial level. That's what you're doing. And you don't need a lot of set theory. Isn't that right? You need clever use of simple set theory. I mean, there's some very clever uses of simple set theory in elementary graph theory. Yeah, yeah, yeah. But let me finish my phone, John. You can react. So, there is this second part in mathematical practice where they have learned to use the set-theoretic language. And then, of course, if you ask them how do you define it, if you look in textbooks, they have to state the official definition. They've learned, let's make use of the set-theoretic language, for a good reason, I mean, that's a big success. I mean, there's no doubt about that. So, they state the definition of a graph in the way you did.

40:00 But my hypothesis is that they still think about graphs in the way James and I described it before. And there is a gap between these two things. And it's not easy to get rid of this gap. So you briefly said, well, unlabeled graph, I can still make sense of that in terms of an isomorphism class, something like that. But strictly speaking, that's not true, I think. So an isomorphism class, for example, has members. Right? That's why it's a class. An unlabeled graph doesn't have members in that sense, okay? An unlabeled graph has nodes and edges. But for an isomorphism class, where are the nodes and where are the edges? And by the way, I do not mean the nodes and edges in the members of the isomorphism class. I mean the nodes and edges of the unlabeled graph, the isomorphism class. And I don't see them, you see? So I'm just claiming there is a difference. There is also one other reconstruction that you could use. You can say, what I define as a graph in set theory, that's the unlabeled graph. That's also sometimes the people's way of dealing. And then if you look at the label graph, then you label the nodes of a graph in a set theoretic sense. So you have an additional labeling mapping, right? We have a limited amount of time. Yes, yes. But in standard reasoning, this would be a kind of double labeling. And that's still not what graph theory is, I think. would say. So there are these two parts of mathematical practice. We were sensitive to the former, you were sensitive to the latter, which we would call a set theoretic reconstruction. Okay? Which is interesting, but if we can show that there is a theory of graphs in the first sense, an axiomatic theory perhaps, okay? Where this turns out to be not incoherent at all, then I think we would be closer to this first part of mathematical practice. And it's not incoherent what we've seen from the theory. So I don't think there anything problematic about it. Can I just... The brief response is you haven't given a definition even of these dumbbell graphs or dot and line graphs. The lesson of the 20th century in mathematics is you've got to define things properly. Like sets, you mean? You define what sets are? No, you define... But then I don't have to define what a graph is. But then I don't have to define what a graph is. Wait till I finish. Mathematical practice says you take these really simple ideas

42:30 and set theory, like function, set, and so on, as fundamental, and then you define structure in terms of those. Now, the problem, I'm making a very simple logical point here, which you could put in more conventional terms than this. These diagrams don't correspond to particular graphs. I mean, from a philosophical point of view, don't want to say that I've got a particular graph actually inscribed on this whiteboard. You don't want to say it's a graph like this. Well, I'm making it, when you say it's describing the isomorphism class or the concept or the isomorphism concept if you like, for this particular graph here, you're using it, it takes on a status that's more like say a free variable, a logical status more like a free variable In fact, if you forget the fact that A was supposed to stand for arse and E for elbow, and just take those as letters, and then you add the additional condition that A is not equal to E, that rather clumsy notation performs the same logical function as that nice dumbbell. but you should never confuse the diagram that describes something with the thing itself it's just an absolutely fundamental mistake maybe we should move on so we've got a number of people on the list, so I think Stuart for us next thank you, I wanted to ask this report for James remember that? I like that distinction you made between Hexides used for for for identity identity contextual identity yeah yeah and so this is a sort of a side meant to be a supportive side comment that if we want the traditional metaphysical route with exeities then we have to take the Caesar questions seriously you know whether the natural number two was the same as the real number two and whether this set is the same as this number and so on Because, you know, they're excedes and they really look the same and different. Whereas that question doesn't even come up with the identity that you're actually,

45:00 with the sort of contextual identities you're calling it, that we actually invoke. Right. That seems to be one difference. And also, if you do have a structure in which you have both the reals and obviously then the rationals, then you can identify the rational two with the royal two because they're in the same structure. When you've got a clear structure, then within that structure, you can make sense more. No, of course you have to, but to take sort of a traditional problem that goes back to ancient Greece, you know, with Plato's, well, take the three-graph and the two-graph, you know, to use the more modern version of it, you know, no edges, right? So three elements, no edges, right? Two elements. Now, are the two things in this graph the same or different from the two things in this one? Or, you know, any two? And you think, well, that's not a question, right? But if they had word axiades, then it would have to be a question, and then we have to say it's unknowable or, you know, unscrutable or something like that, right? And another sort of supportive comment is that even, this actually relates to something that John said, but when you were describing the graph that does have a non-trivial, doesn't have a non-trivial, the rigid one, right? Even there, you're presupposing identity, at least in the meta language, because you said, Well, this one relates to 2. Well, what does that mean? Well, there's x, and there's y, and x is different from y, and it relates to x, and it relates to y. So you can't even handle the, it seems to me, the rigid ones unless you're allowed to involve the anatomy. You know, some call primitive, if you want. I mean, how else can you even describe the structure? Well, you could describe it with wiggly lines like that. But it would be a mess. You wouldn't be able to understand what it meant. it like that scribe you still have to want to say that they're different yeah so you'd end up the picture is much clearer than the proper than the set theoretical so identity is really bound up with the whole enterprise but yet it's not the sort of ex-deedy identity that's the uh that's what the metaphysical promise is and the only point i really wanted to make with that example was that you could have absolute discernibility even if you didn't have any intrinsic property of status Yeah, absolute within the structure. Within the structure, yeah. And not necessarily from one structure to another. Right.