Identity indiscernibility & ante rem structuralism (contd.) / Richard Pettigrew response / discussion

0:00 And each structure exemplifies itself. That is, the relations of the structure hold of its places when the latter are construed as genuine objects. Now, if this makes metaphysical sense, and I guess it may be a big EF in some circles, then the dependence relation and the slogans rank you on structures is that of constitution. A structure is constituted by its places and its relations in the same way that any organization is constituted by its offices and the relations between them. Now, this I'm getting from my metaphysical friend, Gabriel Usguiano. The constitution here is not data meteorology. It's not the case that a structure is just the sum of its places, since the places have to be related to each other via the relations in the structure. That is the relations usually matter, except for the degenerate cases like the cardinal force structure and some of the graphs. I think of an anti-ramp structure as an organic whole consisting of or constituted by its places and its relations. Now, to help clarify these matters, the metaphysical theses, consider an objection raised by Jeffrey Hellman. This was alluded to earlier, but I don't think we fully went through it. So here's a complaint he makes. On Shapiro's view, numerals, you know, the places in a unique, he has places in Scarecourt, in a unique archetypical structure answering to what all progressions have in common. But the places of the interim archetype are entirely determined by the successor relation by and derivative from it in the sense of being identified merely as the terms of the order of induced by form. Now, in the case of an in-rate structure, this is a structure that you're sort of from already pre-existing objects. We understand that a particular successor relation in the ordinary way is arising from the given relata, reflecting an arrangement of some sort. But if the relata are not already given, but depend for their very identity upon a given ordering, what content is there to talk of the ordering? What can succession mean if we're extracting from all in-ray cases, and if we can't speak of relata without making reference This, I submit, is a vicious circularity. In a nutshell, to understand the relata, we must be given the relation, but to understand the relation,

2:30 we must have access, already have access to the relata." End quote, right? It's a mouthful. The first sentence of this passage is correct. That is, numerals on Shapiro's view, numerals denote the places in a unique structure. All right? I presume that the talk at the end of how the relata are to be understood, and what must be given, and what we have access to is only a metaphor for the underlying metaphysical themes. It's not a matter of being given anything. Hellman attributes to me the view that the relata, the objects of mathematics, depend for their very identity on the relations of the structure. In other words, the relations are metaphysically prior to the places, and the relations somehow fix the places. Now, despite Helman's charge, a view like that might make sense for the example, actually the example he's talking about, the natural numbers, or the ordinal free structure, the one I talked about about a minute ago, for that matter. But the view that Helman attributes to me makes no sense for the finite cardinal structures or the degenerate graphs, the graphs with no edges. The four places in the cardinal four structure, for example, cannot depend on the relations of that structure, since there aren't any such relations, I mean, except for identity and non-identity by which three places can be met. As above, the metaphysical view is that a structure is constituted by its places and its relations. Neither of these is prior to the other. So, priority is not well founded. Sorry. Again, the slogan that the objects depend on the structure can be understood in this spirit. And the dependence here is the dependence of constitution. Now, while Rama said this was a in closing, and I know I'm I'm going to have a few nights here, but when I start to get scholarly, and actually I learned in my heart that my commentator knows something about Plato. While we're on the subject of Platonism, let me conclude with a brief look at the old master to revisit a theme in my book. Some writers in ancient Greece distinguished arithmetic from a subject that they call logistic, which is the practical discipline of calculation. So logistic concerns measurement, the practical discipline of calculation, so it concerns things like measurement, business dealings, and the like. Now Plato, who wouldn't much care about that last bit, makes a very different distinction. So he still has two subjects, arithmetic and logistic. But they're both pure.

5:00 Both arithmetic and logistic are theoretical disciplines concerned ultimately with the world it being. Arithmetic, this is a quote from Plato from the Gorgias, deals with the even and the odd in reference to how much each happens to be. So if one becomes perfect in the arithmetical art, then he knows also all of the numbers. For Plato, that's arithmetic. For Plato, logistic also deals with the natural numbers, but differs from arithmetic, quote, quote, insofar as it studies the even and the odd with respect to the multitude they make both with themselves and with each other, also from the Gordon's. In short, it seems that Plato's thought that arithmetic deals with the natural numbers themselves, while logistic concerns the relations among the natural numbers. Now, according to Jacob Klein, which I see that my commentator brought along, Plato's logistic, quote, raises to an explicit science knowledge of relations among numbers, which precedes, and he must precede, all calculation. Now, I've always found these passages puzzling, ever since I was an undergraduate reading Plato. The main thing I found puzzling is, what the hell is arithmetic? What exactly are you supposed to do when Plato's arithmetic? Remember, logistic is the study of the natural numbers and their relations to each other. Arithmetic is the study of the numbers themselves. So what are you supposed to do? So what on earth is there to know about the natural numbers besides their relations to each other? What could count for Plato as a theorem of arithmetic as opposed to logistic? It's not that I'd expect Plato to be an A theorem structuralist. Although, I mean, if he were to be a structuralist, that's the kind he'd be. But what could he possibly have in mind for arithmetic to study the numbers in themselves? Now, I take it that Plato's logistic is what we would recognize as arithmetic, the mathematical theory of the natural numbers. Perhaps Plato's arithmetic, this is where I'm going out in ice, is not a mathematical study at all. Rather, Plato's arithmetic asks for a philosophical account of the natural numbers. It inquires after the metaphysical nature of the numbers. In developing, or at least inting, at the description of natural numbers as places in and an anti-run structure, a structure constituted by those places and the arithmetic relations on them, perhaps I'm engaging in Plato's arithmetic,

7:30 or at least I am if I'm getting it right. Thank you. Right, well, I'm here pretty much in the role of skeptic, as you mentioned as John is as well. We've both committed something like what I've seen described as limited structuralism, so I guess we're sort of curious and antidotes to all the math physics that's being done. But in this role, I have this increasing sense of unease as I read the draft that Stuart said to me of the paper he was going to give, because I find myself, until these dissection plates at the end, agreeing about what he was saying, which is an uncomfortable position when you've got to respond to someone's paper. But... Isn't this great? That's right. But I realise this is for two reasons, right? Well, I'm sorry, really for one main reason. that most people who are limited structuralists are so because we find something wrong with the alternative they find something wrong with the metaphysics or the epistemology of something like Stuart's view and so they become a limit of it. But I'm not a limited structuralist for that reason I adhere to the view because I think it's just a right interpretation of math So, I agree with all the metaphysical bashing that went on in Stuart's paper, and particularly his rebuttals to the objections that have been raised from metaphysicism. And I also agree with the one interpretive part of the paper, which was the stuff about I and minus I. The reason for that was it's exactly the sort of account that a limited structuralist would have given, giving, or would give, or would want to give, of I and minus I. So that's the book I want to talk about today, because it seems to me that once the n-to-n-structuralist has conceded

10:00 in that, particularly as the case of I and minus I, that the free variable approach is the correct approach. It seems to me that you go back to the motivation for the theory at the beginning, and find that the motivation is undercut by a set of arguments he himself today, in the forum, sure. So, let's go back to, I just want to take you through what I take to be the motivations for antiretstructuralism, which come out of the Maserat paper, which anyone who's familiar with this debate is familiar with what numbers could not be. So we'll start with the definition. This isn't quite how Banasarabh set it up, but something like this is what's going on. We're going to say that a triple, n0x, is a data construction, if n is a set, 0 is a member of n, s is a map from n to itself, And then the three conditions that we're used to imposing on data structures. First of all, just that anything in n doesn't map on x to 0. Second one, n is 1 to 1. So for all n and n, n is n plus x to n. And then finally the principle of mathematical induction. and I'll just put that like that but the details of this are important the point to be made is that Benazirah pointed out that at least if push my positions would have to say if push to say something about this symbol that they use use the black board face if push to say something about that they would have to say something like and it's a data construction they may not say it in those words they would agree with someone else with that. They would say that, but the point is that if you look at the mathematical practice and such things mathematicians say, you'll never be able to get any further specification of what M is supposed to occur to in that sense. And that's about, for instance, the point that if you take Z to be the set of

12:30 finite, sorry, malo-ardenals, and take Vm to be the set of finite polynomial-ardenals, then there's no reason for counting either of these, or indeed neither of these, to be the reference of that. This is Bonassarap's point, that is the example that he gave. And so you can see the Bonassarap paper has something like a puzzle. The question is, how do you interpret the optimances of the mathematicians that use this symbol of that work they say? And there are two solutions that come up. One of them is the one that Stuart, most famous for defending, and that's the anti-round structures position. And the solution that that offers is that n functions as a proper name. This is exactly what I thought you couldn't have N doing, but it's also a problem by taking N to refer not to something like this, where the elements of the underlying set have intrinsic matures independently got to refer to something like the universal shared by all data construction okay so or Michael Resnick's version of the view is that it refers to patterns shared by all different structures so that's one way of doing it that's the anti-ram way of doing it and the other way of doing it is as I usually call it a ray-structionist way, but let's call it a limitless, since that's what Einstein's called it. And that way takes n not to be a proper name, but to be a free variable. So we've already talked about free variables a little bit today, but let's just give a quick example of how free variables function in that. So this is sort of Theorem that you may teach the first year, like undergraduate group theorists, is right. If G is a finite group of prime order, then G is cycle. How does the proof go? So while the proof usually starts off by saying let g be a final group, probably more than let little g be a member of its underlying set, then the orbits of g has to be a subgroup of g, by Lagrange's theorem, and the primeness of g, therefore it has to be the whole of g, and so g generates, little g generates g, so basically any non-identity element of g will generate g, so it's like a book.

15:00 And that's how the free variables are used, and that could just have been quoted straight from election on group theory to first years. And in this, G is functioning as a free variable, right? I take it back to be an uncontroversial interpretation of what's going on here. A bound variable. No, it's bound. Universal quantifiers are fantastic. Okay, okay. It's the same thing. Okay, sure. So what the What the Eliminative structuralist is claiming Is that this symbol Is functioning in man Exactly as G is functioning here It has one small difference While G might be used as a free variable Introduced by any other formula Other than G is a finite group of prime order, the free variable by word phase n will never be introduced by any formula other than n as a data construction. So you could say, and I guess this is something like what you were saying when you talked about i staying as a down parameter, especially like some different class of terms, you could call this a dedicated free variable. It's dedicated in the sense that whenever we use it, we don't have to state this condition, unlike we have to state the condition of symbols like G. We don't have to state this condition. The reason we have a dedicated symbol for it is that whenever we use this symbol, we're meaning that. Okay, so that's the situation with regards to the anti-Ren and in-ray structuralist and this symbol N. So now let's move over to the case of I. In that case, of course, you have to lay down your complex field structure axioms in order to introduce the symbol that we're face C. And then, of course, as of the case of N, you have the Banaskov-style puzzle

17:30 because there are lots of different systems and terms that satisfy the complex field structure. and again you have the question are you going to be an anti-reg structuralist or an in-reg structuralist about C but the point is whichever way you go on that the same sort of problem arises for the symbol I because we introduce that by such assertions as I as a member of C or not by such assertions but just by this assertion I as a member of C and I squared plus 1 equals 0 Okay, that's the definition of this. And, but of course, Vanassarab's problem, or something analogous to it, crops up once again at this level. Because it's a theorem, whichever way you interpret C, it's the theorem of complex analysis that are exactly two of these things. But there's nothing you'll find in mathematical practice that specifies which one it's going to be. So you've got to solve this. So, again, the in-array or anti-REM solutions that will crop up. However, in this case, as it's evident in fact, it shows in the in-array analogy to the in-array structuralist solution for this. The anti-REM structuralist solution doesn't work for this. You can't say that I is going to refer to the universal shared by all things that satisfy this We've specified the facts about ions and member of the complex playing that complex, made up of universal, in particular on the ANSI-REM use, made up of particulates, the same places in the structure are particulate. Okay, so this is just to draw the analogy between the two. And then to put forward the following argument that this, you see, because I think as Stuart mentioned, Hellman, in justifying going down the minimum structuralist route, or a limited structuralist route, attacks the anti-round structuralist position. And that's the way that this debate has often gone. It's been taken to the default position that one should be an anti-Ram structuralist. And not only if epistemology or metaphysics becomes too obnoxious to you, should you become an in-ray structuralist. I want to use this example and the fact that Stuart's, I would say, conceded that this example must be done in the in-rate structuralist way, to show that one should also go the in-rate structuralist way about other seeming proper names in mathematics, such as these, which have been taken by anti-run structuralists to be proper names.

20:00 Okay, so a couple of reasons why you might think the evidence of mathematical practice points to Warren's anti-amstructuralism in the case of these. One of them is, these seem to be used interchangeably with definite descriptions, but as Stuart might have done that himself. So is I, but that shouldn't be in terms of proper anatomy, it should be in terms of free variable on construction. And the other objection might be that you don't state these in the free variable form You don't state them in a form analogous to the group theorem at the start At the start where you say if G is a finite group of prime order So you never say if N is a data construction then and so on and so forth Instead you just state things like right, but that, again, is true in the case of time. So it looks to me as if now we have no uncontroversial cases of proper names in mathematics, but we have one uncontroversial case of a free variable that looks like a proper name in mathematics. So I would suggest that this shows that instead of the default position being anti-RAM structuralists against which one has to argue if you want to be in-RAM structuralists, it should be the other way around, but the default position should be in-RAM structuralists interpretation against which you have to argue if you want to be anti-RAM structuralists. Okay, so that's all I want to say about that. Do I have time to say something quick about your Plato? Yeah. Yeah? That's it. Yeah? Okay. Just because I've advertised, I'm not going to say something. I feel obliged to say something. Unfortunately, I'm not going to say something terribly. But there is a distinction between theoretical and practical arithmetic in Plato. So when Plato says that arithmetic is just the study of the number in itself, it looks as if there's nothing to say because there's just one number there, and anything you can say about that number in a theoretical realm could only be about its relation to other numbers, and so it would have to fall under the ages of logistic.

22:30 But Plato does distinguish between what he calls the arithmetic of the crowd, which is one thing, and that of lovers of wisdom, which is another. And when Pratarkus asks him what this is, he says the distinction is not small, Pratarkus, that those who deal with number, some kind of units which are in some way unequal, such as two-armed camps, two-headed cattle, two other of the smallest or the largest of all things, while the others, which is the lovers of wisdom, I presume, would not follow their practice unless someone posted the unit such that not one single unit of those in a myriad had differed from any other. So you've got the idea, you've got two rounds of that arithmetic there. So there is at least this to be done in arithmetic. It is to say of two sets that they are of the same size, And then afterwards, in order to do calculation, that does involve the relation between sands, which is the study of the disease. So there is something to say in Plato's arithmetic. But rather than saying something exegetical, I more wanted to point out that now that Plato has been brought into the debate, And the real point to recognize when we didn't wait to be in the philosophy of maths is that for him, numbers weren't what we call numbers. They're much closer to what Stuart calls cardinal structures. Actually, they sound like graphs. Finite sets, how about finite sets? Finite sets, they distinguish about all elements. Yeah, sure. So there's some kind of sets of pure units to that, a finite set of pure units for a place to show. So the last thing I just want to say very briefly was that this looks to me like it opens up a question of whether we're really able to interpret our use of number words and numerals when we're not speaking as mathematicians correctly. We've always been, in philosophy of math, we've been interpreting number words as though used by the masses or the crowd, as of the same, referring to the same things as number words and numerals as used by number theorists. And it's numbers of wisdom. Absolutely. Stuart Leighman, think so?

25:00 So, I would just say that the reading of Plato suggests that there might be a distinction to be made between the reference of four, for instance, in the case, used by a number theorist, which does refer to the basis of the fourth place, what I would say is the use of the three variable ranges of the fourth places and data construction, and four as used by one of the proletarians, who use it to mean something or to ascribe the property of formants, or something like that to a finance side. So that's all I wanted to say about Clayton. First of all, the idea that you had of thinking of ending INR as variables or constants or parameters, And the same would also apply to numerals, which I would think to be the paradigm of proper names, 0, 1, 2, 3. So they all end up sort of falling under that. Since I took the major advantage of anti-run structuralism over the limited versions, or at least over Hellman's version, is that it takes mathematical language literally. And if your interpretation is right, then it's going to narrow that down a bit Because it looks like those things are singular terms under that reading. I never thought of interim structuralism as the default, sort of taking the high ground. It's true that Hellman goes after it, but his book was published in 89. In 97, I've got a fair amount to say about the in-ray versions too, and so the debate goes on. So I don't know where the sort of burden of proof lies, sort of whichever one gives the nicest. And the official line in the book, actually, is that, in a sense, the three versions of structuralism are the in-ray, the two in-ray versions and the anti-rem version, are in some sense equivalent, you know, inter-translatable to each other. And the anti-rem one is the one I like the most because it's the most extravagant, but I don't know if there's anything that, you know, anything that's sort of default about it. All right. Anyway, thanks. All right, so we've got 25 minutes of questions, so let's go.

27:30 Thank you. Yes, Steve. So this is a comment in a sort of kind of naive word. I mean, the comment is that James has referred to the no-priority view in the boldness when the comment that Eddington, in the 1920s, had a very similar view of structures and about physical objects, but it was based in part of the reflection on the next group theorem. The idea that the group elements and the transformations, they come as a package. He expresses it quite nicely in a debate with, well, a response to Braithwaite, because Braithwaite's trying to beat him up with the Newman problem. He's just going to say, I just don't see that it applies. What's another problem? Well, we can talk about it. So I'm not unsympathetic to that. I think it's an interesting way forward, but the poor abuse metaphysician might have the following kind of worry. Yes, she is. There's a circularity. I'm sorry, it's a bit of a synthetic, but it might be a kind of circularity. I thought the whole point of structure, part of motivation for structure, this mathematical structure, was to answer the question, okay, so what are numbers? Okay, they're places in the structure. Cool, so what's a structure? Ah, it's places and relations. Okay, so what are places again? And, you know, we're not getting anywhere. And it's no good to say, well, access and reference, those are just metaphors. Because the metaphors, there has to be something behind the metaphor. Otherwise, they're empty metaphors. I realize the fashion of metaphysics is getting out of hand. It's not really fair. It's inevitable. It was. No, structuralism, especially Instagram structuralism, is a thoroughly metaphysical view from the start. So it's all in good fun. You know, metaphysical questions have to be taken seriously, not necessarily the standard intuitions, though, that, you know, the metaphysicians impose on us. Those are up for great. That's okay, that's okay. I just want to know what you're putting in a replacement. And if you specifically have a no-priority view, I mean, if you're saying, okay, so this then sort of should feed into our notion of dependence, and, you know, we should be thinking of the places as straightforwardly dependent on the relations in a naive way. Then, again, I'm sympathetic, but I then say, okay, so what is it there? What is the, you know, maybe this goes back to Catherine's worry about sort of instantiation. What's the terminology? What are the relations between the places and the relations?

30:00 I thought I was at least trying to at least start on that project, right, in the last section of the paper, well, besides the play we want to do, right, the last section before that. I'm relatively new to this, but, you know, I think it's a start. This is a minor metaphysical question. Actually, I'm not going to be a passionate metaphysicist anymore. Not just because everyone seems to be doing it, but also because we don't want Catherine to feel sad and numb. so this is a minor medical physical question I just wanted to see if I could get clearer on the distinction that you presented in response to Fraser between your kind of what you mean by structure and what traditional platelets mean by universal so if I understood the distinction is that traditional kind of tonic universal like properties they are forms of objects Okay, so I suppose there are two things that I wasn't aware that the traditional Platonist couldn't describe properties to systems. Probably. Okay, so that's one thing. The second thing is that I'm not sure I understand the distinction between individual objects and systems, because most individual objects are in fact systems. So, you know, I'm an individual object, but I may have component parts that stand in certain in relation to my cells, and so on and so forth. But there may be some objects without internal structure, but they're . So objects are systems. On the flip side, you know, systems generally speaking, as far as we've been talking about, particular systems, it seems that they're objects. So I guess what I want to know is what is the distinction between objects and systems that underwrites your distinction between structures and traditional? I can answer that because obviously, I mean, I take a structure and a system that they refer to as a singular term, right? It's a one in some sense. You know, it depends on your perspective. If you're looking at it as a unified thing, or if you're looking at it in terms of its parts, or the objects that make it up. I don't know if they're just that same thing or not, but I don't think the distinction between things which are in effect unities and things which are systems is sort of built into the fabric of reality. Well, generally speaking, we regard these things as both.

32:30 Yeah. So you can look at it one way or look at it another way. Right. So what we do is an integral object or as a collection of space-time points or as a collection of molecules. Right. So that doesn't seem to have a metaphysical distinction between what you mean by structure and what traditional platelets mean by it. So a structure is the form of a system which already viewing it as a plurality, as a collection of objects. So even though the thing that you're viewing that way could be thought of as a one, but you're not using that perspective. I don't know if that helps, but if you're going to sort of brush over the distinction between sets and objects, then the view is going to start to come down looking like traditional lateness. But I don't have a problem with the dust. The all that just doesn't strike me is, you know, is that bad? Yeah. a token. Just to say, first of all, what I was saying before, all of these places, I don't really get it. This kind of thing is the kind of thing I was, not the details of what you said, about this kind of project. I just want to ask you a more specific question about what you say about dependence and constitution. So the structure is made up of the places and the relationship. So often, in other instances where something is constituted, often you You think of the constituted thing as depending on the components, right? And either someone's just different, right? No, the constituted, so the structure is constituted by, oh, oh, so you say that- You want the components to depend upon the structure, right? The structure itself, it's like, I don't know, it's like an integrated whole, which is made up of its components. Right, and- But it isn't like you can sort of take the components and look at them separately. Sure. So it's not like a theoretical sum. Sure. No, I see that, but I'm not going to. Are you making any terms about what depends on what there? Are you just saying this is? No, I'm not. Right. Other than, right. That is, the parts depend on the structure as a whole. Right? And the structure is composed of these. Right. So yeah, so for the placing plus the relations together go to make up the structure and at the same time depend on it. But I'm not saying that's a confectioner thing, but that's supposed to be.

35:00 Yeah, it doesn't sound right when you put it that way together. Oh, no. It's a tough thing. I mean, we're all going back to medical stuff. I mean, sometimes you get that really logically, right? So the parts of the whole, but nevertheless, the whole is somehow is better than some of the parts of the part of the stuff. But this is if you could somehow think of the parts as separate from the whole. I mean, you could sort of take six and one and two and three and, you know, and say, oh, yeah, here's one. I'm going to put them together. No, sure. Sure. So, you know, this might be a bad analogy, but it's something like an emergent property and philosophy of mind or whatever, like where you don't get it until you have, you know, a short remark then a question to Stuart the remark is just as Stuart said, it's hard to say where the advantages are of N-to-Ram versus in-rate structure but one would be, I take it concerning N, natural numbers that I really don't think in arithmetic people are thinking of statements about natural numbers as statement about So, I take it, if you ask them, I bet most of them wouldn't even know what a data kin structure is, or simply isn't in system. And they've done arithmetic, you know, a long time before this data kin came along, right? And they, what you suggested would make arithmetic being parasitic on set theory, whereas I take it most people's, people, standard mathematicians, would, if anything, think of it the other way around. First, you have natural numbers there, in a sense, given. And then you have, perhaps, sets of natural numbers where you regard natural numbers as raw elements or something like that. That's not what we do in set theory, but that's what they would think of as far as natural numbers are concerned. So I think it would be a small advantage of interim structuralism to make sense of that intuition. Right. Would you like a quick response to that? I think, I almost think it's a bit suspicious to appeal to what you think mathematicians would say about it, partly because they have thought about it so little and usually, I'm a mathematician so this isn't going to be offensive. Well it is but you're one of the offended, yes.

37:30 So it's partly because of the use of something like the definite use of the definite article and something like that. If they were pushed and suddenly had to think about it, that may be what the response would be. But as Stuart's example of his computer engineer friend, if you're pushed on the complex number, you suddenly just think, oh no, it's just there. He didn't give me a philosophical answer, he just said that's cute. And then he went on to talk to his son. But when asked the first time, he said, well, it's the square root of minus one. Yeah, sure, I'm sure it's the square root of one. Mathematicians would say about the natural number. So I think it's a bit quick to use that. But that's all I would say. It's hard to tell. It's a matter of interpretation and you told us to get across My question to Stuart So there are two cases Let's for the moment assume interim structuralism is right and there is this graph with two nodes and no edge Let's assume that That's true anyway And there are no Higgs-ITs that distinguish the two things So we just have our two nodes And then there are two cases say, there is a structure, now we want to introduce names for these nodes. And that's basically the story that you gave. That nicely explains... Yes, exactly, that's the same story, right? But I was just wondering, it could also be that sometimes we do not have to introduce these names afterwards, after dealing with the structure, but, right, they think that, so to speak, the names together with the structure, they could, right from the start, come as a package. Because once we accept the existence of this rather with the two nodes and no edge, then it's no problem to have a bigger structure where you have this and a semantic value mapping that maps one name to this thing and another name to that thing. And if one was worrying and would say, well, there is no thing like that, then probably this is for the reason that you wouldn't accept the original structure in the first place. But if we take that for granted, then the bigger structure is easy. It's just a hybrid now. It's a semantic mathematical thing, if you want. And it could be presented as a package. And the way of doing it would be by axiomatization, right? You have an axiom involving I.

40:00 You're assuming it denotes one of these, but that's all. Yeah, so I didn't have to. Yeah, you didn't have to do it the way. I mean, typically, I don't know actually how it's presented now. It's been a long time since I took the class. But I remember when I was introduced, whether it's part of the axiomization or whether it's brought in. It could be in any case. It could be either way. It doesn't matter. But you can axiomatize the complex numbers without I. I introduce I as a name, or you can do it with I, in which case you'll have a non-trivial onomorphism. If you do it with I, you won't have a non-trivial. Oh, you won't have a non-trivial, right, because the I would be in the same part of it, so that's right. Yeah, but you would have a non-trivial onomorphism if you forgot about the I, right? So, in a sense, the point is still there. Yeah. It doesn't matter. Well, the onomorphism, I mean, to be an isomorphism, that's to reserve the relations of the structure one of the relations is that that constant has to note the same thing. It'd be unlike, like in the natural numbers, zero always has to note the first thing in any two models, but yeah. Oh, sorry. Actually, it's a question for Richard. I think it's one of the difficult situations In this case, you have the actual geometry which determines your use, in particular use, of left and right. So this, I think that would be more akin to what Hannes was suggesting by introducing Einstein's expertisation of C. where you have you can't there are just two models of your language, left and right are in your language and there are two models that are two distinct models of your language rather than you using left and right as a free variable to range over either because you do have an attention yourself to what left refers to when I think, it refers to when I think But that's dissimilar to the case of I minus I, right?

42:30 Because there's nothing more that I'm thinking about I that distinguishes it, even in my own showing. I think that's helpful, but I did wonder if a human body part was okay. You don't have to use a human body part. You can just define a pair of an anti-morphs in a perfectly mathematical way. How many of you define the left? Yeah, it was. So, if it's okay to use a very hard way to screw on it, why isn't it okay to use a symbol in the case of that, is that really simple? So the suggestion is that you... It's the symbol, as opposed to the one. Well, because when you use this, you're somehow trying I think to refer to something in your that corresponds to this, whereas when you're using I, you're not trying to do anything like that. You believe when you use left that you're picking out something particular, and that if you could see, if I could, for instance, see your representation of these things, I would see whether it was the same one. But when I use I, there's no question of me worrying about whether you're referring to the same one, It's just a genuine free variable. I'm going to test any of those patients here. I suppose there's a platonic realm of this. Sure. And there's this platonic realm. And now there's my left hand. And I suppose, I'm not sure what sort of anti-rame, in-rame problem is what. I'd have to be very famous to it. But I suppose that by thinking of my left hand, I'm actually, there's a relationship with the platonic left hand. Right. it will say, whereas when I write the symbol I, that means there's no relationship with it at all. And you're comfortable with that, are you? Have I walked into a trap here? Yes, I think unless this has got a consequence on that. But yes, I think so. The thing that fixes it is the Anceli interpretation. Though it's hard to say. that fixes it. You'd have to give some. Yeah, yeah, exactly. People left on the list,

45:00 Oliver Pooley and then John. John gets the last word. I don't want to say anything. I just don't want John to say anything. I want to say something which probably is very simple, but I'm only moved to say it because you were saying, well, you think the structure is the same person, but the place of the structure is in particular, and you said things to suggest that maybe it was a bit of a mess, and the way I was thinking of it, it didn't seem to be a mess at all. Colors are universal, but we also can talk about colors in singular terms for colors and we can consider, you know, theorize about colors. Sure, the color between this and this, right, yeah. And it's not clear why exactly the same thing is going on when we, in singular terms, talk about numbers and we're thinking of them in the algebraic structure to terms. But if that's right, then it seems that you should also admit that the place in the structure isn't universal. It's part of universal. But when you think, so the structure is universal, which is exemplified by various concrete systems, but why not think that the particular relancer in the concrete systems are, when the system is exemplified by the structure universal, the places in the concrete structure are exemplified by the use. They're sitting in that slide, yeah. Yeah, I mean, I made that distinction in the book. There's two different ways of talking about structures. One of them is what I call places or opposites, which is where you're thinking of the places that's occupying and potentially being not occupied by objects. And then there's the places or objects perspective, where you're thinking of it as an object in its own way. And I think both perspectives are needed because sometimes we think of the structure itself as exemplifying that or other structures. So you say the even numbers exemplify the natural number structure, you sort of think of the numbers as objects, and then you take a system of part of them, and then you see what structure exemplifies them all, or it's the same one.

47:30 Yeah, so I'm comfortable with that, but that's sort of back and forth. Catherine, I think I'll just point with you that you shouldn't be both at once, so you should say that the places are of the same category as the structures themselves. but you can talk about the structure and if it takes in another sense to think of it as a university, you can't find them at the same time. Well, I've said there's two different perspectives. And it's important that you can suffer, because otherwise you start contradicting yourself. Now, when you say the even numbers exemplify the natural numbers, how can that be? So when the even numbers exemplify the natural numbers, four is sitting in a two spot. So you have to be able to have these sort of two ways of uh is that okay or like final question john well i i thought it was interesting that uh there is a kind of ironic uh conclusion to the debate because you and uh and richard seem to be heading in the same direction actually and and essentially i can describe the way the difference struck me It was just that you had this nice way of looking at how to treat I as if it were one of these kind of restricted variables or extra constants. And what Richard did was to point out that the structures themselves could be treated this way. Exactly. Okay, so let's suppose that we've reached this happy conclusion where everybody's agreeing on this. Then the question becomes, what function do these abstract or anti-rim structures perform? explain anything. Maybe we're just stuck with them because they're there. That's a long story. I've done epistemology. There's a lot more to philosophy than just philosophy or mathematics than just that. So both views have problems, explaining how mathematics is known and how the languages to be interpreted and the like. So what work is it doing? Well, can I just put in the last word? We're re-running the platonic thing about forms, right? And what you call anti-remonstructions or structures are like platonic forms. And classically, the problem was whether the form horse was a horse and so on. You grasp the nettle and say the form natural numbers.

50:00 Okay, you're saying, you're taking the view that the form horse is a horse. Well, not the form horse, not that one, but the form natural infrastructure is a natural infrastructure. That one is, yeah, but not horse. Okay, so all I'm saying is it looks to me like we're getting a rerun of something really old and reputable. So the old news part is, as long as it's not bad news. I've got some logistics of my own to sort out. But first of all, I'm going to say, can you not hear a total acquisition? You hear that? Thank you. That's one of the ones. Right, now, thanks, George. Oh, thanks. First of all, those of you who are speakers who are going to the hotel, I'm going to get a bunch of taxis to take you there and copy maps that will get you to the restaurant which isn't that far away. So you're all OK. who are coming to dinner, but aren't in the first category. Here's the department. If you go out of the department, it's a broad...