Identity indiscernibility & ante rem structuralism

0:00 Thank you, and thank you to OIC and James for organizing this. This is a wonderful conference. I look forward to hearing the rest of it, hearing the other papers. I often could stick a little epigraph on my papers in the chapters of the book and so on, but it just occurred to me that this is probably the first time that I've ever actually put an epigraph to a group by somebody younger than me. I have these people that are alive, like, you know, I've used 101 from church when he was still alive in his second neurologic book, and John White has in the Rolling Stones, but anyway, there's a passage in the forthcoming paper by Honest and James that I really like, and here it is. The suspicions of metaphysicians play much less heavily with us than the implications of mathematical practice. So I'm going in, well, I'm going in for these sort of half metaphysics, but realizing that I have to engage in it as well. All right. Now, my book, my 1997 book, Philosophy of Mathematics, Structure and Ontology, spurred a small but lively and interesting discussion concerning structures that have non-tributable automorphisms. We've heard a bit about some of them already. Part of the source of this is some slogans that were ill-chosen at best, It wasn't at best, and I'm like false at worst. On my view, mathematical armies are places and structures, and at least on one of the articulations of the view, these structures exist independently of any non-mathematical system in the example of mine. So I borrowed the term anti-rem to characterize the structures. So here's one of the embarrassing slogans. The essence of a natural number is its relations to other natural numbers. Number two, for example, is no more and no less than the second position, actually we already had this, than the second position, the natural number structure six is the sixth position. So this passage, and others like it, suggests that one can characterize each mathematical object uniquely in terms of the relations to other objects in the same structure. Actually, I even said that. So I'm embarrassed to say.

2:30 each office is characterised in terms of how its occupant remains to the occupants of the other offices in the structure. Now this seems to suggest a strong principle of the identity of individuals. Define a property to be structural if it can be defined in terms of relations with the human structure. For example, the property being a prime number is a structural property of arithmetic since it's definable in terms of addition and multiplication. The property of being the number of my daughters, or of being one of James Leidman's favorite numbers, is not a structure. The property of being the number of complex square roots of minus one is not a structural property of arithmetic, although it is one in complex analysis. So the suggested principle, one might think, is this thing that I'm calling IND down here. for any obvious x and y in the same structure, if x and y share all of their structural properties, then x is identical to y. Now, a structure is said to be rigid if the only anamorphism on it is the one based on the trivial identity function. Actually, you know, we've already been over this in the first five minutes. Now, arguably, this principle IND fails for structures that have that aren't rigid. Mike, pull it down a bit. Oh, yeah, I kept telling... You can't see. When you put it on, you can't see with boards. Yeah, no, that I can't even see unless I look at it. Mike, even when I move away, I don't know. So, all right, so structure is rigid if it only has one bottom office of the trivial one. And arguably, the IND principle fails for structures that aren't rigid. The most commonly cited example is in complex analysis. The function that takes a complex number, a plus bi, to its conjugate, a minus bi, is an automorphism. And it follows from that, just to take one example, that for any formula in the language phi x with only x3, phi of i, if only if phi of minus i. are indiscernible. They share all of their structural properties, at least in complex analysis. So, the anti-RAM structuralist is committed to holding that I equals minus I, contradiction.

5:00 Another off-site example is Euclidean space, where things are even worse. Any two points in Euclidean space can be connected with a rigid translation, which is an automorphism. So all of the points share all of their structural properties. Antigram structuralism was told that there's only one problem. Again, QED, or that is, QED, not QED, R. So if antigram structuralism did have this consequence, it would indeed be absurd. Now, in the chapter in my book on epistemology, I introduced these things that I call the finite cardinality structures. So the cardinal four structure, for example, for example, has four places and no relations. So it's a somewhat degenerate instance of a structure, but it's no less a structure for that. The cardinal force structure is exemplified, for example, by the system consisting of the legs of our lazy catagon. Now the cardinal force structure is the worst offender of INDs possible, the principle I have on the last slide. Since there are no relations to preserve, of every bijection of the domain is an onomorphism. Each of the four places is structurally discernible from the others, and yet, by definition, there are four, and not just one, such places. So, yeah. Now, I learned from, oh, may I sort of already do it, but I was reminded by Honest and James's paper that the four-cardinal structure actually is, or it's like a motion to a graph. And that's, of course, mathematical. Now, I would think that my original wholehearted acceptance of the finite particle structures is some evidence that I had no truck with principles like IND, despite the old chosen slogans scattered throughout the book that seem to suggest otherwise. All right, but let it be known that I now have no trucks with the identity of interturbals or with related principles like IND. Now, if we put aside the old chosen slogans the motivation for the identity discernibles and for IND strikes me as metaphysical Not that that's bad Now, after after noting that, so this is, you know for someone who actually doesn't seem to participate generally in metaphysical discussions, after noting

7:30 that the complex roots of minus one not distinguished, or quote, this is a quote from John Burgess's review, are not distinguished from each other by any algebraic properties. Burgess points out that on my view, the two roots are, quote, quote, are distinct, though there seems to be nothing to distinguish them. End quote. So the complaint here, right, that John made, seems to involve something like the principle of sufficient reason. If something is so, you know, i and minus i are different, or the two square roots are different, so if something is so, then there must be something that makes it so, or at least explain why it is so. Now, frankly, I'm not too sure what's being demanded here. The fact that it's a zero-much complex analysis that minus one has two distinct square roots seems to be enough to distinguish them. What else do you want? Concerning the more homogeneous cardinal force structure, what else is needed besides the stipulation that the structure has for distinct places? Luca Carinin, one of the fellows about the start, besides Ferguson-Helman, brings a lot of detail to the issue. He proposes, Carinin does, this is in his 2001 paper in the PM, I forgot the title of it. Originally, he had a rather old title like D&T Rem Structuralism is Dead. So why did he change his story? Why did he change the... Why did he change the term? I don't know. He got a little less old when he actually met us and started talking to us. So anyway, he proposed a general metaphysical thesis that anyone who puts forward a theory of a type of object account of how these objects are to be individuated. That's his word. So let L be a distinct linguistic practice, and let A be an object within the purview of L. The claim is that an account of individuation, can you pronounce this name, for L would specify, quote, the fact of the matter that makes each object A the object it is distinct from any other object in the purview of L by quote, provided a unique characterization thereof.

10:00 Now, in my structure and identity, my contribution to the first contribution to Fraser's line, I point out that if the individuation here is supposed to be done with a formula with only one free variable in a countable language, then hardly any mathematical theory can meet permanence requirement. There are too many objects and not enough formulas. You can get a little more mileage if you use sets of formulas, but even that's going to run out. The more natural route, and the one that Yuka takes, is to wax even more metaphysical and attempt the individuation by invoking properties and relations instead of formulas. Now, why think that the formulas of a given language exhaust the resources for our metaphysical tasks? Now, to go back to a topic that was raised in James' thought, if each mathematical object has a hexeity, a property that applies it alone, then the job of individuation is done trivially. And I think that's really where Yoko was going. I think he wanted to argue that mathematical objects have to have these hexeities. and therefore because hexeities aren't structural then structuralism is done for. Alright, now so the existence of a hexeity for an object of A provides the fact that A makes the object it is, distinct from any other because only A has that hexeity. The problem, of course, is that it's quite well a problem for me. For him, it's the feature. It's virtually analytic that SCDs are not structural properties, so that the anti-rep structuralist cannot invoke this trivial resolution and the individuation test. Now, some authors have sort of entered the tussle by proposing metaphysical principles that are weaker than permanence in the situation. But still we need Burgess's request that the theorist find something in which it's any distinct objects. And we've been over this already this morning, not this morning, two talks ago. The weakest of these is a requirement

12:30 that for any distinct objects A and B, there's the error of flexor relation R such that R, AB, R holds between AB. James had proposed that a few rounds ago in this debate. So complex and Euclidean geometry passed this test, thankfully, but the cardinal force structure and many graphs don't fail it unless non-identity counts as an irreflexive relation. Well, I mean, non-identity is an irreflexive relation, but because of whether it counts as one in this context. Now, among the works in print of which I'm aware, Only Jeffrey Ketlin's most recent well-fine trait yet. Of the works that are in print are forthcoming, of which I'm aware, only Jeffrey Ketlin and Mike Garvin Leideman agree with me that there is no individuation or distinguishing requirement at all, or at least none that requires the structuralist to provide anything beyond an ordinary mathematical axitization of the common structures. We may have reached a metaphysical standoff. one side, so this would be Karaman, Tim Button, and perhaps Burgess and Geoffrey Holland, invoke intuitions, often sort of called by thinking about ordinary physical objects, and formulate metaphysical principles inconsistent with some of the slogans of anti-run structuralism. The other side, me, ours, rejects the metaphysical principles in question in part because they conflict as interpreted via anti-ramp structuralism. Now, as I see it, the goal of mathematics, the goal of philosophy in mathematics is to interpret mathematics and to articulate its place in the overall intellectual enterprise. So saying something about its subject matter, its methodology, its applications, and so on. Now, as suggested in my instruction identity, my contribution to, one of the contributions to Frazier's recent volume, One desire of the enterprise is to write an interpretation that takes as much as possible of what mathematicians say about their subject as literally true, understood at or near face value. And I call this the faithfulness constraint. Now, to be sure, like this thing that I'm calling the faithfulness constraint is not absolute. It's one constraint among many.

15:00 Hellman, for example, I think, confidently argues that his modal reinterpretation of mathematics philosophical problems that literalists, like my president and me, must face. This, he argues, this Jeffrey argues, justifies a non-literal reading of mathematical language. As with much philosophy, if not science, ordinary thinking, it's a holistic enterprise. Metaphysical principles, I admit, are part of the mix as well, despite my occasional debachment. But they pay their dues like any other philosophical doctrine or theory. So, in the first instance, why should one think that such things as sex, natural numbers, real numbers, complex numbers, and Euclidean points are objects? Well, to make a very long story very short, this comes from the faithfulness constraint. At least that's a kind of facial reason. In the most straightforward regimentation of the theories, linguistic terms like 6, pi, 7 plus 4i, all function as singular terms. Faithfulness would at least suggest that they are singular terms, at least as a first guess. More importantly, I think, more importantly than that, I think the first order variables of the theories range over the respective objects. And I take it, some of the sentences in the theories are non-victuously true. Of course, all these conclusions are controversial. There are battles to be fought with reconstructive nominalists, like Helman and Chihara, who do not take the language at face value. And there are battles to be fought with fictionalists who deny the non-victuous truth of the sentences. And this is not to mention constructivists and anti-realists and so on. on other occasions on other grounds. For now, I'll just assume that the straightforward realism plus faithfulness leads to this thesis that says numbers of points exist. And they're in the range of the first order variables and true theories. So they're objects. So I'm comfortable in following the neologicists in taking the existence of singular terms and true sentences as implying, or at least strongly suggested that there are objects denoted by these terms, that each such term denotes an object. And at least front of facial, what looks like a singular term probably is one.

17:30 When it comes to ordinary objects, one might think there's some sort of a converse to this. That is, one might think that to be an object is to be the sort of thing that can be denoted by a singular term, at least for ordinary, medium-sized physical objects. Although it will be tedious and pointless, there seems to be nothing prevent us, that is, us, the community of language users, from giving names to every physical object, or at least to any given physical object we encounter. Anyone who wants to can go down to the beach and start giving names to the grains that Sandy finds there. You know, Sarah, Candela, John, right? And we can name the snowflakes of raindrops as they fall if we can work fast enough. However, I'm not sure to broach, at least broach it and get away from the other topic of this conference. I'm not sure if we can name each of the subacomic particles studying physics. At some point, our ability, even in principle, to single out individual specimens of some kinds of objects, glasses. To return to the matter at hand, I submit that the converse of the Neologesis Principle fails for mathematics. It's simply false that to be an object of mathematics is to be the sort of thing that can be picked out or denoted by a singular term. Even if we invoke the usual idealizations on lifetime and attention span, then there is a unique name for each natural number. But arithmetic here is the exception, not the rule. There is simply no naming any point in Euclidean space or any place in a finite structure and in some graphs, no matter how much we idealize on our abilities to pick things out. The objects are just too homogeneous for there to be a mechanism, even in principle, for singling out each place, as required for reference, as that relation is usually understood. But, again, stay tuned. We'll get to I minus I in a little while. Still, I would insist, real numbers, members of the entire hierarchy, points in Euclidean space and places in finite cardinal structures and graphs are legitimate objects.

20:00 They're in the range of the first order variables and coherent and thus true mathematical theories. For this reason, I would follow Quine and not the neologesis here, and take quantification of single ontology rather than reference. So reference is necessary but non-sufficient. Or, no, sufficient but non-necessary. One of those. Suppose solely for the sake of argument that mathematical entities such as sets, points, and numbers of all sorts are different in substantial ways from ordinary medium-sized physical objects, in interesting ways other than the fact that mathematical objects are not themselves ordinary medium-sized physical objects. One might want to deny the label object to the mathematical specimens, or else to suggest there's some ambiguity in good work in the number of times. But my preference is to keep the notion of object unequivocal between mathematical, scientific, and ordinary discourse, and to adjust any metaphysical principles about objects as such to suit them. All right, section two, the role of identity. Although this round, I think we've been over it. How's that? Good, all right. So now in the book, I wrote that, quote, Quine's thesis is that within a given theory, language, or framework, there should be definite criteria of identity among its objects. There is no reason for structuralism to be the single exception of this, end quote. Because that was an equal to me. Admittedly, this can be read as a demand for a non-trivial definition of the identity relation on the places of each structure. This pronouncement, plus the other slogan, is one of the embarrassing things. Plus the other slogan suggests that, like the identity in discernibles, which is above our a gentle heart. in discussing this embarrassing passage my contribution to Fraser's volume I said this so I'm talking about that so this is me talking about me talking about that passage

22:30 what I meant or think I meant or should have meant was that if we want to develop a theory of structures then there must be a determined identity relation between structures there is no room for a view that seriously as objects and leaves the identity relation between structures indeterminate. Surely the same goes for places within a given structure. When it comes to mathematical objects, places within a given structure, identity must be determined. Now, it doesn't fall from this that identity can be defined in any non-trivial way. I submit, as I was suggesting this morning, or not this morning, two talks ago in a conversation, that ordinary mathematical practice presupposes identity. And following faithfulness, the faithfulness constraint, anti-rep structuralism can follow suit. In a sense, identity can't be defined. Suppose first that we're working in a first-order language without identity. This is sort of the trivial, quite a logical point. Let E be a binary relation symbol that's supposed to represent identity. and let the gamma be any collection of formulas in the relevant language. If gamma has a model M at all, then it has models in which E is not identity. Indeed, for any part of kappa, there's a model M' equivalent to M, such that there are kappa many distinct objects in the domain of M' all of which bear E to each other. To use a language of metaphysics, but in a rigorous way, one just adds two pockets of the objects in the domain of the model. Now, of course, first-order logic with identity has a special logical symbol, which is required to be the identity relation in each model. That is, with first-order logic with identity, identity is presupposed. Now, in second-order logic, identity is sometimes defined as follows. Okay. The handout is really just the same as the slides, as you might have guessed right now, but I still need to change that because it's the only way I have to slow down. All right. All right, so A is identical with B, just in case for every NSXH and all the index B. Now, in standard semantics, the right-hand side of ID does, in fact, express the identity relation.

25:00 But this becomes on standard semantics that identity is presupposed there. To show that a formula, that the formula on the right for every x, x, a, if and only if x, b expresses identity, we consider a property or a set, usually a set, that holds an a alone. So if in fact it's a property that we're talking about here, then it's an xeity. but normally the standard semantics for secondary logic is done in set theory and are done with sets and then it's just a singleton set. So P holds of A and nothing else. In other words if B is different from A then not P of B. Now in non-standard or Hinkins semantics, the thing on the right for every X, X, A and only X, B is an equivalence relation but a D not B identity. the situation is exactly the same as the first order logic without identity. If M is any Hentgen model, then for any cardinal kappa, there's a Hentgen model m prime equivalent to M, such that there are kappa many distinct objects in the domain of m prime, such that if any B are two of those objects, then m prime satisfies for every XSA, and only if XB. Now, as with model theory, So with mathematical brackets. To pursue a point that Jeffrey Ketlin made, the identity relation is presupposed throughout the enterprise. In characterizing the rigid structure of the natural numbers, John said this this morning too. I don't know what he talks about, he said this morning. To characterize the natural numbers, we invoke a non-logical successor function. In calling it a function, we presuppose that in any intended interpretation, each number has a unique successor. That is, if B and C are both successors of A, then B is identical to C. There is simply no way to say that the successor relation is a function or that it's one-to-one without invoking an identity or something else that presupposes identity. The same thing comes up at the level of metamathematics. It's a theorem that there is only one group with five elements up by isomorphism. That means that for any two models of the theory, there is a one-to-one function from the domain of one onto the domain of the other that preserves the group operation,

27:30 which is itself as a function. If we're not entitled to the identity relation until we define it, and remember that we can't define it, we cannot even state that two groups are isomorphic. The practice in the algebraic tradition is to define a structure by giving axioms. These employ non-logical terminology appropriate to the structure plus a sign for identity. I think that's what John called the morphology. The latter, identity, that is, is the sign for identity, is just another non-logical term. It's more like a sign for conjunction or the sign for the universal quantifier. We insist or presuppose that A is identical to B is to hold an interpretation if and only if A and B are the same object. Now, one of the arguments in my first book, I sure brought along the, it's one of these epigraphs, but from church. But one of the main arguments in my first book, the book on second-order logic, is that an advocate of second-order languages with standard semantics presupposes no more than is involved in ordinary mathematical practice. Something analogous applies here. identity relation, we assume no more than is presupposed in ordinary mathematical practice. All right, now I'll move on to talk about I. So this section's called Speaking of I. I thought I'd give you this really pompous title like The Truth About I. Now, many years ago, little anecdote, right? who's probably, he might be the smartest person I know. But he's a brilliant computer engineer with a strong sense for philosophy and usually the patience to pursue it seriously. At the time, he was explaining the basics of complex analysis to his young son, another sort of budding prodigy, talking about, I mean, he was about like eight or nine. He's not my friend. So he was explaining the basics of complex analysis to the young son, talking about the various properties of the complex unit. So I interrupted and I asked him, I said, well, what is I? And he looked at me in a puzzling way as if we were setting a trap.

30:00 He answered tentatively, the square root of the minus one. Then I asked him, well, which square root? There are two of them. He thought about this for about three seconds, and responded, that's you, and went on talking to a son. Now, in the philosophy of mathematics, there's at least a potential problem with the language of complex analysis. The term i, little i, does function as a singular term. And of course, the linguistic role of a singular term is to be known or not. But, of course, the linguistic and mathematical communities have done nothing to single out a unique object for this term. They can't. As my friend's first answer indicates, the complex unit is often defined informally with a definite description. Let's say, i is the square root of minus one. I seem to remember one of my first teacher's assignments. It's the square root of minus one. Now, it's surely reasonable to insist that definite descriptions require uniqueness. Suppose, for example, that somebody said that she pitied the child of the father of this paper. As in some contextual parameters, something's gone wrong since the author has three children. If we follow Russell, what she said is false. and if we follow others, the attempted assertion suffers a failure of presupposition. So why doesn't something similar happen in complex analysis? Now, I just learned last week that this is actually an instance of an issue, an interesting issue in philosophy of language. It's called the problem of indistinguishable participants. That's the sort of thing that actually does happen in ordinary language all the time. So the standard example they use is something like this, that when two bishops meet each other, now when a bishop meets another bishop, he blesses him. That was it, right? So you can't really tell sort of who the, which bishop is blessing which, right? But that still sounds like a perfectly grammatical, ordinary English expression. Now, the solution to this, I think, is to rethink slightly the role of singular terms at least in mathematical language. A good model for this, for the reflinking,

32:30 is the role of parameters and the role of existential elimination and universal introduction in natural deduction systems. Suppose, for example, that someone's doing a derivation in, yeah, so, is in a natural deduction system and reaches a conclusion in the form exists at sphinx. resting on some premises or assumptions, depending on how you write that, depending on which book you use. Sometimes you have a little fish line here. Then, typically, the next step is to make an assumption, 5b, where b is a term that doesn't occur in the formula. Then she proceeds to deduce a formula psi in which b doesn't occur either, and which doesn't rest on any premises containing b other than the one we introduced, Y-A. Then she's entitled to discharge the assumption and then have psi rest on whatever exists X, Y, X, Y, S on it. Plus whatever other promises were invoked along the way. It's sort of standard stuff in natural deduction. Now, what about this term B? In some natural deduction systems, it's actually, well, I shouldn't say most, but most of the ones that I've used, all the ones I've used, The term B is supposed to be a constant. Sometimes it's a free variable. A few books actually have a special category for singular terms. They're used just for this purpose. They're called parameters. So it's either a free variable, a constant, or a new thing, a parameter in another category. In either case, though, b functions grammatically as a singular term in the derivation. That is, when you're going from 5b downward, it functions just like any other singular term. But it doesn't have the semantic role of a typical constant. That is, it's not a proper name. We don't designate a particular object T in the domain of discourse and say that BD knows T. Now, in actual reasoning, in mathematical or natural language, it would be strange, or suppose it's a singular term, which is called the most common thing. I think, you know, the Lennon book in 4 says it should be a singular, the sentence should be a constant, right? So most people tell you, most books that I use tell you to use a constant.

35:00 It would be strange, in actual reasoning, it would be strange to say the least to use an existing singular term in that rule of B. Suppose, for example, that I was doing a deduction in real analysis and I got to a conclusion in this form, this explanation. And then I noticed that the term zero doesn't occur at any of the undisgarged premises of my deduction. So then I say, okay, assume phi is zero. now if this fits the letter of the rule for invoking existential elimination, then it'd be infelicitous or, well, stupid the term zero already has a role in the language to denote zero and it may well mean that flying doesn't hold at zero now similarly, suppose I'm thinking about basketball and I get to an intermediate conclusion that someone will be traded next month It would be weird to assume that LeBron James will be treated to start the process of existential elimination. But what's usually done, in mathematics at least, is to introduce an unused letter just for this purpose. You know, we'll say, suppose 5n. At some point, you might remind the reader that n is arbitrary. And 0 and LeBron James are anything but arbitrary. Especially LeBron James, he's really special. to better model how the inference is deployed to better model how the inference rule is deployed in real life it might be better to follow some authors and introduce a new category of singular term for use and existential elimination and while we're at it also universal introduction because that's very similar that seems so these items have the syntax of singular terms, the grammar of singular terms, but they have different and semantic roles and constants and variables do. In some ways, parameters function as variables. In others, they function as constants. In the case of existential elimination, for example, we have it that some object in the domain satisfies fun. The role of the term B is to denote one such object. The rules of engagement require the reasoner to avoid saying anything about B that does not hold That is, the B is supposed to be arbitrary.

37:30 Now, this is a new proposal. Actually, I have a sort of vague memory of seeing this, but I'm too old and I have no idea remembering where. But if you do have parameters available, there's no need to require the introduced formula phi b to be an assumption, to be discharged later, right? So typically, right, you put that down and you say, assume phi b, you know, you're doing natural deduction. But there's really no reason to think of it as an assumption. The rule of existential elimination could be simplified as follows. And again, I remember a book, seeing a logic book that did this, but I have no idea where or when. So I could just be making it up, it could be wishful thinking. But here's the rule. From a formula in the form exists that's phi x, one can infer phi b, provided that b does not occur previously in the deduction. That would mean you wouldn't have to discharge it later. You wouldn't have to go back and write that phi. That's essential elimination, at least in my experience. It's one of the hardest rules to teach. I mean, this is in fact how you do it when you're using tableau proofs. It's exactly that. Oh, that's where I saw it. All right. All right, so the inferred formula, so it would be taking the tableau method and putting it back into natural conduction. The inferred formula, 5B rests on whatever premise is an assumption the existential plane rests on. The system would still be sound, although admittedly the model theory might get a bit trick. Now, I think this version of the rule fits mathematical practice better This one never really explicitly notes the discharge of the premise 5b anyway. Right, now we need just one more step, and then we're home. Now we let b stay in the language permanently, continuing to play the same role. This, I submit, is a rational reconstruction of the actual use of i and minus i in complex analysis. In the algebraic closure, John Burgess once suggested something like this to me, and it may even be exactly the same idea. in the algebraic closure of the real numbers we conclude that there is at least one square root of minus 1 and so we have x x squared equals minus 1 so we let i be one such square root and go on from there we have an i squared equals minus 1

40:00 we might note in line with existential elimination that there is nothing to be said about i that does not hold up every square root of minus 1 this is as it should be are insurmountable. Now, it seems that, to me, that the community of language users has gone on to allow slight abuse of language, if it is an abuse, remember the problem of the distinguishable participants, has gone to allow slight abuse of language and declare that i is the square root of minus one, you know, just what my friend said. They also say that minus i is the other square root of minus one. Notice, incidentally, that notice, by the way, that the two square roots of minus 2i are 1 minus i and minus 1 plus i. And as far as I know, no one has been tempted to call one of those the square root of minus 2i. And the other one, the other square root. Alright, so that's the end of that. topic, where I'm going to stick my neck out with all these metaphysicians around here. So metaphysics, universes are particulars. So just what is an ancient structure, and how can objects be constituted by one? Now consider another one of the uncomfortable slogans that I proposed in the book. one we just got from, oh, I see. Structures are prior to places in the same sense that any organization is prior to the offices that comes through it. The natural number structure is prior to six, just as baseball defense is prior to shortstop. I know I shouldn't use that example, but. Or US government is prior to vice president. I had wrote the book before I started talking to it. You know, I realized it was part of the world that doesn't know what baseball is. I also said many times that the individual natural numbers the places in the structure are not independent of each other the problem is to articulate this notion of dependence now in his contribution to this event Oystein makes an interesting proposal so let me try my hand in one as well now first of all you know you kind of do

42:30 a little warm up in philosophy you say what you're not doing The priority or dependence invoked here is not epistemic and it's not dog-sastic. It doesn't have anything to do with how structures in their objects are known or how they're learned. It's a metaphysical dependence and so, alas, I have to know why it's metaphysical. Now one fairly obvious way to understand priority is in terms of essence or possible existence. To say that an object that A is prior to B is to say that B could not exist without A. Or actually, in O.S. King's talk, we've already gone through this and shown why this isn't going to work anymore. Or to say that the A's and B's are not independent of each other is to say that there cannot be A's without B's and or vice versa. Now, despite the proposal that I approached tentatively in structure and identity in Fraser's volume, This clearly will not do here, since mathematical objects exist of necessity if they exist at all. So there is no sense of the natural number structure existing without its places, nor vice versa for that matter. Nor is there any sense of the number four existing without the number six, or so say our intuitions. Now, speaking of Fraser, he argues that, his sort of famous conclusion here, is that anti-mermstructuralism is either bad news, if committed to the identity of indiscernibles, or old news, it's just traditional Platonism. So bad news, identity of indiscernibles, or old news, Platonism. Now, I take it that the charge of bad news is addressed above and dismissed forever. Now, as for being old news, I did concede in the book that anti-repstructionalism may be a variant of traditional Platonism, at least concerning its ontology. It may be only a matter of emphasis. On almost all views, in line with what I call realism and ontology, natural numbers exist. and so the natural numbers form a system under the usual aerodynamic relations the structure of that system is the natural number structure what if anything is distinctive about anti-rem structuralism other than its focus on the structures rather than the individual objects now I've suggested previously and I've taken some flight for this and I'm sure I'm about to take some more that an anti-rem structure is a type of anti-rem universal

45:00 you know, the name I got from, in that it's a one over many. The same structure can be exemplified in multiple systems. The difference between structures and the more usual kind of universal, such as properties, is that structures are the forms not of individual objects, but of systems, collections of objects organized with certain relations. So let's consider a simple example, the ordinal three structure. This is the form of any system of three objects So a categorical accusation is straightforward, just one of the rigid ones, you know, so we're not invoking the other problem. So the ordinal free structure has three places, each of which is filled by an object of a system that exemplifies. One such system is my children in birth order, all this to be honest. So Rachel occupies the first place in the structure, Yona occupies the second, and Aviva the third. In this system, the requisite relation is that of older than. The places of an anti-arm structure are kind of offices, which are occupied by various objects and various exemplifications of the structure. And I call this the places or offices perspective in the book. So this is where you're looking at a structure as the sort of thing that can be exemplified in different systems. The metaphysical innovation, if you want to call it that, if you want to call it metaphysical, if you want to call it an innovation, is that there is a legitimate perspective from which one can construe the places of an 18-run structure as on-flight objects. And I call this the places or offices orientation. In metaphysical jargon, although structures are universal, the places in them are particulars. I know this is going to strike any metaphysician as sort of greedy. All right, so the structures are universals, the places in them are particulars, or can be construed as such. That is, the places are not, as Fraser suggests, bundles of universals. Places are constituents of universals. So why think that inherent parts of universals can be particulars? Surely this runs against metaphysical tradition. justification for my claims is the rather humdrum one given above. There is a perspective from which genuine singular terms denote places of anti-run structures, and more importantly, places are in the range of bound variables and true theories. The perspective in question

47:30 is that of pure mathematics, at least as interpreted here, well, in the end.