Putnam's Paradox

0:00 Okay, so I'm going to start off just with a quick review, really historically, something from Quine, and a bit on the way Puckman presented it through Lewis's mouth. Then I'll say a bit about how I got involved with the whole topic, which I think leads to a slightly different slouchment, and that takes me to the whole argument and a collection of questions there about does the whole argument bear on public's paranoia or not. Now I did think to, I worked on your papers best on this, and I thought perhaps I should and I thought, well, that's not very nice. Chaps just arrived. I'm not prepared. That's right. Chaps just arrived, and, you know, we have all these questions. So I didn't do that, actually. But I can do that next week. But I've learned it up from something. Okay, this past week, your thinking is very different from mine. I'm wondering if you have some pretty straightforward reasons why my thinking is on the wrong lines, and then I'd like to try and wrap it around. Anyway, so, and we'll get through these, I've just brought them a few to have a little bit of text to look at. The first one, this is just from Pursuit of Truth, because in Quine's hands, in Scrutability, what was going right back to the word and object, and it was something to do with his thesis of indeterminacy of translation, which is rather important that whilst you had sentences translate holophrastically in these holes, if you try to break them down into terms, where the terms are referring in it, then you get into problems and so-called analytic apophyses and so forth. So Gavagai was his great example, not of intertermistic translation, of course, but of intertermistic reference and scrutability reference. So he was talking about it much later, and I think there is a real question about what Quine proceeded I think it's clear, as it were, not good. But come ontological relativity, and what he's doing here in the pursuit of truth, I think it's really a bit puzzling what he's saying. So here's an example of what he says. There's a rabbit, that's Gavagai,

2:30 and it remains key to the sensory stimulations by which we learned it, even if we reinterpret the term rabbit as denoting cosmic complements The term does continue to conjure up visions appropriate to the observation sentence to which the term was learned, and so did it, but there is no empirical bar to the reinterpretation. The original sensory associations were indispensable genetically in generating the nodes by which we structure our theory of the world. The thought that matters by way of evidence to the theory is the stimulatory basis of the observation sentences plus the structure that the neutral nodes serve to implement. The stimulation remains as rapid as ever, but the corresponding node or object goes neutral in the result of the grads. OK, now, I've marked a passage on the other side of the page about Bose-Einstein and Fermi and some Bose-Einstein and Fermi should have certainly been a rack. I haven't noticed that before. It's been Einstein for a long time, so I haven't noticed that before. But I'll be coming, actually, because I am talking on Monday from the show, and suggesting take Catherine's place, because Catherine Brady, unfortunately, is going to be unable to come. So I thought I'd talk on quantum classical statistics issues, which is actually related. And there he's saying that what happens there may force us to give up particles, and actually numbers as values of variables all together. So I'm just very much denied that that's required as of Monday, just to note the two passages are rather side-by-side here. Now, Quine regards this claim about its credibility reference to being trivially proven with elementary model theory just to take this into space and put a one-place predicate the name, so this is on the side of syntax, then there'll be some interpretation for this language, some model for it, whereby F of sigma is some set, and if A of sigma under the interpretation of sigma, if it is contained within it, then F of A is true, so this is elementary semantics, but now one notes that under any permutation of the universe of discourse here, f, then one induces a new, I'll call it f-star sigma, a new interpretation, such

5:00 that a f-star sigma belongs to f-star sigma, if and only if a f-star sigma belongs to f-under-signal. Okay, and the point here is, what is A of F-star sigma? Answer, it equals F of A of sigma. Okay, so A under the original interpretation signal was some individual in the universe of discourse, and F of that individual is what the name is mapped onto under the new interpretation generated, as it were, by F and from the original one. Okay, so Quine calls these f's proxy functions, and the point is here, it's because one reinterprets f to be that predicate that's true of all of the things that, that is true of f of the f of the things that f was previously true of. So actually, I think that's fair enough, and quite in regards, it's trivial, and he's never made a big deal about it, although in his logical relativity he did. He made something of a big deal about it in the context of inter-termist translation to serve that thesis of inter-termist translation. Now, of course, Putnam came along and made a huge deal out of it, apparently without realising that it was more or less what Quine had been saying. Now, Putnam did present it in terms of Lanham Stolen, and indeed Quine and also for Relativity then proceeded to say quite a bit about Lanham Stolen. So I think that these are two slightly different things in itself, and I would like to focus in the skolan business and i think requiring when he wrote the preface to was focusing rather on this simple argument rather than um and the papers that he came back to answer i think in 94 and again it was this that was at the focus um so i made it wrong

7:30 I think one can usefully detach this from issues to do with Lernheim-Skobin and try to get a different mix. I think if you've got an infinite model, then two are sort of similar. I think that Luna Fallon is only used to show that you don't have to worry about differences of the internet quite an hour. Yeah. And that's . I don't think it does much more than that. That's right. Okay. Now, Pudden didn't quite present it like this, although I think one can fairly quickly get more as to what he had to say about it his point was what fixes the intended reference because it's first thing was intended and then you know you have these induced interpretations and then the idea is what you put a lot more here you put in everything here and it's all more theory and you still can run the same permutation argument and still end up with unintended interpretations so the theory is true but it's what makes it true and what we're putting it is fa you know f is the uh is on the mat you know a is the cat so the cat is on the mat but what makes it true while you commute and you find out that it's because a cherry is on a tree that the cat is on the mat is true that somehow what makes what we say true can be completely unrelated to what we're talking about um but then another way of putting it is that you can have all the theory you like as long as the model has got the right cardinality there will be some assignment of sets relations in the model that will make true enough to the theory so you know truth three you always have as long as you've got consistency i think that's just another way of saying the same thing um and then no nice golden strengthens that i guess to say that you don't have to worry about

10:00 okay um lewis's way of putting it is a little bit a little bit different um and here i just got the page out of Lewis's paper top left he's talking about global descriptivism and I think well there's quite a lot that goes into this local descriptivism starts off with local description that just sounds like empirical linguistics, really, but then Lewis pushes it to say we're in the limit, you know, one doesn't just aggregate familiar theories, one just goes to totality of knowledge, total science, and to call that global descriptivism. And I think what Bass has said was right about this. Already the problem is there with local descriptivism, and going to global descriptivism doesn't really change it very much and already in the global scriptivism we've got what the past was anti-nominalism. Now, okay, but let's just see how Lewis presents it. I've marked the left-hand central passage. Well, Lewis says that global scriptivism forces a paradox, and so here he's should descriptivism as false, or Putnam's incredible thesis is true, or there is something wrong with his oppositions of our whole line of prose. Unlike Putnam, I resolutely eliminate the second and third alternatives that one remains must therefore be the truth, and the descriptivism stands refusing. It may be part of the truth about reference, but it cannot be the whole story. There must be some additional constraint of reference, some constraint of the might that we are unlucky in our theorizing to eliminate all the allegedly intended interpretations and then a lot of people would be a bit flummoxed as to what Lewis meant by having something more to reference because what Lewis goes on to say is that there are elite properties that somehow

12:30 there's a business of the world carving the world up at the joints there are these special properties and relations and and then one feels like saying well so what you know how does that help us in getting to them in the language and Lewis hasn't really made very clear how we're we're supposed to get, and then somehow the world forces them upon us or something, and then various people accuse Lewis of speaking mystically or something, and come and respond with a thinking print set to Lewis, and then Lewis counters, like saying, come on, form the names if you want, but, and I for one actually lose grip a little bit on what to think about what Lewis is saying. Well, excepting that I do have my own way of arriving at this whole problem, and then I seem to find myself a little bit in Lewis's ballpark, so let me just say a bit about how I arrive at this problem and then go on to the whole thing. I arrived at this problem because I started off for unrelated reasons, well, for reasons that came from quantum mechanics, and strangely got me to Kant's and Kongo counterparts, And from that I got into issues of identity and discernibles, and that led me to think that we, all of us, have been rather unappreciative of just how powerful the principle of identity and discernibles is, the correct principle, being the only logically viable principle which Hebel and Benayes introduced in the 1930s and Quine then made something of in the 1960s and that he might be greatly ignored. Now, I think everybody here knows a bit about this, which for another reason I've spoken about it on two or three occasions, so let me not go on about it too much. But what it led me to the conviction of is that the arguments that most philosophers have taken to show that one can't describe, there are problem situations that one can't describe, like Black's two spheres of iron, or Strawson's checkerboard, or various, supposedly, problem

15:00 situations. It seems to me that one can describe them, and one's led to this business of group discernibility, perhaps relative discernibility, and what most philosophers have been stated about is absolute discernibility which is a very important logical category of reference but it's only one and in many ways the more interesting categories of reference are to do with relative discernibility and these are to do with where one has situations where one's got some complex predicate in say particular two free variables that apply to two objects where this is near effects of connection okay and of course what could be concealed in this is all kinds of qualifiers it could be referenced by bound qualification to a whole lot of things this could be highly complex perfectly but if there exists such a predicate which is irreflexive then you've got discernibility now if an addition is asymmetric If, on the other hand, you've got a one-place predicament that applies to one out of a collection and to no other, then you have absolute discernibility. The point I want to make here is that the issue about not being able to refer to weak one out of two weakly discernible objects is that this doesn't really depend that much on issues of identity. You could just use the equality sign or rather negate it. And this would do just as well, except, of course, you might be asked, why would you write that down? But the point is you can write it down. And in terms of the expressive ability of the predicate calculus, you will have many situations the fact that there's a plurality, but you can't refer to any element of the plurality uniquely. Okay, so this point about the kind of inability to refer to one out of the collection has nothing to do, I think, with the principle of identity in the syllables. It's just built into the calculus, and again, it's been rather ignored. Anyway, so what all of this led me to was a general take on description.

17:30 The physics puts us in a position to provide descriptions of the world in terms of objects. And it helps to formalize as far as questions of objects when we are debated. We do have, I think, a great achievement with Frager, Russell, Quine, that whole line in terms of what is ontological equivalent. It's appropriate to use that framework for resolving disputes about what would be the critical anthropology of the theory. Of course, it's very vocabulary dependent, and here the suggestion is that the right predicates and ones that are invariant under the physical theory that one is looking at. And then it's a matter of applying that in various cases, turning gravity, special relativity, general relativity, classical statistical mechanics, problems of statistical mechanics, you name it, and that's what I've done. And it seems to deliver very satisfactory research repeatedly in the case. And it seems that people describe the world as we would wish it to be described, with no sceptical challenges arising and so forth, although I think that was a slight modification to make about that. And in the course of all of that, of course, in looking at the progression through returning gravity special relativity to general relativity, one is led, among other things, to the whole arguments and one I think that's something very remarkable in fact the entire history very remarkable and it's to do really with the history of the variable and it's not really what one thinks that's that's the point there's a a naive take on it, which most physicists will buy into, but eventually they get involved with subtle enough questions they may have to deal with at some point or another, and I think Newton had to deal with it quite early on, and so forth. Okay, but now let me just approach this bit by bit, because I do think there's a lesson

20:00 for semantics, generally, and that's the question that I have about Putnam's Parallels. Well, actually, I'll tell you all, before I get started on the whole argument, let me just say this. If you are going to base the principle of identity in certain ways, it seems to me really that there's a bit of a difference to Putnam's Parallels, because what it's saying is that You don't individuate concepts by the extension, not if you are in the business of trying to determine what objects are in terms of predicates. So why would that matter? No, no, no. How do you derive it? Well, if one's using predicates to determine whether one's got what are the objects of would be those which are, on the predicated basis, discernible, then I don't think you can start off thinking that the predicates are extensionally defined by whatever objects there are. Now, that might be a red herring. I don't know if that's a red herring or not. Because perhaps that only speaks to, I don't know, the context of discovery or something, of justification. But it does seem to me that one's involved in a slightly different point of view than traditional model theory. And if you think of the objects as values of themselves characterised by concatenations of predicates, then it's pretty clear that you can't put the objects around without carrying the predicates with them. You're in a situation where it seems to me it's plausible that these sorts of permutations of the interpretation, the universal interpretation, do not represent distinct states and affairs. So that's a general thought that just comes out of using a principle of identity in syllables.

22:30 And I'm not sure if it's really hearing or not, but anyway, it certainly links up with what is the lesson of the whole argument, so let's come to that. Now, the first person to suggest that the whole lot of them could bear on France Paradox, I think it was Tim Mordland, but that was known. And I've just got a page here from his paper, which is in the essence of space-time, which is sorry, it's three years. he talks about the random line solution to the whole dilemma so perhaps I want to go through that a lot so the whole argument is this if you've got some manifold here's a point here on the manifold and suppose you've got a make a scalar field and everything becomes totally elementary you've got some scalar field on the manifold and it has some value at the point P suppose I have a different autism on this manifold which is a map which is differentiable in fact then, and let me denote that age, ok, so that's a map from the manifold onto itself Now, I can define a new field. Let me denote it H star phi. You can see why I was using the star notation here. I submit the new field as taking values on Hp equal to phi on P. okay so this thing is a new physical field on the manifold takes different values at the same manifold point whereas Phi on the point P to be like three H star five on that very point takes a different value that Phi took on H so now you've got a new field and you do that for all the fields in the manifold Now suppose you've got a set of differential equations which these fields are supposed to satisfy. I suppose that set of differential equations is generally covariant, meaning that if phi, and then I could write a lot of other fields down, but let me not bother, is a solution.

25:00 set of equations. If this is the case, then they're generally the moon. And then that looks crazy because what's happened, you can imagine diffeomorphism that is the identity everywhere on the manifold except in the little region here. Everywhere else it's the identity. But in here it's not the identity. And now if this This is true, that Fries is a solution and NH star Fries is a solution, and that means that I can fix the values of the fields everywhere in the past that this inner region is known for, and then by choosing my diffeomorphism, I can make the fields any value, I can give them any value I like in this little region here. So a theory which is generally covariant is somehow deeply unacceptable, physically speaking, the Cauchy problem is not well posed, you can fix all the data you like in the past, and anything can happen in the future. Now, there's a Ramsey-fine solution to this. So says Morgan, I think it's really true. Ramsey-fine isn't quite the right word for it. Of course, the Ramsey sentence does get mixed up on the paradox in various ways. I think there's a solution in terms of using bound variables. I think that's really what we're getting at here. So that's not enough either. I don't have the first page, I'm allowed to have it. Let me just say what the solution is. The solution is to take all of these fields, these solutions, these diffeomorphic solutions, as a good one. And I want to go into that in a little bit more than a minute. Make the equivalent in what sense? Right, that's why I want to turn them in, absolutely, because it's not pure at all. But in some sense, one takes these classes of solutions as an equivalent class.

27:30 Now, what Mordorin points out here is that's, well, Thornton says that's not acceptable. It can only come about if you were going to, if you were going to not have names P for points of the manifold, you were just going to have some expression like this, this, this, and so forth. I think this is all a little bit mistaken, so I don't want to go into it. But what he then goes into is that the substantivalist is going to resist this move to an equivalence class. Substantivalist is going to insist there is a factor of the matter as to what the value of the field is at a point key of the menatron. And then the comparison is with Uttman's paradox, and that's the bottom left. To take a very general example, you can see that any theory according to which both I am the Eiffel Tower and name substances, there will be some maximum description. description given in the vocabulary for theory. If the theory is true, this description will constitute a model of the theory. Now consider the description results from this one if we exchange the terms named Tim Malvin and Eiffel Tower, wherever they occur. The new description contains such sentences as T.M. was constructed in 1889, scandalizing Corinthians, blah blah blah. There is a passive interpretation of this new description which constitutes a case of really renaming the Eiffel Tower and applying different conventional linguistics to these objects. In the passive interpretation, the two descriptions have the same content. But we are interested instead in the active interpretation, according to which the new description is just plain false. In the active interpretation, Eiffel Tower refers to the very same objects and various descriptions, and since the Eiffel Tower is not five-delayed, it is not true. Okay, so he develops this point fully, and let me not go into the details of it, because it will take more time than I've got. But, well, just to comment the bottom right-hand penultimate parallel today, he says, if the whole dilemma is generated by such a broad metaphysical picture of substances and attitudes, it should be counted equally general terms. The more traditional metaphysical response immediately presents itself to oppose the doctrine of fair particulars and excellence, namely essentialism. So what Maudlin is tracing this through to is, look, if you're going to consider some bare particular underlying all the properties that the object might have, and permute the bare particular, you're going to be led to a situation where somewhere in the ballpark of that is parallel, and one needs to oppose that.

30:00 and the right way to propose is essentialism and essentialism is somehow not going to allow us to detach the object from certain of its properties okay so there was i think the first person really to to make a connection with I think I haven't probably stated it in as clear a way as it should be said, and that's probably because Morgan doesn't say it that clearly. Let me say it a bit more clearly just in terms of the mathematics of it or the formal structure of it. Let's see. I think it comes out best, actually, in a coordinate-dependent formulation, where, let Now, under a coordinate system phi, sorry, x is going to play much the same role as an interpretation and under that coordinate mapping really of fields on the manifold on two functions in our form the case whereas before and the signal we had a map from syntax under some set you universe of interpretation and a collection of subsets of that set, and then one-place predicates can map onto subsets, two-place onto a set of all the pairs, three-place predicates onto, etc. Now, instead we're mapping functions in the manifold onto functions in R4. I'll put a subscript X, and the condition is that if I can put this is to mean that the scale of field five the point P takes value of Delta then this will go over to scale of field five such with X meaning that the coordinate and scale at the point X of P takes value in depth I assume Delta doesn't change okay now now I can apply again this diffing organism on the manifold and I

32:30 get a new scalar field h star pi and likewise here I'll get h star let me call it x star x indeed, which will take values in the point x of h of p, so this will be true just when this is true, okay. Phi in the coordinate system h star of x at the point x of h of p will lie in delta just when this is 1 to 2 to 1 to 2 to delta. But the thing exactly parallels what goes on with the form of syntax, it really is identical. And I think that's the reason to think that we ought to learn something from the whole argument vis-a-vis paradox, trying to set these two things up side by side, and they look formally identical. And that, of course, makes one quite interested in what is the correct solution for the whole argument. And I said that it's to do with taking the equivalence class. Now, let's look at just what that means. Here's Einstein. On this first clear statement of what it means, Einstein himself was thoroughly confused about this for two or three years, and it served him up in discovering the equations of GR until 1915. more or less happened in 1912, but then he got confused about all of these points. And this quotation answers from Arthur Van Norton. Let me just read it out. This requirement of general covariance, which takes away from space and time the last general of physical objectivity is unnatural, will be seen from the following reflection. All our space-time verifications invariably amount to the determination of space-time coincidences. If, for example, events consisted of the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points. Moreover, the results of our measurings are nothing but verifications of such meetings of material points of our measuring instruments with other material points, coincidences between the hands of the clock and the points in the clock tower, and the observable events happening in the same place at the same time.

35:00 The introduction of a system of reference serves no other purpose than to facilitate of the description of the totality of such coincidences. We allot to the universe four spacetime variables in such a way that for every point event there is a corresponding system of values of these of the variables. To two coincident point events there corresponds one system of values of the variables, i.e. coincidences characterized by the identities of the variables. If in place of the variable is x1 to x0, we introduce functions of them as a new system of values. So systems of values are made to correspond to one another without ambiguity. The equality of all four coordinates in the new system also serves as an expression for the space-time convergence of the two points. As all our physical experience can be ultimately reduced to such clear senses, there is no immediate reason for preferring certain systems of coordinates to others. as opposed to sort of re-righted as a requirement of general primerance. Now, there's a couple of things to say about this. One is that Einstein makes some pretty striking claims, you know, maybe, kind of, the first sentence, particularly in Vars, where I'm not talking about this, and related, he somehow demonstrates general primerance as some sort of a priori or any way, the only empirically proper thing to do, because all we have available, for instance, is worse than it. So he's making some rather strong epistemic things, and drawing some pretty sweeping metaphysical things from it. And that's what everybody's focused on when they look for this passage, which is what Morton presents. Presented this form, young, has little force. Its conclusion is not the most interesting kind of a passage, which is that space and time have lost the last round of physical objectivity. Fein rightly calls it a suspicious-looking verification as Darwin. So what Norton is saying there is fairly typical and I think it's missing the point. And look at what Einstein spends his time trying to explain. He's trying to explain how to think in terms such that P is not physically meaningful. He's really trying to explain how to work with this extraordinary situation where values of field-equals isn't physically meaningful. And what Manchin's

37:30 talking about is that it's the value, what you can say is that phi has value in delta where some other Psi has value in, well, it could be something else altogether, in O, you know, in the sigma, where some fuel Chi has got value in, you know, O and the go, and so forth. But that's all, those are the only sorts of statements that one can make. That's what Einstein is trying to explain in this answer. And it's, well, I think it's quite a radical constraint on how one expresses oneself. Coincidence is like it's a diffeomorphic invariant. Okay, if it's true that psi takes value, and I'm going to go where psi takes value, and that remains true, and there will be different models. Okay. And what we have to understand from the whole argument, and this is actually, I think physicists agree on this, so I don't even know what to be questioning it actually, I really don't. It is diffeomorphic equivalent models, a model that can be mapped into another by a diffeomorphism. Our equivalent is that diffeomorphic invariant structure is known what is used to describe things from reality. And that diffeomorphic invariant structure is to be described in the ways that I've just said, where one says things like where the phi field has values and delta and the psi fields have values and sigma. And this is, I just come back to it, it's a very extraordinary kind of discipline if you like, it's very different from what we used to. I don't really know what to call it. I mean I'm tempted to call it, it's a bit more like an intentional language. I'm not sure if that's right to call it an intentional language. But how do you decide which of these fields is mass, for example? Well, sure. I mean, that's, sure. But do you have a question? I'm not saying this is unproblematic, but, you know, I think that's a different sort of problem. It sounds like a very good point to me, the point you just made. Well, I think it's a point that's got strangely ignored in all of the huge literature on the whole of what people have not looked much at, is what's really involved in embracing

40:00 the physicist's solution of it. And I think it's not very different, and it seems to me that when one says things like this, that we feel, this is, the Chifo taken values predicate and at the point P is a bit like an object and what one has the mode of discussion of talk here is that that which satisfies one predicate satisfies another predicate that's the canonical form of description if the whole argument is must take this as our model and if you're going to talk about physical objects like that, that which is a C is also a G, or that which is an F is also an H, then I'm not sure if Patton's paradox has the same significance. I'll try to say a bit more about that in a few minutes, okay, but that's the thought that I had. And then another thought is, well, there's several things to say at this point. One is that the entire literature in the classroom space and time about this has pursued a lot of questions that I think are a bit tangential to the real significance of the whole argument. One of them is so-called substantival solutions to the whole argument, which more or less rejects what I think physicists are saying. And I think that's doomed. I mean, I'm sorry. I just don't think philosophers are going to be in a position of finding some meaningful understanding of mathematics here that physicists are rejecting. And Maudlin, for all its brilliance, and very much, but he seems to have the view, and I don't entirely understand it, that a unique diffeomorphic model, and unique model is somehow correct, and all diffeomorphic models are not correct, and this is somehow forever unavailable to us to know which is the correct and unique model, and so I just find that a bit mystified, I don't know how he can be even claiming that. went into this sort of issue, Jeremy Butterfield, another person, spent a lot of time along similar

42:30 lines, which I think are rather fruitless. But even someone like Robert Menasiewicz, who in other ways I think has said some very cute things about this, seems to have a problem with what I'm calling the physicist's solution. And another kind of issue that's come here people anyway say that whatever the whole argument is telling us it's not a general truth about physics it's rather specific to general relativity it has no bearing at all on on the traditional philosophical debate in space-time it goes back to and the usual accountants given there and modeling for example gives it is that you can't define whole difference in Newtonian gravity, instead of, this isn't the symmetry of Newtonian gravity that defies a solution of traditional things, but rather it defies a solution than the Galilean transformers. And Galilean transformations do not include things that have the identity everywhere and, you know, on the tribulation of the region. And so, therefore, it doesn't apply to Newtonian gravity, nor does it apply to special relativity and something very special and very specific to general relativity. And they use that to sort of compartmentalize the thing, not allow it to have any philosophical lessons which go back to things like Leibniz and so forth. I didn't think that's right. And just to give an example of how it's not right is Newton's theory of gravity, the symmetry group isn't really the Galilean group, it's an algebra group, and it's important for Newton to use an algebra group, the so-called Galilean Newtonian group, that's phrased due to Ellers, This is a vectorial plus T, narrowly, plus an arbitrary function of time, totally arbitrary

45:00 This is rotation, it's going to be a tiny panel. Sorry. Thank you. Well, I can take different planes at different times. But this thing is really amazing, because what it corresponds to is the absolute accelerations in Melbourne. And Newton needed this because he wanted his system of mechanics to apply to objects in a uniform gravitational field, which is really what this is giving him. He needed that because he wanted to treat the moon as a GIFT. and then you realize that, well, if you look at the difference equations for Newton's equations, look at the differences, so you write down Newton's equations and then you go to x double dot i minus x double dot j for two particles. And then you find those difference equations are invariant under this. and what this allows you to do is exactly to set up the whole of you and then what's more you even find it in the correspondence with blood but explicitly sites this is an example so I think this is very straightforward as a matter of history the it's just not true that this doesn't bear on the history but of course it may be that nobody properly understood it And as I say, Clarke didn't really understand it, he might have said he didn't understand it. Can you just repeat the point how you get out of the room? I was just about to say, Clarke, I was just about to say, Clarke and the Correspondents misses the point of the symmetry because he says, oh, accelerate, stop the whole world, and you'll feel it. That's right, that's right. He got it wrong. So can you set up the analogy with the whole argument? Oh, the way it goes with the whole argument is that you define all of the particle velocities and accelerations up to some time, but then can you say what the velocities and accelerations are after that time? And the answer is no, because you've got this, you can just transform this totally arbitrary function of time.

47:30 so you can you can yes it's immediate but what and of course the solution is the same we go to the invariant quantities and then those all become properly tied down but that's just exactly the same thing but I think more than it's not just that you actually have the whole of them in 18th century terms, although Clarke didn't see the slander, and Glidelis didn't correct him. Actually, it's an interesting question why Newton didn't correct him. Newton's supposed to have read these last couple of letters from the class. It's corollary six that Newton goes through all of this, and it's clear that it's the equivalence principle, and it isn't really Anyway, it seems to me that if you look carefully at what goes on with how to define a coordinate system in returning gravity, even in special relativity, special relativity is very hidden, but certainly in returning gravity it's very clear that you cannot, you can't make progress with You know, you don't sort of assign coordinates to these particles in some mathematical heaven, some R4. You know, and you know not how you'd assign the coordinates, but you just suppose that you have. I mean, this is the fallacy that physicists have labored under, that you can just somehow give coordinates to particles or whatever. You just can't do it. What you have to do instead is dynamically model the distribution of particles. Yes, using some fictitious R4 and some never-never-land coordinate system on R4, but then you factor out the symmetries and get rid of the arbitrariness involved in setting up the coordinate system. And the way that that works is, the simplest example is probably with Galileo, when you've You've got a projectile, and you set it in motion with loss to u, and there's some sort of word of parallelism. It's good old ballistics, right? That's what physics always got paid for. And you've got some height h here.

50:00 And then you say that the distance traveled s is equal to ut. But then you also know that h equals half a t squared, and then you work out the relation between s and h, and you get rid of the t, which he didn't have because he didn't have a clock. Okay, and that's the way it goes repeatedly through the history of mechanics. It's the same for Newton. What he did, he proved that a particle in an ellipse with central forces, that's what and then you used the angle through which something rotated here in relation to some other angle, or some other planet in a certain system, to deduce an empirical relationship. The problem of the system drops out of the picture. And that's how it's always done. Now, I'm making these emphases because I want to go on now to the John Stancher, because John Stancher is a man, most about anybody, who has thought a lot about this, and has written a lot about it. A tremendous influential perspective, and so forth, is Mr. Einstein. He's having his Annus Mirabilis. He's speaking at 30 different occasions. including right now in fact at this very moment in London as we speak here that's right and I'm a great admirer of John we have our disputes and I do think he's wrong about general covariance and he of course is the one who's educated the whole community into it so very probably I'm wrong but still I have my And it feeds into the issue of Patton's paradox, so she would just read what he has to say about Patton's paradox. Let me give a bit more introduction. John Stanchel's claim is that insofar as you can set up analogous arguments to the whole argument in the turning case or in special activity. It's bogus. He never properly responded

52:30 to my claim about the corollary six and the means of Jupiter and all that, so I think actually he can't deny that, but what he's very repeatedly said is if you try to do your turning gravity as a diffeomorphic, you know, differential geometry, you have a manifold the same apparatus as you have in general relativity, and you get generally covariant equations of motion, that the whole argument in that setting is a sham, okay? That's what he says. And the reason it's a sham is because in Newtonian gravity and in special relativity, and in every theory prior to general relativity, you could have an independent way of individuating particular points of space or space-time. So you could say what the value of the field was at a point of space or space-time, because you had an independent way of getting at what that point of space or space-time was. And then shuffling around values of fields at points becomes unacceptable. claim and I think that's not right and reason being that I agree that it seems as though one could do that you know you have the laboratory bench you you the same system of coordinates to events independent of some theory that one's studying and thereby give independent meaning to coordinates and then the independent theory of studying the values of fields that coordinate points become something very meaningful yes that sounds good I agree that in principle you know I can imagine doing that. But one just never did do that in real life physical theory. You never did it in the internal gravity. Or if you did, you had to learn to correct. You know, some people use sidereal time for a long time. But you have to correct for that. And the way to correct for it is to produce a dynamical model within the theory itself.

55:00 So this is what John is absolutely insistent on that the great difference between GR and any other theory is that uniquely GR, it is impossible to define some hyper-pigmentation, independent of the dynamic because of just the method itself Okay, well I grant that the general relativity of course is that all the more difficult to produce a physically meaningful kind of coordinates. But I don't think it really changes the situation. Even in Newton's gravity, as I say, you add in principle to solve a dynamic problem in order to move into coordinates. What is a rigid body? You've got to apply some dynamics to it to tell you is it a rigid body or isn't it, and to correct very definitions in it that may come about is something a good clock in order to say you've got to dynamically model it and in order to see whether it's good clock or not so repeatedly you although you think to use brute pieces of apparatus to tell you what coordinate values are and that doesn't actually get you very far now in limited circumstances it does if you're looking you can use the laboratory bench. And it's not a bad external system of all of this. So in limited situations, you can use unlimited situations. Okay, I'm sorry I've gone on too long, haven't I? Should we just see what John says here, though, about Hutton's paradox? Because we'll get to the essence of the dispute about it. He says, okay, we can run a version of general covariance in the whole argument at the level of set theory. Of course, set theory is close to model theory. So he's got the world bottom left paragraph, the world SR is called permutable. Whenever RA is a possible state of the world, then the permutation of RA is also a possible state of the world. Every permutation. Now what R is, is a complex of relations, and he takes them always to be N-ray relations with N-places. A is an n-tuple of names, think of them as just n-tuple of elements of a set.

57:30 Okay, so, and then the possible state of the world is that here's a collection of elements of some sets that satisfy some ensemble of relations R. Now, does the permuted collection of elements of the set satisfy an ensemble of relations, or equivalently does the original n-tubles satisfy P-R, and P-R is the primitive permutable relations. Okay, and now this is the concept of relational structures that corresponds to covariance. And now he's giving a little example. This is what I want to get at. A little example helps me to do the meaning of the definitions of P-R into the permutable world and of the trivial identity. Okay, suppose the world consists of just two objects, a cat and a cherry, and only one possible binary relation, R A1, A2, and A1 between them. R holds if and only if, A1 is on a mat and A2 is on a tree. If the commuted relation PRA1A2 equals RA2A1 holds if and only if A2 is on the mat and A1 is on the tree. We start by assuming that there is a possible state in which the cat is on the mat and the cherry is on the tree. That is, our cat and the cherry is a possible state. The permutation is cherry cat. So if there's also a possible state in which the cherry is on the mat and the cat is on the tree, and none of those are a cherry cat, is also a possible state, then this one is beautiful. Now consider P-R and P-A, and P-R is the possible state of the world in which, well, the cherry is on the tree and the cat is on the mat. What was P-A? the permutation of the cat and the cherry. So in other words, if you both commute cat and cherry and take PR, the original relation is N. So you do them both and you just get no difference. And which it says, which I think everyone will admit is trivially the same possible state of the world in which the cat is on the mat and the cherry is on tree, that is to say, trivially identical to the possible state RA.

1:00:00 All right. Now, this is the absolute crux of the matter. The Puppet Structure acknowledges that people have said the whole argument has got something to do with Puppet's Paradox. He's just stated Puppet's Paragons, but he calls this turkey trivia nothing to do with the whole argument. So he wants to have a non-trivial version of the whole argument working here, not something that applies whatever set theory he adopts. And the way he puts it is it's the top right-hand corner. This example depends on the fact that the definitions of a cat and a cherry have nothing to do with the relation R of being on a mountain on a tree. A cat and a cherry define independently the relation R, or its influence. To put it another way, the relations R and PR are relations between things that have individuated independently these relations. You know, that sounds like what he was going on about with why the whole argument fails in the case of, say, the tone of gravity is for shrugging, because you've got independent ways of individuating points in space-time, and if that's the case, and you change the values of fields at those points, then you've got a problem, you know. So in that sense, you'd better not have general covariance, that's what's important in those cases, when you've got an independent way of individuating points. Now, when you've got an independent way of individuating the cat and the mouse, that would seem to correspond with the fact that it's not okay to commute with them. Cherry is not on the mat and the cat on the tree, okay. But then what he goes on to say is, see, I always muddled what he's saying in his top right hand paragraph. He says this example depends on the fact that definitions for category have nothing to do with relations between a mat and a tree.

1:02:30 So that ought to be to say that r cat mat A1, A2 can't be understood as saying the same thing as R of A2, A1, but of course that's not what goes on with the whole argument, is it? I mean with the whole argument you can sort of drag along of R, you know, it's not that, it's not that 5P equals 5HP, okay, it's that H star You've got to drag both the phi and the other fields and so forth in order to get to the identical situation. If you just dragged some of the fields and not others, then you'd have a different situation. So I think there's something mistaken here about calling the attention to the fact that and satirically being able to specify the cattle and the cherries. He goes on to say, But now imagine a world in which a cattle being a cherry depends on whether one is on a mat or on a tree. That is, a case in which the individuation of certain things depends entirely or essentially on the relations between them. In other words, apart from these relations, the distinction between them cannot be made. You may say with a slight abuse of language that this is a case of things between relations of the relations between things. Okay, he goes off to have a piece about Marx, which of course is a great love of John Statshaw. I don't have time to go through this now because I see I've been speaking for much too long. The final page on the handout from John that I've got for you, names in the meta-language arose by any other name. way he explicitly addresses the Putnam paradox business. But there he's saying that all that's going on is that we're changing the names with which we speak about cherries and cats.

1:05:00 So this is bottom left, page 2.6. He says the cherries on the mat and the cat is on the tree means that the cat is on the mat and the cherry is on the tree. On the other hand, if we also mat and the assertion of the cherry's on the tree and the cat is on the mat means that the cat is on the mat and the cherry's on the tree. In other words, just what is made before we remain named, what is renamed has nothing to do with any change in the positions of the cat or the cherry. Shakespeare's wrote as they remain unperturbed by all this renaming. And then he said to you, it seems to think that the inscrutability of reference, which I profess from acoustic conventionalism, that follows from the permutability of names constitutes the general form of the whole argument. As we have just seen, commutability of names has nothing to do with what happens to the objects and the relations, so per se it can have no bearing on the whole argument, which is what we have seen as concerning the permutation of objects and relations and not of their names. Okay, well, I'm sorry I've spent so long to get to the point now, but I think what's happening here is that in general In the Putnam Paradox, one's got syntax, and you imagine holding it fixed, and you imagine different interpretations of fixed discourse, different interpretations, and then you find that you have all these unintended interpretations. Now, if you think of this as the manifold, and this is the coordinate-dependent stuff, then the thing becomes really over-tripping. It's not that you've got different ways that the world is, and our discourse can't sort of grip the world. it's that we just change coordinate systems and talking about a fixed manifold structure that's fine okay and what what John Stachler has just done something similar he said that except you can sort of invert it you can go back and forth on these things and I don't make an apology for this this isn't because I think I muddled it's just inherent in the situation. You can also speak about holding the world fixed and

1:07:30 changing the syntax. And that's what John was just talking about. You just change the names of the syntax. You just move names around and you hold the world fixed. And big deal. and he's saying that's just semantic conventionism. Right? What words do we choose to use? Now, I think what's going wrong on both those trivializations, both of the whole argument and Putin's paradox, is that the way to trivialize it is to hold the thing that you think matters fixed. And so you could hold the world fixed and change the names. say it's trivial. Or you could hold the manifold structure fixed in the fields and just change the coordinate system you're going to use, and then that trivializes the whole level, and this is, sure, it's difficult, it's a big gift. But the real bite of both paradoxes comes from an appreciation that you can work at both bites. So, for example, you can hold the coordinate system fixed and by a different mechanism on the manifold generate new fields, and then it has bite. Or you can hold the syntax fixed, and the propositions and so on, and change the interpretation. And then it seems that the world could be different from what we think it is, and there's nothing we can say that can express the difference. So there's only one who can appreciate both of these, but I think you can deal with them properly. And I think the statute here was making a mistake about . I don't think it has necessarily been mentioned or something that solves it at all. and the reason he's making this mistake is because he's determined that the whole argument just can't be so trivial as Putnam's paradox looks to him you know it's he seems well I think I've said enough without giving you more quotation Let me just sum up then. I've tried to diagnose where Stature has gone wrong in making the claim that he has. And so I think the right thing to say is that the whole argument is only the latest in a long series of arguments about the nature of coordinates in physics.

1:10:00 And I think that where they all leave us is to the same end, that we describe objects in terms of that which satisfies certain field values coinciding with such and such a field takes value, or such and such a field takes value, or such and such a field takes value. and that it then becomes an invariant way of description and one can do something similar to Patton's paradox if you say that the object is that which satisfies such and such a predicate and such and such a predicate and such and such a predicate. Now that sounds very much like a property bundle of objects doesn't it? And if it was that simple to solve Patton's paradox, just so you've got these bundles of properties, then you've said it. So there is more going on here. I think property bundles does get to a part of it that is a bit less, and I can just quickly say what I think is less, and look, take a triangle, which is not symmetric, and you've got three points, right? Now, if the points themselves, I'll tell you those are three atoms. Now, if the atoms are exactly like, you can take the view that many of the atoms will make a difference. The physical situation will change, and that seems to be the same of all problems. So, do I manage to refer to an object by saying it's the object which is at the right angle? And the answer is I still haven't succeeded in referring to an object. That's the point. So this is why it's not enough to say it's a property bundle. I mean, it's something slightly different. One quick way to see this is, suppose I just use Turquoise Symmetra predicates, so I can use any permutation of these things and use the predicate in shape. So, if I just say the object which is at the right angle, I haven't said whether it's object A, object B, or object C, so I haven't figured out any other. If I say the right angle, then it's different. But then it seems I might not be talking about the atoms.

1:12:30 and I think that's that's the real bottom line right now I'm getting myself puzzled because I it seems then that the right solution is to say that the pattern position that becomes the object it's good so now all of a sudden I hope to conservative solution one wasn't saying anything too strange one was just saying perhaps that the predicates are not individually by the extensions you know something to change the extension change the predicate rather extension individually by predicates predicates come first that's what I feel like I'm not sure if it's a map I think I may also have to further say It's not that the object is that which satisfies certain predicates, it's that, it's the bundle of predicates, I guess. It does sound like a problem, but it's the bundle of predicates that is the object. Then it's clear why you can't move the objects around, or rather if you do, I think there's probably something right about saying is a solution. But if it was as simple as that, people would have said it, so I wouldn't have said that. Well, forgive me for getting on so long. Thank you very much. I was thinking that you would like this last part, because part of the solution could be to say there are no objects, that the right angle is singly occupied. Yeah, that's right. You feel that too, that's too un-conservative. I do, I do. And I feel like it's really kicking them out in a way.

1:15:00 Look, here's another way of doing it, it's to say that you can push the analysis of language to extremes. And that's fair enough, because afterwards you tell me when I was talking about the cat, you know, my cup of coffee, actually, I'm talking about Matthew or something, I feel like saying, look, I'm holding it now, I'm relatively close to it. You start to bring in spatial temporal stuff, after all, and then if you're going to put that into the language, this is the more theory, you start putting more of that into the language, and of course, you've got a whole set of complex, spatial temporal functions, you're halfway, you know, you're halfway to your physics. And all of that's just pragmatically in place, normally. was normally just pragmatically engaged, then you end up with huge amounts of theory before you know it.