12th UK Foundations of Physics — Reflections on EPR & Structural Realism

0:00 So, if you're trying to find out the colour of a particular point, you've got to align your apparatus precisely in that direction. and their argument is that you can't align an apparatus with infinite precision which would mean that you can't actually reliably ascertain the colour. I mean, of course you've got a 50% chance of getting it right by chance and that's not measuring it. I guess I meant measurable Oh, you meant measurable right the black coloured region set. It is in fact a closed set. Sure, but I mean the set of inequalities are unable to do that. Oh, it would be in general highly non-would not be measurable. It depends whether you're talking about Petowski. It's always measurable for May and to the Clifton because they assume countable sets. But for Petowski very badly behaved. Thank you. We're going to, during the coffee breaks, Steve asked me to make sure everybody knows that there is tea and coffee provided, I assume. This is one particular outlet where some people are absolutely, say, Obviously he does, and I would say obviously, clearly, he doesn't. I mean, the trouble is that however you kind of wipe things over with complementarity, the discussion, you still perhaps have to face the fact that if you measure, let's say, let's use Bohn's model, if you measure S-Z here, measure S-Z here immediately, it does seem to be some sort of non-locality there. So I don't think Bohr can, however he can wipe away the general structure of the EPR paper, I think he still has to stand up and fight his corner on non-locality there. I mean, he can't be, there is non-locality, but this is, as I say, very much, very key differences here.

2:30 a small question, you mentioned there is this transition what about the transition in all sports particularly with regard to confidentiality from KOMO talk to EPR, what was driving that how does that transition affect his final response Einstein only you can help get it. And I think what Fein I think Yammer mirrors what probably Einstein and Bohr thought and Oprah and Thorpe they lost this chapter it's only I think in retrospect Fein is able to look at it and say well it wasn't such a big transition at all so I mean as we said in the very first quote I think Bohr felt it was an enormous fault from the I must admit that you know this talk of struggling writing gradually coming to terms with things. I wonder to what extent really what he was coming to terms with was how he was going to put things over, getting away from actually EPR's argument itself over to, well, putting it more in the context of the solving discussion. I think it's quite different to 1935 than it is in 1937. and it's not because I think there is a problem. It's imposteristic. We're going to put emphasis on experiments and it relates to the length of the film. There's multiple emphasis on, you know, the falls of space, blah, blah, blah, blah. So, I'm wondering, two questions really. What, if you could comment on what was driving that shift in war? It says, since the historical shutters don't know, how much it was driven by his interaction with the lines in that period. In what sense are the traces of the earlier, I'm saying, complementarity in his response to EPR? I was a little nervous when one said, well, Bohr's responsibility is just positivistic. I don't think it was, but it was about history. I'm certainly not saying complementarity is just positivistic, so I'm not going to at least say that. I thought the two big changes in Bohr's thought were reversed, actually. I mean, up till perhaps, and this is obviously hugely unrealistic, up till 1935,

5:00 he very much had this disturbance model in his mind. 1935, EPR, I think, removed that physical buttress to his philosophical thoughts, perhaps moving then in the direction of the 1940s to this, what's it called, and so on, and EPR might make, it's not so much in my view the transition from 1927 to pre-EPR, the transition perhaps caused by EPR, which was the main development of Bohr's I wonder if there is or could be an element still of disturbance in Bohr's reply to EPR and this goes back to your previous topic about non-locality Now, there is a rational deconstruction for the statement about it being a question of an influence in these very conditions that make it possible to define it. And that's often held to be a very obscure statement. But there is an interpretation of it, which is purely physical. It says if you are going to use the measurements of x1 and your knowledge of x1 minus x2 to deduce the value of x2 from the other part of y, then in order to make sense and to correlate the two values of x2, you actually need a visible connection between the two sides of the experiment. an iron bar which will tell you what values of X are being correspond to values of X have been. And therefore, you shouldn't be surprised if there is actually a visible disturbance, because there is a connection. I can't remember where the argument comes from, but I think why did you take it to you? That would be interesting, yeah. Not as such, I think so. Okay, also let's thank the speaker again. Our next speaker is Robert Bishop on non-linearity and indeterminism. Yes, the title is perhaps a little bit misleading because it sounds like I'm going to say something about, ah, I can show you how you can get indeterminism out of non-linearity. That's not exactly what I'm going to do. What I'm basically going to do is to talk about an argument that's been in the philosophy of science literature since at least, I guess, 91 that I've been thinking about for a while and has also made its way into certain free will discussions.

7:30 although not to talk about free will at this particular meeting. What I want to do is basically lay out the argument and then talk about certain sorts of problems for it that I think at least cast doubt on its soundness and then see what you would have to think about that. Okay, lay a little bit of background just very quickly. let me say a couple of things this is sort of the standard framework in which these arguments have been framed not always explicitly but basically we're dealing with certain sorts of theoretical models in physics what do we do there? well we have equations of motion they describe the time rate of change of the variables that we take to be the key variables characterizing the system Then we need to specify initial conditions and boundary conditions for those equations. And then on top of that, we're dealing with some kind of state space, right, which is an abstract space of points. And these points represent possible states of the model system that we're interested in. And we characterize the states by the instantaneous values of the key variables. Okay, so this is the standard picture of doing things in nonlinear dynamics and so on. I thought, well, I should say something very briefly about what I take to be a crucial assumption, at least in these discussions, that's implicit, that I want to make explicit. it's implicit in the sense that it allows you to do certain sorts of things but I'm not sure that the authors I'm going to mention are even aware that this assumption is in the background and this assumption itself could be as well an entire talk but I call it the faithful model assumption and it basically has two pieces to it one is that the equations of motion that we're interested in somehow faithful represent the target system See, I guess I have some sort of physical system that I'm interested in. Maybe it's a Nam-Tribbon pendulum or something like that. I have a set of equations with which I'm modeling that.

10:00 And the second piece is that the state space that I'm using for my analysis facial represents the physical space of my target system. And this assumption allows me to do certain sorts of things like move perhaps somewhat sloppily in between model talk and system talk but in a way that seems to be licensed by the fact that things are faithful. Now, I'm simply going to assume this assumption. I'm going to drop on this assumption in this discussion. I don't want to try to defend this assumption in the context of non-linearity, because I think that's not possible to do. That's a different talk as well. Okay, so that sets some of the groundwork. And then basically, I'm just going to use a standard notion of non-linearity. We're going to have a normal form for a set of differential equations, write some set of variables X, have a matrix F. And as long as this matrix F doesn't contain any variables or functions of the variables, then I have a linear system, otherwise it's non-linear. So a standard example that people often give in these discussions, we're going to talk about chaos or the Lorentz equations right here and then you see that here we have functions or at least we have the variables appearing in the matrix. So we have a nice clean way of telling when we have a linear or a nonlinear system. Okay as far as determinism is concerned I'm just going to use what I take to be a core feature of determinism. It's often called unique that given some state, this state is always followed or perhaps preceded by the same history of state transitions, right? So you can imagine determinism being bidirectional if you want. And this basically means that in the context of, say, a model, if my model is deterministic, it will repeat the exact behavior over and over and over again as long as I specify the same initial and boundary conditions, right? So, okay, that's the piece that we're going to focus on in terms of the arguments that I want to discuss. Now, as far as I know, I call this the sensitive dependence argument. As far as I know, Jesse Hobbes is the first person to mention it in the philosophical literature.

12:30 Stephen Keller picks it up, puts it to use for certain sorts of applications in his book, In the Wave of Chaos. And then some other people have discussed it. And principally, it's found its way into free will literature because people who have been interested in developing libertarian versions of free will that seem to require some use of indeterminism have thought that this might open a kind of door from the quantum realm into the classical realm and this argument does seem to open that kind of door okay so how does it go this is this is kind of a sketch of the way Hobbes lays the argument out so the first point is I'm going to I'm going to he would use chaos but I'm going to use nonlinear dynamics now and and refer mostly to non linearity and I'll explain a little bit of make that trade, but for nonlinear models exhibiting sensitive dependence on initial conditions, right? Roman mentioned this yesterday. Trajectories starting out in a highly localized region of state space will diverge on average exponentially fast from one another. All right, so this is just a kind of standard assumption that happens in chaos discussions, for example, where you characterize this rate of divergence by a global Ivanov exponent, right? Now, usually people don't say on average. They usually say exponentially, but technically they should say on average because the global Ivanov exponent gives us exactly that. An average value for the growth of the exponential divergence between trajectories. Okay, I'll say a little bit more about that towards the end. The second piece is that, well, quantum mechanics places a fundamental limit on the precision with which physical models can be specified in terms of what kind of initial states you might be able to give them in state space. And it basically places a limit that's related to Planck's constant. So if you have these two pieces, then the next thing that seems to follow from that, according to Hobbes and Kellert, is that given enough time in this quantum mechanical bound, on some neighborhood epsilon for the initial conditions,

15:00 two trajectories of the same chaotic nonlinear model, will have final states that are localizable in some much larger region called a delta in state space. Therefore, the idea is that quantum mechanics will actually influence the outcome of chaotic models, leading to a violation of unique evolution. The idea being that, right, infinitesimally small differences may be amplified by chaotic dynamics in non-linear systems, and surely quantum mechanical fluctuations, for example, feeding into the initial conditions of some chaotic system should be able to follow exactly this line of thought. Okay, so that's basically the argument, and that's pretty much all, well, there are some defenses that are given of this argument by Hobbes, but I won't discuss that here because that's another talk. They trade on a number of misunderstandings. There are other ways I think that one can try and justify or make plausible the argument. Keller just takes the argument over and puts it to use for his purposes. What I want to do here, though, is discuss some problems for the argument. And I'll simply start with what I call the obvious criticism, or at least I think it's the first one that you would tend to think of if you're thinking at all about quantum mechanics. And that is, if this is to have any force against unique evolution, you have to presuppose that we have two isomorphic models in identical states, right? in their history of state transitions. In other words, instead of having one-to-one correspondence between initial state and some final state that my model should give me, I have one-to-many correspondences, right? That's the sort of thing we're going to look for in terms of a breakdown of unique evolution. However, on some interpretations of quantum mechanics, it's not clear that we can talk about some sort of concept of identical states. So now this produces a kind of problem because you might not have precise values for initial conditions. So there would be no two sets of initial conditions on which models could agree, right, if we're talking that way, that we could look and see some history or some difference in the history of state transitions.

17:30 So the upshot is that although Keller and Hobbs don't seem to realize this, this, the sensitive dependence argument has to be relativized in some crucial way upon a version of quantum mechanics that you have to choose that can give you what you want. And you also have to put in some sort of solution to the measurement problem that also does what you need to make it go through in a way that shows you that quantum mechanics somehow renders a system exhibiting sensitive dependence indeterministic. so just in a kind of very broad brush quick abstract way if you have a no collapse kind of view that just has strange revolution which would be a deterministic version of quantum mechanics it's not clear that we could get any kind of indeterministic breakdown via sensitive dependence on initial conditions so that would be one way of saying that look the argument doesn't seem to work at least in some sort of general way It can only be examined if it works at all in certain specific cases. The collapsed view has a number of rabbit trails one could go down. I'm just going to illustrate, again in a very broad way, two things that one might think about. I'm going to think about a kind of toy system where I have a chaotic pendulum. It's a damp-driven pendulum that's exhibiting sense of dependence on initial conditions, and I'm going to fire a photon at it. Now, if you don't like having to fight through the details of the electromagnetic interaction between the photon and the pendulum, like one of these detailed Borean stories where Bohr always thought Einstein didn't look at these details in the mouth, think of something like silver atoms passing through the Zeamont's blue, and some are spin-up, some are spin-down, and just translate it. into that, but the idea is that if we look at versions of quantum mechanics that have genuine collapse of the state vector, there are at least two ways we could cut up the system and specify it in terms of its initial conditions. One is we could just treat the photon plus pendulum system at some initial point before the interaction takes place as the initial state.

20:00 Now, there's a kind of cheat here because I have to somehow say that, well, I mean, it seems I can say clear things about preparing my photon in the same sort of pure state. But one might worry with sensitive dependence on initial conditions whether I can actually get my pendulum back to the same identical starting state again. I'm just going to assume for the moment that that's something that's possible to do because that's an assumption that Kellert and Hobbes make. If that assumption doesn't hold true, you've got a much messier discussion, and it's just the worst for their argument. So in some sense, I'm trying to give them enough ground to try and make things go through, and it's still somewhat problematic. So if I treat it in a case like this, where I start with some photon of pure state, put the pendulum in the same state, unique evolution is going to be violated because I always start with some initial state for this entire system. But every time that the photon hits the pendulum, the collapse of the state vector means that I'll transfer some amount of momentum But each time I get some different momentum or some different jump in momentum that's transferred to the pendulum. But I start out with the same initial state in terms of this kind of separation that I had. It could be the same state, but I would get different trajectories. So in that sense, one might be able to argue that, yes, unique evolution is violated. The sense of dependence argument would go through. On the other hand, if I say, no, I think it makes more sense to simply take as my initial condition, whatever happens at the moment of interaction, so now I take the jump in momentum, right, as being part of the initial conditions, and I specify that, then unique evolution is going to be preserved because I'm going to have different initial conditions every time I run the experiment, and the way unique evolution is defined is if you give me a different set of initial conditions, I should expect that I would get different trajectory. and state space every time. What I need is the same initial conditions giving me different trajectories. So this is one sort of class of problems, kind of quantum mechanical problems, with this particular argument.

22:30 Mainly, we have to say something, we have to commit ourselves to some version of quantum mechanics and some version of the measurement problem, and we need to say something about how we're going to cut the system up in terms of what's a measuring device, observed, you can imagine the pendulum is acting like a measuring device for the photon, for example, etc. But that's not all. There's more that we might worry about with this argument. Now, what I've said so far isn't a way of saying that, well, look, a sense of dependence argument simply doesn't go through ever. What it basically shows is that this idea of amplifying up quantum mechanical influences is simply much more subtle than what Kellert and Hobbes had in mind when they were doing their arguments. But there are further problems that seem to complicate things even more. Like, for example, if I'm thinking about dissipative systems, and my little toy problem of thinking about a damp-driven pendulum means that I have dissipation in terms of friction, right? That is actually going to place lower limits on what it is quantum mechanics is able to do in terms of my classical system. So if I'm thinking about something like sliding friction, right, at the pivot point of the pendulum, I've got two plates, or at least I've got two metal surfaces that are sliding back and forth. That's where the friction comes in, the friction term, right, for the equation. And there's a major role played by friction in terms of molecular bonds. So, provided, and there is evidence that shows that friction and sliding surfaces also exhibit sensitive dependence on initial conditions, so provided that I have that available to me, it seems like minimally what I would need to require is that the jump in momentum that the photon gives, or whatever quantum process I'm interested in gives should cause at least minimally one bond to break or perhaps to form on a time scale that's due purely to the quantum mechanical effects and not something that's due to the effect of sliding friction, right? So it's going to place a kind of limit on the time scale, right?

25:00 Because it has to break the bond, say, before the mechanical process would break the bond. Well, the kick has to be amplified up enough to break the bond's strength, right? So if I could get this kind of delay or breaking of one bond, that seems like that would be sufficient given since the dependence, right, just some small change can eventually be amplified up, and that might influence the trajectory of my pendulum. Okay, well, one might say, well, look, quantum effects are too small to influence a system like that. That turns out to actually not be true. I'm not going to go through the papers, but we have good experimental and theoretical evidence that indicates that quantum effects, for example, from electrons, actually do end up making the dominant contributions in friction under appropriate circumstances. You might say, well, what are appropriate circumstances? as well, you just allow the friction to go for a little bit, and you'll see that over time as the surfaces begin to get worn down, worn down sort of in quotes because the molecular picture doesn't ever look like it's worn down, but there's a transition such that you see that effects due to electrons and other sort of quantum mechanical processes become the dominant effects. So it's plausible that a quantum effect could somehow influence a system like this, but the quantum contribution would have to be amplified, as I said earlier, to sufficient magnitude on a short enough time scale to break one molecular bond. So that seems to place pretty stringent conditions on the quantum effects influencing some sort a concrete nonlinear system. Okay, well, there's more. There's probably even more than I'm going to say here, but this will be my last sort of complication to this argument, and that's that everything was premised on what I would call the usual story regarding global, you know, global life amount of exponents that are used to characterize the on-average divergence of trajectories from one another, or the on-average growth of uncertainties in some small set of initial conditions.

27:30 But this story, if you look at how Lyapunov exponents are derived, presupposes an infinite time limit and that we're dealing with infinitesimal uncertainties. The reason why people typically use the infinite time limit is because if you're looking at the surface of of an attractor, the Lorenz attractor, or the Annon map, or any of these sorts of things. The Jacobian matrix that tells you how it is that vectors transform as you go from point to point along the surface of the attractor is a constant in the infinite time limit. That's why you simply get a life exponent as a global measure of the rate of divergence of the trajectories. Just work in the finite time limit, then that Jacobian matrix changes from point to point, and you have to actually calculate the local Leibniz exponents. And some people have done this. It's quite messy and complicated, but it has its own sorts of interesting surprises. The other point to worry about is that, well, we're just thinking about infinitesimal effects, and I'm interested in a finite time, right, or my photon and pendulum system is only going to be operating for a few minutes in the laboratory before I detect my effects or a few weeks, but it's going to be some finite time scale, then the quantum contributions better be finite and not infinitesimal. Otherwise, I'm not going to see some change because I'm not going to wait around for the infinite time limit to give me an answer about this. So the appropriate way to analyze this kind of experiment, it seems to me, is to use the local finite time, lifelong exponents. But if these vary wildly from point to point as I move along the attractor, then this opens up more possibilities for the quantum contribution to actually have no effect. One possibility, for example, is the jump in momentum may actually grow slower than is necessary to break the bond. um that's good because this is basically what i want to end with um so here's the artist rendering of the butterfly picture for the lorenz attractor people have actually shown that you can find little regions on the attractor for example where you have negative liponov exponents which means

30:00 that the local Lyponov exponents mean that the uncertainty actually shrinks in these regions. There are other regions where it might be zero. There's some where maybe it grows more slowly or more quickly. So that means that if if the jump actually grows slower than what we might expect by the calculation we would give for the global Lyponov exponent, we may not be able to break the bond on the right time scale. Or worse, perhaps it gets deposited in one of these regions The local iPod exponents are all negative, in which event the jump in momentum would actually disappear. We have convergence of trajectories in those regions. The quantum effect, perhaps, would be lost. So there's another way in which the sense of dependence argument may actually be in hot water. So, just to sum up, the sensitive dependence arguments seem to be indicating something like sensitive dependence somehow opens a door for quantum mechanics to infect nonlinear systems. And just as a point of clarification, even though originally they were discussed in terms of chaos, it's really the nature of nonlinear dynamics that opens this door, at the same time that it complicates the door. So, sensitive dependence arguments go through it all. through smoothly. There are problems that we have to worry about in terms of quantum mechanics. There are problems in terms of what dissipation might do in terms of of the limits that that places on such the growth of quantum effects in such systems. And we have to worry about these problems about the finite time liable exponents which are tracking actually what's what the growth rate is going to be of any kind of jump that might be applied to the system. So the Abstract reasoning about a sense of dependence argument, such as what Hobbes and Kellert seem to rely upon, isn't going to do the job to either establish the truth of or the falsity of such arguments. You have to look on a case-by-case basis to make some sort of determination. But I wouldn't be optimistic about finding a sense of dependence argument that works.

32:30 I would hold my breath. Well, I wouldn't hold my breath because it might be had to be my health. So, thank you. There are six minutes of questions. Thank you. Let's continue to do some more about Perriotis. One of the things that I've said in account of the escapism you think we're going to do was that if you took any isolated classically intelligent system to contact, then you can even get a lot of timescales to work at a lot of timescales. The systems don't have to be fairly dramatically in the other class of the math. Now, any intelligence you've got in the campus has got to have some sort of the SAS class of these that that makes your systems in order to their classical behaviors because so even in a no-time statement like Edward wrote they'll be as a mechanism for all of that perhaps. Now I'm sure as I can see, if you think about that mechanism will always have to come into play fairly quickly in a heloclassical system to avoid non-classical behavior. So that seems to say that the details of the interpretation I think, well, that would imply a kind of change, at least in the way in which the sense of independence argument is traditionally, if there is such a thing as a tradition for this argument, it's not that old, but the way it's laid out. The way it's laid out is the first thing, the assumption you make is that we can take We take over everything from chaos and non-miniarity that we're used to in the classical world, and then we suddenly realize that, oh, there is a quantum world underneath somehow, some way. What happens if it's looking, and this is pitched as if the quantum world is giving a kind of perturbation, possible perturbation of the classical world. Now, one could reformulate things and say, well, look, ultimately, at the bottom, it's a quantum mechanical world. So now we have to worry about things like what are the analogs between a quantum system and a classical system, the quantized version, et cetera. I haven't looked at that, but my guess is if you reframe it that way, then you've got a different set, you've got a different kind of sense of dependence argument, one that

35:00 perhaps one could have much more hope for than the version that Hobbes and Kellert run. Again, following up for David's question, while I've got sympathy for David's point in the sense that irrespective of the interpretation of quantum mechanics, something in the interpretation will play the role of the probability for multiple equations. But the issue of interpretation would definitely be important according to the work of people trying to put the sensitive dependence argument to. If it is supposed to somehow show that there is indeterminism at its amplified level of Well, then it better be the interpretation of quantum mechanics, which has indeterminism. In a trivial sense, take the Bohm theory, that no such argument will show indeterminism. So, some kind of effective indeterminism, but whether that's good enough or not, depends on the philosophical aims or people who are trying to view that. What I probably didn't say clearly enough is that, I mean, the sense of dependence might actually deliver to you the conclusion that quantum mechanics always infects such systems in some way, but it doesn't necessarily tell you that that's going to be deterministic or indeterministic effect. That's going to depend very much upon whether or not quantum mechanical theory or the interpretation of quantum mechanics is at rock bottom and deterministic. the real, you know, chance is all there is kind of sense. And then from that perspective, and that's the reason why at least some libertarian free will theorists have been interested in this kind of argument, because they say, well, look, we could get something that maybe is useful to us. Whether it's useful or not is another question, but they see it as a route for bringing about

37:30 what they want, in which it meant they would not like, at the end of the day, quantum mechanics deterministic and they got these effects amplified up because it would be what they used to, that's true, yes. Time for one more question. And that's the second speaker. We're going to turn to the last speaker. Today is from Michael Stoltzner and the title of his talk is The Principle of Least Action, Structural Realism or Mathematical Solstice. The German expression for carrying coal to Newcastle is carrying owls to Athens. In this sense, I would like to carry two German and the smaller Austrian owl to Leeds. The object of consideration is very close to what Harvey Brown has been talking about this morning, namely the principal of his action and variational calculus that enjoyed a great popularity and led to beautiful mathematical results in the first two decades of the century, but later got a bit into the background. And one of the messages I want to convey here is that you can learn a lot from that, not only physics and mathematics, but also philosophical. Even then, the principle was quite disputed. Here is Max Planck's exalted pronouncement. As long as there exists physical science, its highest desirable goal had been to integrate all natural phenomena observed and still to be observed into a single simple principle. Among the more or less general laws which manifest the achievements of physical science in the course of the last century, the principle of least action is probably the one, as regards form and content, may claim to come nearest to that final goal of theoretical research. final ideal goal. On the other hand, the logical empiricist Philip Fran was quite laconic. It is not at all characteristic for the orbit a mass

40:00 point follows that along that orbit any magnitude assumes its smallest value. If the orbital curves satisfied another law, there would always be a magnitude that depends on the velocity or acceleration and which is smaller for the orbital curves than for any other curves. Just this magnitude would then be regarded as a measure of the action of nature. And in a recent paper I have discussed how Planck and Hilbert on the one side, it's in the studies, and the logical empiricists on the other side discussed the issue, what was the philosophical significance of the principle of these actions, and to became a veritable shibboleth, whereas Planck thought it was the final goal of theoretical research. And notice that the issue was not so much the ideological one about teleology and vitalism, but that the mathematical universality claim for the PLA for logical empiricists represented an illegitimate border crossing between physics and mathematics. So for them, in Franck world, then there would be other mathematical descriptions. Some of the material I'm discussing here is in the studies papers, but not the interpretation I'm giving of it here. And I think this indicates that popular suspicions one always has about the principle of least action misses the gist of the matter that was not an issue of backward causation nor of material teleology, and what we should really do is, as people in these days, and also the logical entrances, look closer on the mathematics. And I think the general perspectives one gains is that the principle of this action involves some global features, includes possible solutions, and then you have a criterion that distinguishes possible ones and that it has modal aspects and that's a point Jeremy has always been stressing during the last years that there are certain types of modality involved in analytical mechanics and that it is structural or systemic features of the ensemble of positive dynamics that was set up to

42:30 single out the actual one. We go to the third point in more detail during this presentation. So what's the aim here is to show, to discuss whether the PLA can really be thought of an instance of structural realism and I will discuss first Planck's position and argue that there a sort of epistemic structure of realism is combined with another realist criterion or a second or even two further realist criterion to end up at a certain convergent realist position whereas in Hilbert's case you find the Leeds position somehow realized or the Leeds Bristol position realized that he searches for deep mathematical structures but in case three there would remain with mathematical anthology, and this suggests, I think, to put a bit more at a distance mathematics and physics, and to consider the principle of least action as a certain type of mathematical thought experiment that's performed with a whole class of mathematical structures. So here's the general type of variational problem, which I think most of you know, and what I want to convey in chess is that you need basically three things. First, that there is the variational problem, that you have to specific about the interval you are discussing, so that's the global aspect, and that you have to be clear what are the classes you want of possible solutions you want to discuss. Very popular is C2, you can start with PC1, You arrive at C2, and the physics is specified by the function f, which is typically the Lagrangian, which in mechanics is just the difference of kinetic and potential energy. So then you arrive at the Euler-Lagrange equation, but in actual fact, the Euler-Lagrange equations are just necessary conditions for the variational principle to hold. there's a large list of other necessary conditions like this Legendre condition

45:00 and you realize that if this term is bigger than zero then you see also that this has to do with whether Legendre transformation is possible whether you can go to the Hamiltonian formulas whether you hit some constraints The other point where you see the globality most directly is that if there are focal points where all curves have to go, the principle is not unique. Yet the absence of that can be translated into another differential equations. What has occupied people at the turn of the century and later was to find sufficient conditions. and that was the most difficult task of the classical phase that only was first answer was Weierstrass and got a more precise form was Hilbert and the main idea is to prevent such things you have to embed your extramal into a whole field of extramal that's also sometimes called the Maya fields that such thing cannot happen and the nicest is what you Now, in the Hamiltonian picture, what perhaps most people have seen in the Jacobi theory, that you have all the extra miles, somehow like spaghetti, and then walking, going at, with e times t, there are the wave fronts going from one surface, where you have the reduced action function, e goes to one valley to that of the other. So that's the picture you want to have here. And there are some further mathematical intricacies. One might be familiar to you. The Lagrangian can be non-unique. And that's basically in three dimensions the same case which you saw in Harvey's talk. The many difference, I think, is here that the argument of an earlier paper of Marmot, who is also behind this Morandi physics report, whom you cited, That you have different symmetries and that you arrive at different quantum theories. So once you have a continuous spectrum in one direction and in the other you have a few-point spectrum. I also, so that's about what's the relation between the Lagrangian and the equations of motion.

47:30 And here's an example about how you have to relate your modal class, if you wish, or the possible dynamics. as in Hamilton's principle when you don't vary time and all motions take the same time but they correspond to different energies that was why I before was saying well they are the wave fronts moving in the Hamilton-Yakobi theory or you can take Mopotri's original principle where you don't arrive at the Euler-Logrange equation but you have to use energy conservation to get the equations of motion and so alternative motions take different time but have, obviously, the same energy. So, these are already told. So, let me now come to a philosophical proposal that's based on this distinction between necessary and sufficient conditions. And I would like to distinguish three orders of formal teleology. and I call that formal teleology because all the philosophical questions of action principles were often discussed under this heading. So if you associate, you have first to understand what's a sort of an abstract version of the principle of least action, and that's heavily modality-laden, so to speak. The actual motion you extremize is the value of the integral you view in comparison motions, which are near the variations. And you take as expression of a causal law, just normal differential equation. And then you end up that the formal theology is merely a complement of the causal explanation. And the strength of this, that means the difference between the differential standpoint and the integral standpoint, depends much on your concept of causality. If you're rather lax in causality, then there's little space for So, I propose to distinguish as a zero-order teleology that U is uniquely determined within MU, the model class, as first-order teleology, and that was basically Planck's belief that

50:00 exists a functional whose extremal values yield the causal dynamics. So, and Planck believed that really that could be done in all domains of reversible physics. But if you are focusing on what is the reason for sufficient conditions, then you really make the claim that there exists some structural features of the modal class which guarantee that w of u attains its minimum or maximum. see in the position of Hubert. It might sound strange to you that whether uniqueness could really be a criterion, but it was. And that was Mach's point of view. Mach didn't like very much the principle of intersection because in principle there were things that were directly intuitable and the other things which we are theorems. Now, Mach had a very lax notion of causality, just functional dependencies and complexes of functional dependencies were the facts. And he even considered the core of the principle of least action as energy conservation on the one hand, but there's also more general notion involved, more general conception involved. He writes, notice that the principle of least action does not express other than that in the instances in question precisely. So much happens as possibly can happen under that condition, or as is determined, is uniquely determined by them, and besides pet salt. In the case of all motions, the paths actually traversed can be interpreted as distinguished instances chosen from an infinite number of conceivable instances. The uniqueness is decisive. This goes back that, well, all the other alternative motions were prepared pairwise. So to see, here's the actual motion, and to each one on the lower path, so to speak, there's one on the upper path, taking the same time. And Mach insisted that even if there was the action principle everywhere in physics, that

52:30 didn't mean anything that it would have some ontological significance and neither modality has an ontological quality. Now let me turn to Planck, who in 1910 stood in a fierce polemic with Mach. Now, in starting this polemic, Planck emphasized that while originally we had divided the sciences into the fields that corresponded to our primary senses, the relative physics was characterized by unification and by de-anthromography, for anthropomorphization. And our physical worldview has today reached sufficient stability that we could really take this as an instance of realism. So he agrees that Marx's positivism was a good antidote against the exaggerated mechanical worldview, and so he brought physics back to what was allegedly the basis of physical claims. But he overshot the mark by degrading the physical worldview together with a mechanistic one. And Marx does not really touch the essence at all the demand for a constant world picture which does not depend upon the changing epochs and people. And then he goes on to do that in even clearer form as a no miracle argument. This constancy which is independent of every human especially every intellectual individuality is that which we now call the real. But there is now a more specific solution for Planck takes to cope with a problem which is often called pessimistic meta-induction because the whole polemics with Mach is, of course, about the history of science and the changes in the physical world conception because Mach's books always had followed the historical critical method. So that was the point that discussed. So if principles had, I think I forgot this, Plante tells us that in all recent

55:00 concepts, conflicts between facts and theory, were always the principle who helped the fields, not intuitive foundations. And he says the same about his radiation law. Well, interestingly, he argues that it's a great virtue of the principle of least action that it could be applied to discontinuous functions, as they appeared in the radiation law, whereas Hamiltonian dynamics could go away. because the Hamiltonian dynamics, or so we believe, stood behind the argument you could put as a justification of genes theory, which yielded the divergences. So, that was a great hope. The principles survived, but obviously then the principles had to be of a very abstract nature to survive all these ruptures, and they had to be adaptable to scientific progress, otherwise they would face the standard objection. And knowing what was the mathematical problem with the principle of least action, and emphasizing that it was very useful if we didn't know details of the problem, And Planck told us not that it was a deep mathematical structure, but it was, as I had stated in the large quotation on the first page, it was his general belief that there always was such a principle of least action in which the whole content of a field could be cast. But, of course, this principle of this action was adaptable, so he had to plug in various things. He had to find the Lagrangian characteristic of the field. But that was not everything. There was an... And as the mathematics alone didn't make it up, and the Lagrangian, of course, could be considered as somehow not really good for realists, as it was not unique, Planck had another column on which his realism stood, and that were the constants of nature. And he even claims that the constants of nature are replaced as the old substances.

57:30 So there was the elementary quantum of action, the gravitational constant, and many others. So, invariance, as again, when we praise that the principle of least action was an invariant quantity, invariance is again a variety of reality criteria. And, so, each major change not only conserves the principles, and you always could find a principle of least action, each major change uncovered a new constant, like the velocity of light and the quantum of action. And Planck combined this now, in his later writings, into a convergentist-realist argument, when he had an idea that was not uncommon in neo-Kantinism, then, of successive levels of objectivation approaching the general goal of scientific research. So he said, yeah, well, the metric was not, was relativized in the sense that it had relativized outdated concepts of space and time, but we got a new, more absolute concept and there's not relativity everywhere. So let me now come to Hilbert. In Hilbert, the principle of his action is the cornerstone of his axiomatic method, his axiomatic method in axiomatizing all 10 minutes, all natural sciences. And you will see this in a couple And Hilbert's lectures come out everywhere in mechanics, in continuum mechanics, everywhere he started from the principle of this lecture. Klein even wrote about Hilbert's fanatical belief in variation of principles. That's in the letter where he convinced Pauli to introduce Hilbert's formulation of German relativity in the famous Encyclopedia entry. Here are the steps of the axiomatic method. First, you want completeness, so you should be able to derive all the laws... There should be another switch.

1:00:00 There is just one switch, ah, you mean the one can switch to both? That might be a single bulb, so we'll use a single bulb. It seems like if something's going to happen, it's going to happen during this time, so, you know, it's dark if you hear anything coming. Okay, great, so it's now like in a tennis match, I don't know if I got any advantage from you, but... So, you want to derive all the laws of the fact-true domains as the physicist presented to you. This completeness has nothing to do with the completeness in the Goethe theorem, etc. Then you check for consistency. Play it back to arithmetics by defining the appropriate number fields. Then you check for external consistency with other physical theories. And of course after Goethe 2 didn't work anymore, but it still was. I mean, for axiomatization of physics, it was much more safe a basis if there were given an appropriate number field than just structures that were justified in other ways. Analyzing the axioms' mutual independence, you arrive at studying alternative theories by modifying single axioms or at the concept which is pertinent here, deepening the foundations. In 1918, Hilbert describes it as such. Once it has become sufficiently mature for the formation of a theory,

1:02:30 anything which cannot all be the object of scientific thinking succumbs to the axiomatic method and, consequentially, to mathematics. By penetrating into deeper levels of axioms, we also gain deeper insight into the essence of scientific thinking and become more and more conscious of the unity of our knowledge. the axiomatic method, mathematics appears to be disdain to a leading role in the sciences. So, deepening the foundation was to take the fundamental propositions, the basic empirical facts themselves, like the parallelogram of forces or Lagrangian equations, and try to reduce it to deeper structures. Now, that was not proving the basic propositions, but find deeper propositions. And he thought that deepening the foundation was important for increasing the stability of the edifice of scientific knowledge. So, let me give you some types of deepening the foundations from Neubert's 1918 writings, and sometimes I elaborate that a bit more in detail. Well, you could eliminate dependent concepts. You could simplify by deriving one concept from the other. You could establish connections with the neighboring disciplines, even before this was physically justifiable. You could reduce, and that's one of the strongest forms of deepening, you could reduce, and one of the border crossings between mathematics and physics, logical empiricists did not like, You could use physical concepts to mathematical ones. So, in general relativity, there is a reduction of all physical constants to mathematical constants, the integral invariants. And let me make some words about Hilbert's foundations of physics and his axiomatization of relativity. And here I have grouped the four axioms plus two conditions to show you the various levels of depth, so to say. So there was this axiom of the world function, and the claim that H had to be invariant with respect to arbitrary transformations of the world parameters, so that coordinates have no significance themselves. And from that, Hilbert, 1915, derived his leitmotifs here,

1:05:00 and now it's not a proof, it's first wrong and it's badly written, But later on, Ebi Noether improved on that, and so in the later versions, He was always cited what Harry called, and many people call, Noether's second theorem, and the Bianca identities. And Hilbert believed, and that's why he called it light motif in supporting the field theoretical ideal of unity, that in this way, gravitation, from gravitation, he could derive the electromagnetism, because in this Hamiltonian there's gravity plus means electrodynamics. Then he had two further axioms because H was not fixed yet. But his claim was indeed that just from the invariance here, that already followed and the claim was wrong as I said. So, you had this additive form of the Hamiltonian to guarantee that no higher order than second, and no higher derivatives in second order in the G-menu here, and then you have to fix the metric. And then there came two supplementary conditions, I think, which should also be counted as axioms, namely that causal order had to be respected and that the solutions had to be regular. And this singularity issue is as unclear, or it was as unclear for many decades as the general covariance issue which Harvey has talked about. And I would think that that would provide perhaps a nice example for the structural realist in how then relativity could be traced back to very deep invariants from which, of course, not everything followed, although Hilbert fanatically believed that this was possible. But not all deep endings are, and there are problems with that already, but not all deep endings are like that and can be traced back to basic invariants. Conceptual simplification, for instance, might not be unique. So Hilbert lauded both Boltzmann and Hertz for having deepened the foundations of Lagrange mechanics either towards forces, in Boltzmann's case, or towards constraints.

1:07:30 So constraints here means . Moreover, a mathematically deeper level might correspond to a physically non-standard formulation. So this was in the days when Hilbert tried to get a continuum view also on mechanics. The axioms of classical mechanics can be deepened if using the axiom of continuity and so on, and then one applies Bertrand's maximum principle. I think that few people will have heard about Bertrand's maximum principle. There are many original principles, but few will have heard about Bertrand's principle. type of continuity, where Hilbert suggests to have the following axiom of measurement. If for the validity of a proposition of physics we prescribe an arbitrary degree of accuracy, then it is possible to indicate small regions within which the suppositions that have been made for the propositions may vary freely without the deviation of the proposition exceeding the precise, prescribed degree of accuracy. basically does nothing more than express something that already lies in the essence of experiment, and it is constantly presupposed by the physicist, although it has not previously been formulated. And in a way, this accent is too deep to be applied to chaotic dynamics, only if you say, well, I read that as value ascription. F has value F, but if, as it is suggested here, I read it as measuring means that it, for a certain moment here, remains in a certain range, for all times, or at least for sufficiently long times, that you have problems with measuring part existence. Okay, here come the conclusions. I think that while the classical realist has sought for the basic constituents of the world the structural realist can as Hilbert indeed showed profit from mathematical deepening as they expressed in the Princey Glovely's action and I would even claim that this leveling you saw in Hilbert's axiomatization of general relativity for such axon-principle formulations, and it was also a sort of basis of Planck's belief

1:10:00 that you had the general thing, that there was such an invariant quantity as an action, or that it could be formulated with an action-principle, and then you had to plug in the Lagrangian, and then you had to specify further things, so you had the choice of Lagrangian, let's say, you say what this is all about, and with the choosing the interval, we are more specific about the problem. But as we have seen, the mathematical deepening might not always agree with the physical layering. Think of Bertrand's maximum principle. Moreover, I think that deepening the foundations, as we saw in Hilbert's unsuccessful theorem in general relativity requires what I call with Mark Wilson, mathematical optimism. So the structures you find as the biggest must also be mathematically meaningful. And that there's a certain interplay between defining what is meaningful and that you can realize the structures can be found in another quotation of Hilbert on variation of calculus in the 20th problem, where he writes, has not every regular variation problem a solution, and regular means analytic in that case, more or less, so singularity-free, provided certain assumptions regarding the given boundary conditions are satisfied, and provided also, if need be, that the notion of a solution shall be suitably extended. So, moreover, if you look to Hilbert's lectures, especially in the early days, he often applied all these variation principles and theories that were very far from a final state. And so, one might think, well, wouldn't it then be better to take the establishment of such a mathematical structure as a sort of a lack of thought experiments were very much as suggested here. You use the dialectic between conjecture and counter examples and take proofs not only as confirmation of laid out structures, but as providing a certain heuristic and indeed in the heuristics, the action principle had been remarkably successful. And so in

1:12:30 This would even help perhaps to the Leeds-Bristol position that ontic structural realism then might learn also from questionable mathematical ontology whereas otherwise it might well be a, so to say, you go a totally other track than the physicists do. And the principle of Leeds-Action then provides a large class of such mathematical thought experiments, many of which might lead to meaningful structures that are physically fundamental and some perhaps not, and so in that sense it's a partial return to Mach that the basic method of thought experiments is variation. Thank you. Fascinating talk. I just have a simple question about Hobart. Do you think, I mean, given the influence that Philip put on the variance for his gravitational action, in 1916, as you know, Einstein also introduced an action principle for John Wilton, but based on non-invariant action, and of course it's, it's non-invariant, I mean, it's It's equivalent to the Hilbert action, not to a total of different versions. But it's not invariant. Is that surprising or not? Well, I think Hilbert... Well, I mean, perhaps in 1916 probably Einstein already knew about that. I mean, there are earlier uses Einstein does in special relativity with action principle, but in a way that the Göttingen groups regard it as very pedestrian. But just the mere fact that you could do, you could have an action principle of general relativity based on non-variant action. So is that something that you're supervised? I'm not sure. I mean, I mean, Hilbert was not... Hilbert, at places, was very much thinking that these structures were the really fundamental thing. But then he gave

1:15:00 a lecture in the next term where he said, yeah, well, they're still preliminary and we played with that. So I think Hilbert's... Hilbert's attitude there is inconsistent perhaps or it would then have thought well I mean then maybe that's an invariant in a deeper sense in another sense or so I think that that was that was what was one of the messages I might have wanted to convey that there's a certain flexibility in this mathematical stretches also they are very much much deeper and much more solid but some physical concepts which which Hilbert never hesitates to await himself. I thought it was slightly tender to a question. I suppose you can help. Octavio Buena has an argument against structuralism. Octavio Buena, at the University of South Carolina, has an argument against structuralism. runs well that's what follows. The structure really kind of rather smartly points to the persistence of certain structures through the, you know, history of physics and the accumulation of understanding of these structures as a response to the pessimistic meta-induction. So, I'll tell you, well, what happens if we find cases where we get shifts or people advocating shifts to very different kinds of mathematical structures? And the And the example he gives, which I'm not familiar with myself, is of the late von Neumann giving up on Hilbert's face. Now, I was just wondering, can you see this as a consequence of the lure of the deepening? The attempt to sort of deepen the foundation may lead you to mathematical structures that are physically non-standard. So, along the lines of Buena's criticism, that's going to be a problem for any kind of religious view that emphasizes structures, because all of a sudden you get these different structures. Why shouldn't we go with those rather than the standard ones? Oh, I mean, yeah, but von Neumann, I mean, first von Neumann's book was the most influential product of the Hilbert School, much more influential than Hilbert's lectures that had been sitting

1:17:30 in Göttingen for eight decades, then, well, that's not a good example to criticize you because from Neumann, and Mikluf Reddy has shown this in a paper, I think, in 1996 in the studies, which just consequently went on, that he saw there is a problem with the Hilbert spaces, and having already introduced rich algebraic structure, he saw, well, if wanted to maintain the frequency interpretation of probability I have to go further. I have to go to type 2, 1 for Neumann algebras. And now, with the benefit of hindsight of the algebraic structure, that was a beautifully smooth transition, more or less uncovering what was the real point. And if then you look back into von Neumann's book, you see, Oh, yeah, there are already talks about what later became quantum logics that he liked, I think that he liked this, that he liked the, I think the point is that, again, I'm not familiar with this example, but the point is that, in a sense, physics didn't follow von Neumann. that's the point if you're going to be a realist you want to point to your the idea is you point to your best physical theory and you say those are the things i believe in it could be a structure really should point to your best physical theory so those are the structures that correspond to what the world is like so in some sense you know the world you know is represented appropriately in terms of hilbert's place and then there's one neumann's ghosty voice you know so i think Granted, in Hilbert's own work, one can see traces, in his early work, of the later conception of this, you know, the lure of the deep, you know, that Hilbert and von Neumann, in a sense, not suffered from, but were enamoured with you, and I thought, is this your point, can then present problems for, not just the structure really, but anyone who places on what's important on mathematical structures, deep, you can get an awful lot of structures that are very non-standard physical. Now, I think that that's not a good example.

1:20:00 I think that's something you could get easier rid of than those things I have in mind, because you could just argue, well, there's a certain historical fact, simply, that a physicist, after quantum mechanics was laid out there, and von Neumann then casted all that in stone, went on to do quantum field theory and also von Neumann in the subsequent years put all the structure in place that was good for the development of mathematical physics in the 1980s when people who had made their hands dirty like Hagen wanted to know it more clearly the physicists simply didn't look at this what I thought is that there could happen things where the physicists make progress uncovering physically deeper structure that is at odds with what you do very much a la von Neumann finding the deeper structure. So if the physicist would have really cared for the foundations of quantum mechanics and then ended up maybe after EPR, which is something that totally defies the algebraic approach. I think these are the crucial points. Further questions? Okay, then let's thank the speaker. Thank you. I think that's a hard time to study. This is not the only thing I would like to do. This is not the only thing I would like to do. This is not the only thing I would like to do. This is the case of my own thing. I have a lot of life. That's interesting. and the... and the... and the Walmart... and the... and the... and somewhere... now copy it, because it's in some obscure... Make a copy. Make a copy.

1:22:30 No, no, sorry to me. It's like the whole thing. It's like the whole thing. Which might give me a motivation to learn at the end of the day. How much do they say in this picture by half? It's about 6 p.m. paper. Actually, that equation is a little peculiar, because if you calculate out its energy and its energies, it's not for the development of this. Because it's actually a problem. Well, it is, but it's Q1Q dot. Q1 dot Q2 dot minus Q1Q2. If you express that as the other way to the difference of squares, q1 dot squared plus q2 dot squared minus q1 dot squared minus q2 dot squared. Once you take that and construct the energy, you don't get a positive or negative effect. It's not about being positive or negative. So the thing is, it makes it quite much sound. Oh, sure. Why are they not brothers and sisters? Which one? This one? This one? Sorry, write that down. Ah, yeah. Even Max Blage Institute. Yeah. I have this in the studies. That is complicated. Also, if you are interested in the data, then there's a problem where they are. Yeah, that's a study. How long are you in Italy? How long are you in Italy? Yeah, it will be one dollar. You want to see it too? It's the same thing. You want to see it? Then I'll send you the email. Just write me the name. I know the rest of the time.

1:25:00 In which group are you? Each of those are the same. And each one, we take a five of these. I'm going to take a five of them. So if I have to be a negative one, I'm going to take a five of them. So if you get... I'm going to give you some of them. Yeah, I'm going to take a six of them. I'm going to take a five of them. But... I mean, I mean, there's no other thing. I don't know. Yeah, I've never understood that. It stands alone and useless in the requirement. There's nothing that stops you from doing math. This is all the information that you can't afford to do. I'm sorry. I can't. You have to do it right here. I'm sorry. I can't. You have to do it right here. Take a look at that. I'm sorry. I can't finish cutting this down. I'm sorry. I'm sorry. I'm sorry. I'm sorry. Yeah, okay, well I'm going to kiss you on that again. Yeah, yeah. I mean, there's a lot of ambiguity in quantizing a period of time. You're sure that's another operator ordering? Is it? Yeah. Is it? Especially here. Once you get to the Amacard. In terms of the criteria of the choice of operator ordering, and the criteria of the

1:27:30 feature is going to find that. Can you show that the quantization of one becomes the quantization of the other? Why are you doing the ordering or something like that? It's so, so complicated. I mean, there's one good news. I mean, if the quantum system must have this and have this classical limit, so that both quantum systems must have the same classical limit, by the way. Well, at least you have one thing where this relates, and maybe not that expected, but if you have DL, or BQ or 5 dot or so, I mean, this thing can look at momentum, and as I say that's what goes also the road where you see whether you can make the Lagrange transformation. It's just the second derivative. So that's the road where things go wrong with Lagrange Hamiltonian having constraints. It's a similar We've got a similar role, but that's something I don't get on this scale. Neither has a gauge symmetry, right? Neither has a gauge symmetry. They're both global similarities. So the role to Hamiltonian should be unblocked in both cases. That should be shown that there is... You know, in this case, you've just got the... that P2 equals Q1, and Q1 equals P2... sorry, P1 equals Q2. This should be shown. Oh, probably, yeah. You probably have a hematone in both cases. Yeah, but it's... I don't know if you have a hematone in both cases. Yeah, but... Well, I think that... And you can say for certain that they yield different quantum theories. There's no chance that you remember that. Yeah, that's worth doing right. I don't know, that's a platform. No, but you can just check for the gauge groups. Yeah, I mean it shouldn't take longer for me to check that for myself. I think it's clear that the SO3 gets you a beautiful bad state and if you have an effort

1:30:00 to make sure you can probably get the results posted in one direction. You could contemplate that there was a subtlety involved in the choice of inner product that chose a silly choice of inner product with one that would lead you to the other one. Okay, but trauma is not an idiot. So there's reasonable mathematical pieces. Yeah, I think it would be legal. But I didn't check most of that. Oh, okay, it has a series now. Yeah, where the banks are on that. The relevance of this building. What the parts are? Not just over the relevance of the action. We come in and turn right, and just go straight down to the Mohandji. Maybe what they write down there is a Mohandji book. Yeah, that's Mohandji book, because it's still right here. We are up there here, down in place of fields. There's another Mohandji book. Yeah, but it's great. But that's probably where it's about to cite it. And who would add in that? Or to cite this paper somewhere, a big yellow strand. Yeah. You go across that fly, which is a little motorway. I'm already using my falcon anyway. I'm going down, going down, past where people went to the Portuguese restaurant. There's two branches of water students. You go all the way down, across Head Row, and then there's two branches of water students. Yeah. but I can do a these is not great academic bookstores I have to say my cause it's a small book it's okay more distance is quite nice there's the student union I will go student union bookstore I'm not sure it's open it's outside of town that's you want to go yeah that's That's quite good. Okay. What do we think is an adapter? An adapter? Yeah. Um, your best bet is, um... I should have bought one in Europe. I have one because I was from Europe. And I was told it was from Europe and I could have put it apart and gave it to you, but it doesn't work.

1:32:30 You can try at Morrison's. Thank you.