An answer in search of a question - "proofs" of tridimensionality of space / Why 3 dimensions of space aren't enough (contd.)

0:00 So it's only because string theory has this ambition to be everything that we want it to be renormalized. The other statement is that it's not the infinities that bother us. Every infinity can be renormalized away. The question is, are there an infinite number of infinities that need to be renormalized away? So the real criteria that I am putting down, which makes sense, is that predicting any experiment would require an infinite number of input parameters. I want to have a finite number of parameters on the basis of which I can predict any experiment. That's all I want. You want a theory that I can use to predict. That's right. The universe is notoriously uncaring about what I want and what I don't want. That is what I want. Maybe one more question. Okay, I'm a little bit worried about the possibility of string theory, which you emphasized. So, because string theorists have made, again, predictions that, you know... Super-symmetries around the corner, we will soon see it, and then it was never seen. There was no, to this day, no experimental signature of string theory. You said that the testability of these extra dimensions, of course, that's a very interesting and important point. So, do you know whether there are theoretical estimates as to the size of these extra dimensions? Are we close to getting there, or is it just, you know... The estimate is that they're all at the Planck scale and we'll never get there. Nothing that I said should give you the impression that string theory is making a prediction in the sense that this prediction is false by string theory alone. String theory does make predictions. All the predictions are of the form that if you take two particles and smash them together with Planck scale energy, the scattering will look like a certain thing. So these are not predictions we'll ever be able to make, unfortunately. If something happens, it can be better explained by string theory than anything else, such as logarithmic equations, but it might not happen. Thanks very much. Nick Hargett and David Hilbert are getting a paper. Keep in mind that I think they're going to split it up into two 12-and-a-half-minute presentations. And then we'll have a five-minute presentation.

2:30 I guess there's a general question that we're interested in and going to address here, which is something like, how might we, as humans, experience the world as being three-dimensional and linear, so the space we live in has that character? And really what we aim to do is just describe two proposals for how that might come about. I'm going to talk about Ponterey, and Dave's going to talk about Roger Sheppard, the contemporary psychologist. I mean, we're going to make some philosophical and interpretational points about these two people, but we don't have a big philosophical thesis, and we're not particularly endorsing one of these, or I'm not. Anyway, the purpose of the talk isn't to endorse one, but it's probably important to understand what's going on in the talk, because Poincare and Shepard are pretty obviously in disagreement in some ways, and it's not that Dave and I are having an argument up here, but... So just of course, I mean, philosophers of psychology will probably find this an interesting question anyway, but why a philosopher of physics should be interested in it, I think, well, here's how I motivate myself of why I think I find it interesting. We are sort of physical things, systems of a certain type, living in a physical world of a certain type, and that has effects, has consequences for how we experience things and how we come to know things. It may not be a complete story, but it's going to put interesting constraints on epistemology and experience. So, maybe disappointingly, we don't have any of those kinds of things to tell you, but that's why this kind of project is interesting. Why philosophers of physics and psychology should talk to each other. So I'm going to talk about Poincare and, in particular, science and hypothesis, and what's sort of in the background is some recent work on Poincare that kind of really sets straight, I think, what he said, and, in particular, Michael Friedman and Jemima van Menehem wrote a couple of nice papers. Which are in the background of what I'm going to say, and I thought I'd also mention, but also for recent interest in Poincare, Elie Zahar's book, though I think I disagree with a lot of his history. So, I'm kind of assuming most, many of you will have looked at, well at some point, at great science and hypothesis, and I'm talking particularly here about chapters three and four.

5:00 And if you'll remember chapter 3, Boncret describes non-Euclidean geometries and then right at the end he introduces the idea of geometric conventionalism and in the standard version of the book he essentially answers, the last two pages he answers three questions. Why is geometry conventional? Why is geometrical axioms conventional? What does it mean to say they're conventional? And how is that convention made? For us in particular, so for us here I mean for humans, I mean geometry is Euclidean, for Poincare, this is Poincare interpretation of science. Okay, so why are they conventional? Well, there's a sort of disjunctive syllogism, they're not analytic because they're not topologies, they're not synthetic a priori because we can conceive logically incompatible systems in geometry, and that shows that they're not synthetic a priori, and they're not a posteriori, they're not matters of experience. You know, there's a way of interpreting Poincare and thinking of introducing sort of under this standard, under the termination picture here, but really what he actually says is the problem why geometry is conventional, is that there are no rigid bodies, there are no strictly rigid bodies, and I think what he has in mind there is something like the following, since there are no strictly rigid bodies, and since bodies in geometry and geometries about bodies actually describe strictly rigid bodies, there is absolutely no... It's just not possible for the laws of geometry to be true in our world. There are no strictly rigid bodies. And this is different, say, from Newtonian law of universal gravity. I mean, when you actually study a planet and you're trying to calculate a planetary orbit, I mean, you make approximations and idealizations, but it's certainly not impossible that the gravitational forces and motions that you come up with are actually followed by the bodies. For geometry, it actually is impossible for the laws to be satisfied, and for that reason, they're not experimental truths. Okay, I'm not indeed endorsing or criticizing this argument, but that's what Poincare says. Okay, so what are they? Well, we all know that he says they're definitions in disguise, and he compares them to choices of units and choices of coordinates.

7:30 So, to make the point that there's no question, I mean, there's only a question of stipulative truth here, there's no kind of further question about which is the right stipulation to make. There's only stipulative truth here. I want to make another comment here about how I think people often think about Poincaré. In drawing this analogy between the geometrical conventions and choices of units, choices of coordinate systems, there's nothing in the text to suggest that he thinks those three choices have exactly the same character. That the choice of geometry is just like the choice of units, which is just like the choice of coordinates. There's also nothing in the text to suggest that he thinks that choice is, for instance, just a very trivial one of, say, replacing a word in a language with another symbol. These kinds of choices could have all kinds of richness. OK. Let's talk about setting up some background here. How do we make these conventions? Well, they're conventions, so they're not forced on us, but they're guided by experiment and constrained by logic, and the substance of what I want to get at is, well, the genetic aspects of this question. You can answer that logically, but just sort of... Writing down, stating what the experimental, what the experiments are and their results that you're taking into account when making this convention. But I actually want to, I think Poincare actually also addresses the genetic aspect of this question. Namely, how in fact did we come up with the convention that we have? That space is Euclidean, that geometry is Euclidean. And that's what we get in chapter four. Okay, so. This is what happens in chapter 4 of Poincare. Poincare distinguishes two different spaces, one he calls geometrical and one he calls representative. It's important that, I think, through understanding what Poincare is saying, that you realize that geometrical space covers two different Things that I naturally, for a long time, took to be distinct and found very hard to read what Poincare was saying. Literally what one might think of as being abstract mathematical space and physical space. There's no kind of, I have a sort of picture where, yeah, there's three spaces.

10:00 There's the space we use in our heads to represent things, there's physical space, and then there's an abstract mathematical geometrical space. Those last two things are just rolled into one of Poincare. So, of course, geometry in that case is essentially concrete because it's about physical space and it requires a choice because, well, there are competing geometries. So, I mean, that's important for motivating the problem here. And, of course, geometrical space for us is three-dimensional and euclidean. He mentions another space in this chapter, representative space. So this is the one in the head. It's the bit that's in the mind. We'll get to the group in a little while. It's the mental space in which we represent the world. So it's sort of a principle of his psychology that it's, I'm just going to read this, aggregates with the spaces of the modes of sense. So, for instance, ours is going to be the visual field plus the accommodation and convergence spaces, an extra visual dimension, and then a motor space plus a tactile space, and this picture for the motor space is each muscle has a little nerve that tells you how hard that muscle is working at any time. So, it's an important principle of what he's going to say, that we can only represent the world by representing experiences in this space. That's it, that's the only way we can represent what's happening in the world. Okay, so let me just say, so you can think of what's about to come. We're going to talk about having impressions, so... If you're into terminology, having an experience, having an impression, it's going to help you to think of that as just sort of picking out a point in representative space. So if you have an impression, you have some kind of visual experience and a sense of accommodation and some muscular feeling and some tactile feeling. Different impressions correspond to points roughly in this space. Okay, they're obviously not even... Topologically equivalent, so it's not going to be the case that we can just read off any easy way geometrical space from representative space as a quantum vector. Instead we're going to have to also put in regularities here. So that's one crazy problem.

12:30 So basically, we're sort of stuck in representative space, and the basic facility we have is to distinguish two points in representative space, two different impressions, and identify the same impression if we get it twice, so we get the same point in representative space. There's going to be some pretty powerful abilities to make generalizations as well. So from that, and from the regularities that appear in... In the representative space, we get to learn a group. You know what, I'm going to just go so I'm not going to give you the details here, but basically we can tell whether we can distinguish changes in our impressions that are going on, as it were, outside from inside ones, and then using that we can distinguish things that are going on the outside that correspond, we would think of as corresponding to displacements from changes of state, and then we can identify all the displacements because we can correct them in the same way. And essentially then we can learn the regularities. We've got this story that we learn essentially a group structure on a certain class, a certain set of equivalence classes of elements of this, of points of representative space. We get a group and it turns out... I mean, at the end of this story we've figured out their space and we now can think spatially, we know what was going on, these things are describing things moving around in the world. Essentially what we've learned is the group of displacements of solid objects. Juan Correa, this is practically a quote, but geometry is the study of this group, so how do we make that, how do we end up making that convention? But we get to the, we use this story to get to the geometrical space whose rigid transformations form the group we learned. Okay, we've got the first point. I just want to emphasize one thing. It's important in Poincare's story that it's a group that's being captured here. It's important for our point that the way we're capturing the structure, the three-dimensionality, the nature of space, the way we experience it is by learning a group. And not, for instance, by internalizing a kind of internal three-dimensional Euclidean manifold representative space in which we kind of think about things being located.

15:00 Because that's just impossible for Poincare. It's the principle of his psychology that we can only represent in representative space and none of our modes of sense. There is no modal space which has the structure of three-dimensional Euclidean space. There is no modal sense which has the three-dimensional Euclidean structure. So it's just impossible to represent concrete things in the world in three-dimensionality. And so the only way we can internalize the group here, the geometry of space, is via a group. Okay, so I'm in the unhappy situation of being a philosopher of psychology in a philosophy of physics session doing something even worse. I'm talking about a psychologist. But I'm going to start with some phenomena. So there's a well-known phenomena discovered early in the last century called apparent motion. Okay. Yeah, just a sec. It's so cool that we had it set up before. Yeah. It'll, it'll... Everybody can shut their eyes. Here we go. Okay, so this should look to most of you like a single dot moving back and forth rather than two different dots flashing off and on. And by the way, these are very crude demonstrations of stuff you can do very precisely and repeatedly in the laboratory if you have more skill and better equipment than I do. It does actually depend on a number of things, including the timing. This to me looks like two dots flashing rather than a single dot moving. So what we have here in apparent motion is, as the timing is right, you get a percept of a single moving object. Moving on to two flashing ones. It's not specific to dots. You can put any shape. It's the timing that's actually significant. There's some interesting regularities concerning the timing that I won't go into. But importantly for us is the motion need not actually be a translation. It can actually be a rotation instead.

17:30 This is a percept that's developed actually over a couple of iterations here. It looks like a rotation in the plane, back and forth, back and forth, and it can also rotate out of the plane. And again, there's a modestly large experimental literature, and all of these claims are kind of repeatable. And moreover, you can do much more complicated things. In fact, if you're able to use three-dimensional stimuli, which exceeded my graphical capabilities, you can demonstrate motions that combine rotation and translation. So you needn't have just the translation or just the rotation. You can have both, but you can't have that in the plane, and the plane is all that I'm able to actually do myself. Put aside some distracting possibilities, I just want to draw your attention to a couple things. The motions are of abstract shapes. You're not seeing it move because we've seen things like that that moved like that in the past. And the perceived motions don't depend on dynamical cues. It's not that you've kind of figured out what the forces were and predicted the trajectory mechanically. You don't need any such information. So that's just the phenomena. As I said, this was kind of first looked at by the Gestalt psychologists. In the early 20th century. And the basic line I'm pursuing here is, these are mostly inspired by experiments done by Roger Shepard and his collaborators. Now Shepard actually has a theory here, and Shepard is a psychologist who's actually kind of, he says retired, but he was one of the most prominent kind of workers in several areas of psychology in the 20th century, and he had a consistent actually approach to Thinking about questions involving perception and learning, and he's always asking why questions. He's not interested. Although he's a mathematical psychologist, and many mathematical psychologists do the boring thing of trotting out immense statistical and mathematical structures to describe the results of their experiments, Shepard is very rarely interested in merely describing. He wants to know why things are happening. And he's more than happy to postulate lots of internal structures explaining why it is that people behave the way they do.

20:00 So there's two questions you might ask here. Why do we see motion when there is none? Well, the obvious answer here is there's some principle of object conservation, right, that in our world it makes more sense to suppose there's a single object moving back and forth rather than two objects of identical shapes popping in and out of existence in a coordinated way near each other. The more interesting question for our purposes is why do we see the particular motions that we do, and that's in fact the question that will actually lead us back to Poincare. And I'll just also draw your attention, the observed paths are not the statistically most common paths, again it's not just that you, you're not just kind of looking back over your experience and attributing to it the way things have actually moved in your experience. In particular, if you think about most of the moving things you observe in your life, they're living things and they don't follow any such simple path. They very rarely just translate or move. They move in very complicated and hard to predict ways. Although I won't be able to argue this point, it's also not that they don't follow projectile trajectories. The perceived motions don't match any dynamically sensible ones. Now, Shepard's basic approach is that basic cognitive and perceptual attributes, say like the apparent motion you perceive in these circumstances, reflect structures There are a number of internal structures that internalize physical principles, so there's the external physical principles, and then we internalize them by acquiring an internal structure that in some sense lines up with this external fact. And he really is interested in universal physical principles, so it's things like the locally three-dimensional Euclidean structure of space that We acquire internal structures that correspond to, or in the case that I actually know better, in the case of color, it's the regularities and the variation of sunlight in the natural world, which are kind of pretty much the same everywhere, that lead to the internalization of these particular structures. So, in the particular case, apparent motion reveals a structure underlying the perception of shape and motion. And so Shepard's idea is that by investigating the motions experienced without stimulus support, there's nothing that tells you what path it followed. The default principles governing the perception of shape and motion are going to be revealed.

22:30 Alright, so the idea is by looking at what happens when the world doesn't fix it, you see kind of the basic structure underlying all of our perceptions of the world, which then can be overridden if you have actual data that tells you what your trajectory is, that's what you'll see. But in the case of PAN, you have no such data, so you have to see the machinery running on its own. And the relevant physical principle here, as I mentioned, is that space is three-dimensional and Euclidean, and also, in something that's a big difference for Poincaré, in Shepard's view, this has been internalized over evolutionary history rather than individual ontogeny. So... The perceived motion is one that's kinematically, the kinematically simplest motion. So in 3D Euclidean space, for any two positions of an asymmetric object, there will be a unique And in a relatively precise sense, simplest path along which a rigid transformation can carry that object. And this is a result proved by this congenial character, whose name Nick and I don't know how to pronounce. So I'm going to say it's Chasel, which is almost certainly wrong. But I have no idea what it is. And what Chasel proved is that the motion will be a combination of a linear translation along an axis. With a rotation around that axis, in other words, it'll be a helical path, right, where what we're seeing in the plane is the degenerate case where it corresponds to a rotation where we're thinking of the translation as rotations around a point at infinity. And so Shepard's claim is that what we have internally, the internalization of the three-dimensional Euclidean structure of space, is this kinematic geometry with these simplest motions, which are then revealed in the case of paramotion. And again, notice that the motion need not be, and is typically not, the dynamically preferred motion. Things actually don't typically move in these helical paths. Geometry is deeper than mechanics. Okay. So, again, this is an apparent motion. This is not, according to Shepard, it's not the visual system's best guess as to what the actual motion is. So when you see it as moving, according to Shepard, what you're not seeing is kind of the visual system trying to say, oh, yeah, well, I don't really have any information, so I'm going to make my best guess as to how it actually moved.

25:00 That's not the theory. This is kind of the machinery running without input. And one of the main points of the machinery is, in fact, not to tell you how things move at all. It's, in fact, to allow you to compare them to ascertain whether they're the same or not. So it reflects an underlying structure that's used to compare objects at different positions for identity, and also, Shepard thinks, to represent motion. But you needn't think of the main point of this internal structure as kind of computing the motion. It's used to represent the motion, but it's not used to tell you what it is. But it is used to give you a quick and computationally simple way of ascertaining whether two objects are in fact identical or not. These simplest transformations test for object identity and so by having internal structure, In addition, although this is not the main focus of Shepard's work, any motion can be accurately approximated by a sequence of these screw displacements. So we can represent all notions in these terms as well, but the main focus of the theory is actually on, this is a mechanism that allows us to compare objects in order to ascertain of them. How am I doing on time? I'm probably out of time, right? Well, that's good. That's the last slide. So, who cares? So, the internalized structure, according to Shepard, what this all amounts to is, in fact, it's the group of rigid Euclidean displacements, which we've just encountered in Poincare. The proposal... Nick is really uncomfortable with this. It's empirically supported by the apparent motion experiments. I tried to convince Nick that in psychology, this is a genuine example of empirical support. I think it's better than a string theory. And just to kind of make it more plausible that something empirical has gone on here, our claim is, following Shepard, that these are some alternatives that have been ruled out by this data.

27:30 Less plausible, given the data, that the group structure is not Euclidean or not modus Euclidean, that there is no group structure, that the dynamics rather than the geometry is internalized, and that a 3D Euclidean manifold is internalized. For these kinds of claims, you need to do experiments that are much more complicated and much more precise than the kind of very, very simple demonstrations I showed you a few minutes ago. So, Fong, Gray, and Shepard agree that we internalize the 3D Euclidean geometry of space, and they agree that we do it by internalizing the 3D Euclidean group. They disagree, in some fundamental sense, about whether this internalization is a result of individual learning or evolution. And that's it. Are we going to consider some reasons that we didn't know about Newton's equations? I think so. He said they're evolution. He does. He inspires some hypotheses. Is that why he's a room kid? Is it something like hardwired? No, maybe not. It's not evolution, but I'm sure it's something like that. It seems pretty hard to come up with a series to square with besides some hypotheses. I mean, you have to learn that these regularities are there. And how does that get passed on to somebody over a long period of time, I guess, or according to some sort of, like, expectancy, right? Okay, so somebody might have had to... There's not, there's not a suggestion, there's not a hypothesis, there's not a point story that I put up in a sort of evil machine of some kind. Yeah, and actually, in the value of science, it's told even more than it is over at Harvard. So, is there any experimental work or any psychological work that might decide the question whether it's modern or whether it's hardwired, evolutionary? I mean, we do see, of course, you know, dogs seem to understand space more or less well.

30:00 So other animals, presumably other animals could also, on some natural level, learn it. There are these experiments on blind and sighted kids, young kids, to see if they have the same sort of abilities to reason in Euclidean space. They're sort of given a task of looking from table to table in a room. So I think there was one set of work that said there was a big difference. And so it looks like it is learned, for example, with sexual stimuli, sighted and sighted. Well, it suggested that it was working much more carefully and in fact did not really find this effect at all. One kind of experiment, just to follow up what Nick said, is to actually try to figure out some tasks that blind people can do that would reflect this kind of internal structure. One of the problems with doing this is to make sure that you aren't just rediscovering that they're blind. So it's actually a little tricky to figure out exactly how to do the experiment. It's like exposing someone to the right stimuli over a long enough period. You could teach them to internalize a different geometry. According to Pondry, yeah. You could do the experiment then, right? Right. I mean, that's exactly what the show is incredibly showing. If you've had those experiences, you come up with... That's why you've got to be wrong. If someone just like us was in that world, we'd come up with non-Euclidean geometry. Our geometry is non-Euclidean. Maybe he changed his mind, but that's what he says in Science of Hypothesis. So, yeah, and all you have are four-dimensional experiences. Yeah, things just like us. In fact, Heinz von Forster in 1971 was doing experiments at the Biological Computing Laboratory in Illinois, teaching students to see before convention exactly this, learning to manipulate objects through a very primitive computer to learn what those, how they would behave in four different states, and he chose four different tools. I will say this in kind of partial defense of contrary that there are experiments that have been done on kind of changing the spatial structure of the world like by putting on inverting glasses and it's well known that if you're going to relearn this transformation you actually need to allow them to move.

32:30 If you don't let them move around in the world and manipulate objects, then they don't, as it were, adapt to the recurring spectacle. That's not proof, but it's a kind of suggestion in the direction of Poincare, at least. I actually know Shepard wouldn't be convinced by that for maybe good reasons. I think there was one question in the back. Yeah, a question for Nick. I'm worried by the suggestion that Poincare got the geometric space, both abstract mathematical space and physical space. Let me tell you an alternative and we can shoot it down. The alternative would be that he thought he owned it in abstract matter. I think so, I think maybe that, okay. I don't think I was trying to suppress that distinction. I don't mean to say it's taking any stance on an absolute relative debate or anything like that. I mean, even if you were sort of a relationist, you would think, well, there's the space we use to represent the physical space, and then there's all these abstract mathematical spaces. He wouldn't have that kind of distinction at all. There's not one that's the right one for using space, and then we have sort of freedom to... You can do geometry in any other space. Yeah, there's a selection of abstract mathematical spaces, and we get to choose whichever one of those we want to represent the motion of bodies in physics. Understanding the text, so understanding that chapter, I mean, he only ever talks about geometrical space. He clearly thinks there's only one. I think it's kind of hard to, it's unclear what he would say about these other ones we think about. There seem to be these other abstract possible geometrics, right? So, geometric space is Euclidean, but one could imagine an abstract non-Euclidean. Geometry space. It seems, we talked about this, and it seems that the way we would sort of suggest understanding Poincaré there is those other spaces aren't really, they are sort of hypothetical spaces that other beings who've had different species, you know, experiences from us would have had as their geometrical space. So they don't sort of stand on their own legs or something, they're on two legs somehow. Does that help at all?

35:00 The most about quantum mechanics may tell us that they are, in fact, pre-entered, I think, in the history of quantum mechanics. Thank you for your attention. Okay, so today I'm going to be talking about the wave function ontology. This is the name that I come up with to describe the thesis that quantum mechanics is fundamentally about the wave function evolving in three-dimensional space, where n is the number of particles that would exist in three-dimensional space were there such a thing. But according to the wave function ontology, there is no such. I reject the wavefunction ontology, and the point of this talk is to explain why the performance of the wavefunction ontology is crazy. Instead, it's probably better to say why you shouldn't believe the wavefunction ontology. David Albert is one of the people who endorses the wavefunction ontology. As a typical of Albert, he addresses it in a very clear and direct way, so let me just read you the quote. The space we live, the space in which any realistic understanding of quantum mechanics is necessarily going to depict the history of the world as plain and stale battle, is configuration space. And whatever impression we have to the contrary, whatever impression we have, say, of living in a three-dimensional space or in a four-dimensional space-time, is somehow flatly illusory. There aren't a lot of philosophers who explicitly endorse the wave-function ontology, but you get it talking to physicists sometimes, and I was grateful that Sean Carroll actually said,

37:30 What exists is a wave-function. And you hear this sort of thing from scientists, and if you take that seriously, that means that there is a three-dimensional space with a wave-function field or wave-function stuff evolving in this three-dimensional space. I don't buy that picture. I'm going to tell you my preferred ontology for quantum mechanics. And then I'm going to tell you my two fallback positions. If you didn't like microphone ontology, I can tell you where you should go, and fallback position number three is the wave function ontology, and so after giving my two fallback positions, I'm going to explain why I don't like the wave function ontology and give you some arguments against it. So, my ontology. I don't have a good name for my ontology, so if you have any suggestions, that would be awesome. My ontology holds that quantum mechanics is fundamentally about n particles evolving from three-dimensional space. What about the wave function? Well, I want to endorse the eigenstate and eigenvalue half of me. If not for every quantum system, then at least for the system of all the particles in the universe. Or if the universe is too big for you, whatever closed system you're interested in. Whatever system that has a pure state that you're interested in. The eigenstate of the eigenvalue half-link holds that if the system is in the eigenstate of some observable, then the system actually has the property represented by the eigenvalue associated with that eigenstate. The quantum state of all the particles in the universe, that's in an eigenstate of some observable, some funky observable. So by the eigenstate, eigenvalue, half-length, the system, all the particles in the universe, actually has a property that's represented by the eigenvalue. The wave function doesn't exist on its own. The wave function just corresponds to that property that the particles in the universe have. So that's my picture. I like my picture. I don't see any problems with it. If you have a problem with it, then you might be interested in fallback position number one. Here I'm going to reject not only the existence of the wave functions, but I'm going to say, fine, forget about the eigenstate time and value of half length, there's no property that corresponds to the wave function either. Now you might say, this is crazy, there's a problematic consequence here. Systems with different wave functions have different dynamical laws describing their evolution. If the wave function isn't something that physically exists or is manifested in the world, then you would just have to build it into the dynamical law itself. Whether you're going to be bothered by that or not depends in part on whether you think there are such things as dynamical laws. I'm happy with the sort of semantic picture of scientific theories where what's fundamental in the theory is this set of mathematical models. And the dynamical law would just be something that's heuristically useful in describing the set of mathematical models.

40:00 So different dynamical laws would describe different subsets of the space of mathematical models. I'm not bothered by that. But if you are bothered by that, here's fallback position number two. Then let's at least endorse what I'll call the mixed ontology. There is both a three-dimensional space in which n particles evolve, and there's also a three n-dimensional space in which the wave functions evolve. I don't have anything definitive to say about this other than it seems like a strange ontology. You have two disconnected spaces. I don't see what the causal connection is that holds between the particles in one space and the field in the other space. And yet the stuff in the two spaces is evolving in tandem. You have some sort of gnomic connection without a causal connection. You have counterfactual dependence between these two disconnected spaces. It's a strange picture. Maybe you want to say that there's a 3n plus 3 dimensional space and these two spaces are hypersurfaces in that space. I don't know. It gets pretty weird. That's why I spot that position number two for me. Okay. If you don't like any of those pictures, maybe you're a proponent of the wave function ontology. So let me, um, uh, look, turn to that. And actually, before giving the arguments, I forgot, I'm going to do a brief history of the wave function ontology. In fact, I'm going to be looking at, if you look at the people, early, uh, people who worked in quantum mechanics, they talked about the wave function ontology, and they explicitly rejected it. So, Schrodinger. We'll stop there. Schrodinger. He started out trying to interpret the wave function realistically, but he ran into problems once he moved from a one-particle system to a two-particle system, and in this 1926 paper, he spent most of the paper talking about the one-particle system. Oh, and the references are at the back. He spends most of the time talking about this one particle system, talking about more particle systems, and he very briefly talks about two particle systems. His one sentence is talking about the wave function for two particle systems, and this is the one sentence that he says. The direct interpretation of this wave function of six variables in a three-dimensional space meets, at any rate, initially with difficulties of an abstract nature. Let me pick up on this in an 1826 letter to a short and clear, and here's what Lorenz has to say. If I had to choose now between your wave mechanics and the matrix mechanics, I would give the preference to the former, your wave mechanics, because of its greater intuitive clarity. So why does one only have to deal with the three coordinates, x, y, and z? If, however, there are more degrees of freedom, then I cannot interpret the waves and vibrations physically, and I must therefore decide in favor of the matrix mechanics.

42:30 Schrodinger kept working on trying to come up with a realistic understanding of this wave function. He eventually gave up. In 1935, he wrote a letter to Einstein, where Einstein had complained about Schrodinger. He said, Schrodinger, you're interpreting this wave function realistically. Stop doing that. And Schrodinger wrote this back to Einstein, I am long past the stage where I thought the wave function was somehow a direct description of reality. It wasn't just the early people working in quantum mechanics who rejected the wave functionality. David Bohm, an interpreter of quantum mechanics, also rejected the functionality. Here's what Bohm has to say about it. Well, our theory, meaning Bohm's theory, while our theory can be extended formally in a logically consistent way by introducing the concept of a wave in a three-dimensional space, it is evident that this procedure is not really acceptable in a physical theory. Well, that's all Bohm said. Bohm didn't say why it's evident. I wish he had. My read of what he's talking about there is that he says, yeah, it's mathematically viable to represent the theories consisting of objects evolving in three-dimensional space, where in Bohm's theory, by the way, what you have is you have the wave function field in three-dimensional space, and you have a single point particle in that three-dimensional space that represents the location of the end point particles in possibly hypothetical three-dimensional space. Right, because Bohm's theory is a theory where you actually have these point particles revolving in three-dimensional space. So, my take on Bohm is that Bohm's idea is that it's mathematically viable to represent the theories consisting of objects revolving in three-dimensional space, but it's not physically viable. And the reason it's not physically viable is that three-dimensional space isn't an accurate representation of the physical three-dimensional world. I think that's what Bohm would say if you asked him to spell out this idea. Bell rejects Bohm's own interpretation of Bohm's theory. This is the first proponent, explicit proponent, I know about the wavefunctional analogy. Bell says, no one can understand this theory, meaning Bohm's theory, no one can understand this theory until he is willing to think of psi as a real objective field, rather than just a probability answer, even though it propagates, not in three spaces, but in three answers. That's what Bell says about it. There's something weird about Bell's position, though. He says, right around this passage, he says, in this paper, he says, he talks about the wave side having an action on the particles.

45:00 It doesn't say particle, he says particles. Talking about the wave side having an action on the particles. So what picture does he have in mind there? Is it the mixed ontology that he's endorsing where you have the particles in three-dimensional space and the wave side, which he explicitly recognizes as in three-dimensional space, having an action on the particles in the three-dimensional space? Does he think there is this causal connection between these two disconnected spaces? He doesn't spell it out. Did he misspeak? Did he mean to say that the wave side has an action on the particle, meaning that one... Is the point particle evolving in the three-dimensional configuration space? I don't know. I mean, it's not clear to me what the charitable reading is here. And I think that's starting to get at the problems that you might want to think of the wave function ontology as. So, let me turn now, we're at section five of the handout, let me turn now to two problems that I want to raise with the wave function ontology. Problem number one. It's radically revisionary with respect to our everyday understanding of it. We think that the world consists of objects with a length, breadth, and depth evolving in three dimensional space. But according to the wave function ontology, there are no objects in three dimensional space. There is no three dimensional space, at least at the level of fundamental reality. It seems obvious to me that this is radically revisionary. I've had people argue against it. It seems to me like you're endorsing a skeptical hypothesis about the world. In the brain in the bat scenario, for example, you think that there's an external world that you're interacting with normally, but actually just a brain in the bat. This is even more radical to the brain in the bat scenario. Because in the brain in the bat scenario, at least you have a brain. You're right about that. At least you're living in a three-dimensional space. You're right about that. But on the way to function ontology, there's no brain, there's no three-dimensional space. This is really radically revisionary with respect to our everyday understanding of the world. Why is that problematic? It's problematic because I want to endorse a pragmatic maxim, which I think has had much usefulness in the history of science. The pragmatic maxim I want to endorse is... One should not accept theories which radically revise people's everyday understanding of the world, when there are other, at least equally acceptable theories, which do not entail such extreme revision. So if you've got a better theory out there that isn't a crazy one, you know, don't endorse the crazy one. Endorse the better theory. And what I want to put on the table as a better theory is what I call in section 2 my own policy.

47:30 Endorsing that I can say is like half length of the system of particles in the universe. Okay. So that's column number one. And that's actually, like, the main reason I have not to like the wavelength on topology. It's just, like, a really fundamental reason. But, I mean, I recognize that it may not be totally convincing. So, let me raise column number two, which also isn't totally convincing. Our games and philosophy actually aren't. But here's column number two. It's just not clear to me that mental state would exist in the world as described by Haber. Albert argues that the Hamiltonian of the equation of motion of the wave function determines in what way the objects in three-dimensional space represent a hypothetical world with multiple particles. Here's what he says about that. The quantum mechanical world with one sort of Hamiltonian will appear to its not-too-closely-looking inhabitants to have two dimensions, and the quantum mechanical world with another sort of Hamiltonian will appear to its not-too-closely-looking inhabitants, just as our own does, to have three. So it's the Hamiltonians, the dynamics that you're working with, that determines the way the world would appear to a observer. And note that by telling this sort of story, he's pretty much taking for granted that there are inhabitants that are having appearances. In Albert's view, there's nothing that ontologically privileges one correspondence over another. All the correspondences are equally hypothetical. So if some correspondence is settled, if the one that Albert talks about as being the natural one, if that correspondence were to actually hold, if you actually had a mixed ontology with a three-dimensional space and a three-indimensional space and the natural correspondence held, then mental states would exist. But if some funky correspondence held, you could have the same wave functions with the same Hamiltonians, but just treat that as representing a different sort of... And then you would have particles evolving in all these funky ways, and on that picture, mental states wouldn't exist. But the fundamental problem I have with this picture is that none of those correspondences is real. The one that Albert talks about as being a natural one isn't real, and the unnatural ones aren't real either. So why would you think that mental states exist at all? It's not a knock-down argument, but it's just a worry that I have, and Albert doesn't have any story to tell about why you would think mental states would exist at all, given the way functions ontology.

50:00 Okay, section 6. Peter Lewis has a paper forthcoming in VJPS where he attempts to save the way functions ontology from these two problems I raised for Albert. He's familiar with these two problems. Here's one of Lewis's claims, a claim that I want to pick up on, that I don't want to argue about. There's a sense in which the wave function is a three-dimensional object living in a three-dimensional space, in the sense in which it is a three-end-dimensional object living in a three-end-dimensional space. So Lewis is trying to say the wave function is positive, but he's trying to say it's not so bad. There's a sense in which you can understand the wave function as evolving in three-dimensional space. Lewis gives two criteria for understanding the dimensionality of space, and he thinks that there's a sense in which each criteria is right. I want to talk about each criteria individually, and I want to reject them each individually, and I also want to reject the idea that we should take them together. So, Lewis's criterion number one for understanding the dimensionality of space. And this is the criterion that gives you a three n-dimensional answer. Lewis writes, so what does it mean to claim that a system has a certain number of functions? The most straightforward answer is that it requires that many independent coordinates to parametrize the properties of a system. I'll talk about that in a second. Let me just put on the table Lewis's criterion number two for understanding the dimensionality of space. In this criterion is the one that gives the answer that the wave function is evolving in three-dimensional space. His criterion number two depends on coordinate transformation. He writes, in order for the form of the quantum mechanical Hamiltonian to be invariant under the choice of coordinate systems, specifying the origin and direction for three axes must suffice to specify the coordinate system for the configuration space. Here's a nice argument for that. I won't go into it here. We can go into it in Q&A if you want. I'm more interested in whether, assuming that he's right about that, whether that can yield the answer that there's a sense in which the wave function evolves in three-dimensional space.

52:30 I want to hold that the wave function evolves in three n-dimensional space, you know, if it were to exist, it should be understood as evolving in three n-dimensional space, but I'm not happy with Lewis's criterion number one. I'm not happy with the idea that you just look at how many independent coordinates it takes to parameterize the properties of the system. The reason I'm not happy with that, and actually I'm not sure about this, but, because I'm not sure what Lewis means by talking about independent coordinates. I'm not sure what work that Paul of American Independence is doing there. But my worry is that a location in the space of any finite dimensionality, or my worry is that he's not adequately dealing with the following fact. The fact is that a location in any space, in any space of any finite dimensionality, can be picked out with a single coordinate. For example, you have a six-dimensional space, you can pick out a location in that space with a single coordinate. What we would typically represent is a point, x, y, z, a, b, c, representing the six dimensions, that can be picked out with a single number, 2 to the x, 3 to the y, 5 to the z, 7 to the a, 11 to the b, 13 to the c, so you have this sort of coding that you can do. So, I'm not happy with appealing to the number of coordinates it takes. I don't think you should look at the number of coordinates it takes to determine the dimensionality of space, because you can always pick out a point in the space with a single coordinate, regardless of how many dimensions the space has. Okay, here's my problem with criterion number two. Criterion number two is a criterion we call that you look at what coordinate transformations are allowed in the space. And my problem here is that I think the dimensionality of space is a metaphysical issue, not a scientific problem. And criteria number two doesn't use factors. What coordinates are allowed in the space? Scientific matter, not metaphysical one. The dimensionality of the space should be independent of what scientific science we're using to describe what's going on in that space. Or even, I think, to describe the structure of the space itself. Distortion space should just be fundamentally a metric for mathematics, and you don't want to have it depend on things like coordinate transformations. You don't want to have it depend on things like the very idea that the Hamiltonian should be invariant from the choice of coordinates and stuff. That again seems like a time-trivial requirement. A natural time-trivial requirement, but a time-trivial requirement nonetheless. Okay, but Lewis doesn't even want to endorse criterion number one or criterion number two individually.

55:00 He wants to give a multi-criterial concept to the dimensionality of the difference. Where he wants to use both criterion number one and criterion number two. And I'm not happy with that either. From the standpoint of metaphysics it seems to me like the dimensionality of space shouldn't be an ambiguous or vague matter. Is the space both three-dimensional and not three-dimensional? Is the space neither three-dimensional nor not three-dimensional? On Lewis's picture, you're allowed to ask those sorts of questions. Graham Preece would be very happy to answer those sorts of questions, but we shouldn't be. So I think there's something wrong with this sort of multi-criterial account of what the dimensionality of the space is. Here's my criterion for understanding the dimensionality of a space. And this criterion gives the answer to the wave function. This is a three n-dimensional object, which is the position that I want, and which is why I don't like the way of function on topology. And I'm not completely on board with this criterion, so if people have better ones, I'd be happy to hear them. This is a criterion I learned from David Lewis. You take a point, a zero-dimensional space is permanent, and then you define a line, a one-dimensional space, is that which can be bisected by a point, and you define a plane, a two-dimensional space, is that which can be bisected by a line, and so on. This might not work for all sorts of spaces, but it seems like it works for simple enough ones. And on that criterion of understanding the dimensionality of space, you get the result that the wave function doesn't create any dimensional space. So, here's my conclusion. I don't think Lewis can save the wave function ontology from the two problems I raised with Haber. I think that the wave function is unqualifiedly existing in three-dimensional space. There's no sense in which the wave function is existing in three-dimensional space. And if that's right, then the wave function ontology really is radically revisionary with respect to our everyday understanding of the world. And that is a reason not to like it. And I also think it's not clear whether mental states would exist, even given Lewis' picture. Now, I'm willing to grant to Lewis that he does better than Albert in the sense that Lewis has a privileged correspondence between the three n-dimensional space and the three-dimensional space. Albert wanted to allow for all these different correspondences. Lewis wants to build more structure in the three n-dimensional space such that

57:30 It seems like it only allows for one sort of correspondence to the three-dimensional space. But the three-dimensional space is still hypothetical on Lewis's view. So it's still not clear to me that mental space would exist. Let me give an analogy for the classical case. The classical picture has a three-dimensional space with n particles evolving in that three-dimensional space. There's another picture, though, where you could classically have a three n-dimensional configuration space with a single particle evolving in that three n-dimensional configuration space. Now, suppose someone came and said, I think that configuration space picture is the right one. I think that all that exists, you know, and we're living in a classical world, so, you know, and they would say, well, I think all that exists is a single particle evolving in that three n-dimensional space. And then the question is, if they're right, would mental states exist? In that world, would all the billions of mental states supervene on the evolution of that single particle evolving in the three n-dimensional space? I mean, maybe it would, but maybe it wouldn't. And it's just not clear to me, and that's why it's not clear to me whether mental states would exist in a world where the three n-dimensional, where the wave function and the topology hold. It's not clear to me that mental states would exist in a world that is three n-dimensional. That's not a knock-down argument, and the fact that the ontology is radically re-visionary isn't a knock-down argument either, but given these sorts of strange consequences, you know, maybe you would even just want to identify them as slight problems in the way of function ontology, well, I have a much better picture, going back to section number two, where you endorse the eigenstate and eigenvalue pathways, or the... System of particles in the universe as a whole. So given that we have this really nice picture, why should we mess with the wave function ontology? If you don't think it's crazy to do it, at least I hope that you don't believe it. Okay, thank you. A number of points, but first of all, I don't see, I really don't see the issue about mental states being a problem whatsoever for some of the three-dimensional space. Right, so take the Bohm theory. Point in coordinate space specifies the position coordinates of all, say, the particles making up your body. So including all the particles that make up your brain, right? So that one position can correspond to a mental state, a physical state, which for rough purposes we can take to... The supervenience basis for mental states. So there we have a three-dimensional space. A point in the configuration space picks out some configuration of a person.

1:00:00 That person has a definite mental state. Ergo, no problem. Now, the Everettian would be able to do a similar sort of argument, right? It says, okay, we have a three-dimensional space. We have a probability amplitude over positions of particles of people's bodies. That's enough to give us a notion of mental state. So, if you're comfortable with billions of mental states Well look, what do you mean by a single point particle? It's not a single point particle, it's a point in configuration space which corresponds, which is interpreted as being corresponding to the positions in which particles will be found and you can either be a Bohmian and say you do have these positions or you can be an Everettian and say Right, let's take a look at the Bohmian picture. But on the wave function ontology for Bohm's theory All you have is the three-dimensional space. You talk about this correspondence that holds? No, no, you have the three-dimensional space and you have the definite particle position in it. You have the core puzzle. All you have is a three-dimensional space with a single point particle evolving in that space and a field, the wave function field, that pushes that single point particle around. Yeah. And so you're talking about this sort of correspondence that holds, but that correspondence is a hypothetical one because there's no three-dimensional space at all. No, that's not right because there's a configuration...