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NOÛS 46:3 (2012) 525–560 The Status of our Ordinary Three Dimensions in a Quantum Universe1 ALYSSA NEY University of Rochester Abstract There are now several, realist versions of quantum mechanics on offer. On their most straightforward, ontological interpretation, these theories require the existence of an object, the wavefunction, which inhabits an extremely high-dimensional space known as configuration space. This raises the question of how the ordinary three-dimensional space of our acquaintance fits into the ontology of quantum mechanics. Recently, two strategies to address this question have emerged. First, Tim Maudlin, Valia Allori, and her collaborators argue that what I have just called the ‘most straightforward’ interpretation of quantum mechanics is not the correct one. Rather, the correct interpretation of realist quantum mechanics has it describing the world as containing objects that inhabit the ordinary three-dimensional space of our manifest image. By contrast, David Albert and Barry Loewer maintain the straightforward, wavefunction ontology of quantum mechanics, but attempt to show how ordinary, three-dimensional space may in a sense be contained within the high-dimensional configuration space the wavefunction inhabits. This paper critically examines these attempts to locate the ordinary, threedimensional space of our manifest image “within” the ontology of quantum mechanics. I argue that we can recover most of our manifest image, even if we cannot recover our familiar three-dimensional space. 1. Introduction For those of us who take our ontological cues from fundamental physics, the dimensionality of the world we inhabit is something about which we have learned to become quite flexible. The world may appear three-dimensional. For example, tables appear to have just the three dimensions of height, width, and depth. People seem to extend out in one dimension from head to toe, C 2010 Wiley Periodicals, Inc. 525 526 NOÛS two dimensions in girth, and no more. But if the best physics tell us that the space we inhabit really has four, five, or eleven dimensions, we can, without doing too much damage to our sense of what kind of creatures we are and what kind of world we inhabit, come to understand ourselves as occupying a higher dimensional space. In general, we can be convinced that our world is very high-dimensioned indeed. To be satisfied, we may only demand a story about how the familiar three dimensions of our manifest image2 are contained within the higher-dimensional world of that theory. We may consider several twentieth century, fundamental theories of physics in order to illustrate our flexibility on this score. Upon learning Einstein’s theories of relativity, we may be motivated to view ourselves as inhabiting not a mere three-dimensional space but instead a four-dimensional Minkowski space-time. Many of us are able to take on this revision to our prior conceptual scheme rather easily. We learn that we do not after all have just the three dimensions of height and girth, but in addition, a fourth dimension of temporal extension. Although the proposed revision to our earlier view about our dimensionality may have been surprising, the fact that relativity theory allows us to construe the three dimensions of our manifest image as three of the four dimensions of its postulated structure, makes the resulting theory easy to accept all things considered. Several of the last century’s theories attempting to unify the fundamental forces also stipulated facts about our world’s dimensionality that challenged ordinary appearances. The first theory along these lines was the Finnish physicist Gunnar Nordström’s 1914 attempt to unify electromagnetism with gravity, a predecessor to Kaluza-Klein theory, positing four dimensions of space in addition to the one dimension of time. Surprisingly, when general relativity is modified in this way to incorporate an additional spatial dimension, it is able to make correct predictions regarding electromagnetism (see Smolin 2006, chapter 3). This proposal differs from Einstein’s in one way that is relevant to our discussion. According to Kaluza-Klein theory, it is not merely that ordinary objects and people have more dimensions than the ordinary three dimensions we thought they had, but indeed that space itself contains four rather than three dimensions. (In the Minkowskian spacetime that many take to be the metaphysical upshot of special relativity, the total number of dimensions of our universe is four. In Kaluza-Klein theory, there are five total dimensions: four of space, and one of time.) Nevertheless, this theory, like Einstein’s, suffers little tension with our prior image of the space we inhabit. After all, the physicist can find ways to account for the fact that although this additional spatial dimension exists, we never noticed it before. For example, one might conjecture (as Oskar Klein in fact did) that the new, posited spatial dimension is wrapped up in such a way that it is too tiny to notice. As Lee Smolin puts it: . . . we can make the new dimension a circle, so that when we look out, we in effect travel around it and come back to the same place. Then we can make The Status of our Ordinary Three Dimensions in a Quantum Universe 527 the diameter of the circle very small, so that it is hard to see that the extra dimension is there at all. To understand how shrinking something can make it impossible to see, recall that light is made up of waves and each light wave has a wavelength . . . The wavelength of a light wave limits how small a thing you can see, for you cannot resolve an object smaller than the wavelength of the light you use to see it. (2006, p. 39) On this way of viewing the theory, the new dimension is wrapped up too small to see, and the other three spatial dimensions of the theory are just the ordinary three dimensions of our manifest image. Theories promising to unify the fundamental forces that followed shared these properties of Kaluza-Klein theory. For example, recent versions of string theory describe the world as containing sometimes ten, sometimes eleven spatial dimensions. Although before considering these theories, we might have thought that the world contained just the ordinary three dimensions in which tables have height, width, and depth, we can be quite flexible about revising this belief should we become convinced by the accumulated evidence supporting the theory. This is possible because in string theory, just as in Einstein’s theories of relativity, the ordinary three dimensions of our acquaintance are still contained within the theory’s structure. Brian Greene, in his defense of string theory, uses the analogy of a garden hose seen at a distance to illustrate how easy it is to reconcile the lessons of string theory with our manifest image of the world: . . . just like the horizontal extent of the garden hose, our universe has dimensions that are large, extended, and easily visible – the three spatial dimensions of common experience. But like the circular girth of a garden hose, the universe may also have additional spatial dimensions that are tightly curled into a tiny space – a space so tiny that it has so far eluded detection by even our most refined experimental equipment. (Greene 1999, p. 188) In each case we have considered, the extra dimensions of the theory are either additional, non-spatial dimensions (corresponding to time, for example), or are spatial, but are too small to see. Either way, we have an account of why we may have previously missed them. Contemporary quantum mechanics, however, is a kind of fundamental, physical theory that profoundly tests the limits of our flexibility regarding what we may understand to be the dimensionality of our world. As we will see, on any straightforward ontological reading of quantum mechanics, the theory requires the existence of an object, the wavefunction, that inhabits an extremely high-dimensional space: configuration space. And this, combined with the fact that quantum theory is as well justified as a theory can be, gives us at least prima facie reason to believe that we inhabit this extremely high-dimensional space. The problem this raises, that on which the present paper focuses, is that no three of the many dimensions of configuration space 528 NOÛS correspond in any direct way with the three dimensions of our manifest image. It is for this reason challenging to see our world as a quantum world. We are missing the account we desire in order to comfortably view the physical space of our world as higher-than-three-dimensional. We have a well-justified theory in quantum mechanics, but lack an accompanying story about how the familiar three dimensions of our manifest image are contained within the higher-dimensional world of that theory. In the next section, I will say enough about contemporary versions of quantum mechanics so it will be clear why any straightforward quantum ontology suggests that our world includes a physical space distinct from the three-dimensional space of our manifest image. Section 3 expands on the nature of this configuration space of quantum mechanics, along the way clearing up some confusions that often arise in its characterization due to the use of a historically-connected but quite distinct concept of “configuration space” in classical mechanics. This section also shows why our familiar three dimensions do not correspond to any of the many dimensions of configuration space. In the following sections, I examine two very different strategies for finding ordinary three-dimensional space within the ontology of quantum mechanics albeit in a somewhat less straightforward way than was accomplished for relativity theory and the various unification theories. Section 4 examines a proposal of Tim Maudlin, Valia Allori, and her collaborators to reject what I have been calling the straightforward reading of quantum mechanics. These authors suggest replacing this wavefunction-centered reading of quantum ontology with an ontology closer to that of our manifest image of the world. Section 5 examines a more scientifically conservative strategy of David Albert, Barry Loewer, David Wallace, and others to find our three-dimensional world within the wavefunction ontology of quantum mechanics. This requires a somewhat less straightforward way of locating our ordinary three dimensions than one could find in the cases of special relativity and the unification theories. For, as we will see, no three dimensions of configuration space correspond directly to the three dimensions of our manifest image. Albert and Loewer have found an inventive way of viewing the relationship between quantum ontology and appearances that does give us a way to recover most of our manifest image, but as we will see, it doesn’t in the end genuinely allow for the existence of the three-dimensional space they were after. And this raises an important question. In coming to terms with the confusions and ambiguities that beset early versions of quantum mechanics, many philosophers of physics were inspired by the remarkably clear-headed insights of John Bell who in a series of now-classic papers (printed together in Bell 1987), provided several coherent and precise ways of understanding the ontology of quantum mechanics. One thing Bell emphasized again and again was the need for any such ontology to include what he called ‘local beables’: that is, entities with well-defined locations in threedimensional space (or four-dimensional space-time). This was a view shared The Status of our Ordinary Three Dimensions in a Quantum Universe 529 by many others, including (arguably) Albert Einstein and Hans Reichenbach. The question is: do we really need to locate the familiar three dimensions of our manifest image within a theory’s ontology and structure in order to be able to view ourselves as genuinely inhabiting the world of that theory?3 Unlike other fundamental physical theories positing spaces of higher dimensions, it appears that quantum mechanics does not give us a natural way of seeing ourselves as genuinely three-dimensional, occupying the threedimensional space we think we do. Nevertheless, the theory is highly justified and can provide us with a clear and precise ontology. And as we’ll see, despite the high dimensions of the quantum ontology, it is still possible to give an account of ourselves and the other objects of our manifest image using the resources of quantum mechanics, if not an account that saves three-dimensional space. This paper assumes for the purpose of the discussion that the correct version of quantum mechanics is going to be a realist version of quantum mechanics, in particular, one that takes quantum mechanics to be a comprehensive theory of what the world is like fundamentally: a theory that is able to tell us what sorts of (mind-independent) entities the world contains most fundamentally, and what kind of space these entities inhabit. Several authors have challenged this way of understanding quantum mechanics, arguing instead that quantum mechanics should be viewed as descriptive merely of the kind of information we can have about the world in certain contexts, and about how we should update our beliefs about the world over time (e.g. Fuchs 2003). I will not argue against such anti-realist versions of quantum mechanics here. Indeed, it is possible to read this paper in such a way that it provides someone with one more reason to take an anti-realist stance towards quantum mechanics. For if one takes the belief that our world is three-dimensional to be incorrigible, then the arguments in this paper ought to lead one to believe that quantum mechanics is not a theory that provides an objective description of our world. My own view is that we do not have incorrigible beliefs about the dimensionality of the space we inhabit, but rather that our beliefs about the dimensionality of physical space may be corrected by what fundamental physics tells us. The reaction that most physicists and philosophers have to relativity and Kaluza-Klein theory supports this fact. 2. Realist Versions of Quantum Mechanics By a ‘realist version of quantum mechanics’, I mean one that takes the theory to be aimed at providing a true description of a world independent of us as observers. Some approaches to quantum mechanics aren’t realist. For example, as we have just noted, many physicists prefer an information-theoretic understanding of quantum mechanics according to which the theory doesn’t describe the world independent of us as observers, but rather the evolution 530 NOÛS of our states of knowledge as we perform experiments.4 In general, realist versions of quantum mechanics are intended to be descriptive of an object or objects that exist independently of us or any other observer. I use the term ‘anti-realist’ to describe versions of quantum mechanics that have it centrally concerning observers or minds. There are several realist versions of quantum mechanics currently on offer. Since the purpose of this paper is not to provide an overview of these approaches but rather to address the physical structure that is common to all of them, my discussion of these theories will be brief.5 There is now a consensus (at least among most philosophers of physics) that the so-called orthodox Copenhagen account of quantum mechanics is, to put it mildly, not promising. According to this version of quantum mechanics, formulated perhaps best by John von Neumann in 1932, states of quantum systems evolve according to two fundamental laws, the Schrödinger equation and what we will call ‘the collapse postulate’.6 Both laws describe the evolution of quantum systems by describing the evolution of the state of what is called ‘the wavefunction’.7 The laws differ in several ways, the most striking of which being that Schrödinger evolution is completely deterministic, while the collapse postulate is an indeterministic law. In other words, given the state of the wavefunction at one time, t 1 , the Schrödinger equation specifies a unique state for the system at any later time, t 2 . This is not so if the system is instead obeying the collapse postulate. For given the state of the wavefunction at a time t 1 , the collapse postulate gives only chances that the wavefunction of the system will be at any other state at a later time, t 2 . Because these laws give different predictions regarding the future states of quantum systems, the question immediately arises: in which circumstances does each law obtain? Von Neumann’s version of quantum mechanics stipulates that for the most part, systems obey the Schrödinger equation. However, when a measurement is being performed on the system, it is not the Schrödinger equation, but the collapse postulate that applies. The trouble with this version of quantum mechanics is that it contains no fundamental, physical account of measurement. What kind of physical systems evolve according to Schrödinger dynamics? Which according to collapse dynamics? The theory does not give an answer in physical terms, but rather in imprecise, ambiguous, and seemingly observer-dependent language (‘measurement’) that has no place in fundamental physical theory. In Bell’s words: The concept of ‘measurement’ becomes so fuzzy on reflection that it is quite surprising to have it appearing in physical theory at the most fundamental level . . . . And does not any analysis of measurement require concepts more fundamental than measurement? And should not the fundamental theory be about these more fundamental concepts? (1987, pp. 117–118)8 In recent decades, several more promising versions of quantum mechanics have been developed, all of which clearly avoid this problem of the The Status of our Ordinary Three Dimensions in a Quantum Universe 531 Copenhagen view. Three stand out as the subject of serious scrutiny by physicists and philosophers of physics: Everettian quantum mechanics (sometimes called the ‘many worlds view’), Bohmian mechanics, and the spontaneous collapse theory of Ghirardi, Rimini and Weber (hereafter GRW). According to the Everettian view9 , the only fundamental dynamical law governing quantum systems is the Schrödinger equation. As noted above, this is a completely deterministic law describing the evolution of states of the quantum wavefunction over time. There is no collapse law on this version of quantum mechanics, and thus no need to distinguish (using for example, such a problematic term as ‘measurement’) when one law does or does not apply. The theory is thus perfectly unambiguous and precise. According to Bohmian mechanics10 , there are again two fundamental laws. There is the Schrödinger equation which deterministically governs the evolution of the wavefunction over time, and then another deterministic law, one we may call ‘the particle equation’11 that predicts the behavior of something else over time as a function of the state of the wavefunction. The nature of this something else is a matter of debate. Some argue that the particle equation describes the evolution of a system of many particles over time; others that the particle equation only describes the evolution of one particle, sometimes called ‘the world particle’, over time.12 Although this version of quantum mechanics contains two dynamical laws, unlike Copenhagen quantum mechanics, Bohmian mechanics does not suffer that theory’s problem of specifying when one or the other law holds. The reason is that each law of Bohmian mechanics describes the behavior of a distinct entity (or system of entities). The Schrödinger equation always describes the behavior of the wavefunction. The particle equation always describes the behavior of the particle or particles. Since the laws describe the evolution of distinct entities, there is no tension in the resulting predictions of the two laws.13 A third version of quantum mechanics is the GRW spontaneous collapse theory, named after the physicists Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber who developed it in a 1986 paper.14 Like the Copenhagen and Bohmian approaches, this version of quantum mechanics employs two dynamical laws, and one is again the Schrödinger equation. And, like the Copenhagen view, according to GRW, this law does not universally apply to quantum systems. Instead, for any given time, the second law specifies a precise probability that the system will “undergo a collapse”, in other words, that the system will momentarily cease obeying the Schrödinger equation, and take on a state with certain precise features.15 The probability of collapse at any given time is determined by certain features of the system’s wavefunction. The resulting picture of the world is indeterministic – given a quantum system at a given time, t 1 , it cannot be predicted with certainty what the state of the system will be at a later time, t 2 . For there is always some chance the system will undergo a collapse. Nevertheless, the theory is precise and like Everettian quantum mechanics and Bohmian mechanics, the GRW 532 NOÛS theory does not suffer the problem Bell noted. It never mentions measurement, nor any other similarly problematic concept. According to GRW quantum mechanics, when there is a collapse of the wavefunction, this does not occur because a “measurement” occurs on the system. Instead, according to GRW, collapses occur randomly, with the precise probabilities of collapse specified by the theory’s second law as a function of fundamental physical properties of the system.16 On the most straightforward, ontological understanding of all of these realist versions of quantum mechanics, we have at least one law, the Schrödinger equation, that describes the behavior of at least one unfamiliar entity: the wavefunction. What do I mean by ‘the most straightforward, ontological understanding’ of these theories? This may be illustrated by invoking an example from a physics with which we are more familiar. Recall the central 2 dynamical law of Newtonian mechanics: Newton’s second law, F = m ddtx2 . On the most straightforward, ontological understanding of this theory, we have a law that describes the behavior of any material object17 over time. The law asks us to figure out the forces acting on this material object, and from here, once we know the object’s mass, we can calculate the change in position this object will undergo, thus knowing where it will be at future times. The Schrödinger equation also has a nice and compact formulation: Ĥ = i~ ∂ . ∂t Just as the left-hand side of Newton’s second law concerns the forces acting on a material object, here, the left-hand side of the Schrödinger equation, concerns the total energy of the system in question. This is what the ‘ Ĥ’ operator (the Hamiltonian) signifies. On the basis of information regarding a system’s total energy, we can calculate the state of the wavefunction at future times. The ‘ ∂t∂ ’ signifies just this, the change in the state of the wavefunction over time. The only other terms on the right-hand-side are the imaginary number i and the constant ~. Just as on a straightforward, ontological reading, the central, dynamical law of Newtonian physics describes the temporal evolution of a material object’s position, the straightforward, ontological reading of the central, dynamical law of realist quantum mechanics describes the temporal evolution of the state of the wavefunction. At least this is the way it has looked to many authors. Peter Lewis puts the point the most succinctly: The wavefunction figures in quantum mechanics in much the same way that particle configurations figure in classical mechanics; its evolution over time successfully explains our observations. So absent some compelling argument to the contrary, the prima facie conclusion is that the wavefunction should be accorded the same status that we used to accord to particle configurations. (2004, p. 714) The Status of our Ordinary Three Dimensions in a Quantum Universe 533 If we look to a central dynamical law of realist versions of quantum mechanics, the straightforward ontological reading of the law is that it is about the wavefunction. It is possible for one to think here that there has been some kind of confusion about the status of the wavefunction in quantum mechanics. One might suggest that what quantum mechanics really describes is the evolution of a system of particles, or bits of matter. The ‘wavefunction’ is just a name for the overall state of this system of particles at a time. So phrases like ‘the state of the wavefunction’ are nothing but shorthand for ‘the state of the wavefunction of the system of particles’. And when one sees it this way, one views the former phrase as redundant – what the Schrödinger equation describes as evolving is just a system of particles over time, not some other mysterious object, the wavefunction. The preceding is a perfectly reasonable thing to think at first. However, it is important to see why this sort of eliminativism about the wavefunction is ultimately untenable.18 It is not just that the Schrödinger equation superficially looks to just be about this thing, the wavefunction. Quantum mechanics has to invoke the wavefunction because there are certain states, what Schrödinger himself first called ‘entangled states’, pervasive in nature, that can only be captured by a physical theory that countenances such an entity as the wavefunction. So, there is also an argument for taking the wavefunction with ontological seriousness, as the (or a) thing theories of quantum mechanics describe. This argument can be summarized in the following way: (1) The laws of quantum mechanics permit the evolution of systems into entangled states. (2) These states cannot be adequately characterized as states of something inhabiting our familiar, three-dimensional space, but rather must be characterized as states of something else spread out in a higher-dimensional (configuration) space. (3) So, (from 1 and 2) there exists something that must be characterized as spread out in a higher-dimensional configuration space: call this the ‘wavefunction’. (4) So, the wavefunction exists. In order to grasp the force of this argument, we’ll need to better understand the concept of entanglement and be familiar with the nature of the configuration space of quantum mechanics. However, this could take us too far away from the main goals of this paper, so I’ve included a more thorough explanation of the argument above in an appendix.19 The important point to take from this argument for now is just the following. In Newtonian mechanics, it was natural to view the laws as being about the evolution of states of material objects in three-dimensional space because these are the types of things that can be in the kind of position states the laws describe. By contrast, in quantum mechanics, many states the laws 534 NOÛS allow (and indeed are pervasive in our universe) cannot be seen as states of material objects in three-dimensional space.20 In particular, what the laws of quantum mechanics require is the existence of some entity whose states are specifiable in a space of higher dimensions.21 The wavefunction satisfies this requirement. The straightforward, ontological reading of realist quantum mechanics therefore is that the wavefunction is the object whose behavior the Schrödinger equation describes; it is an object that inhabits this higher-dimensional configuration space. Thus, configuration space should be thought of as at least one (if not the only) fundamental, physical space posited by quantum mechanics; and the wavefunction should be thought of as at least one (if not the only) fundamental object posited by these theories. I say ‘at least one’ here because, you might recall, although the Schrödinger equation is the only fundamental, dynamical law of Everettian quantum mechanics, and both laws of GRW also concern the behavior of the wavefunction, in Bohmian mechanics there is one other fundamental, dynamical law: the particle equation. I said earlier that I won’t take a stand on the disagreement about what this law is about. According to one side of the debate, this law describes the evolution of the positions of a set of many particles in ordinary, three-dimensional space. According to the other side, it describes the evolution of the position of one “world particle” in configuration space, i.e. the same space the wavefunction inhabits. The following diagram summarizes the different ontologies and spatial structures that are read most straightforwardly off of these several versions of quantum mechanics. Realist versions of quantum mechanics: DYNAMICS Everettian quantum mechanics Schr ödinger equation GRW Bohmian mechanics Schr ödinger equation + indeterministic collapse law Schr ödinger equation + particle equation With a oneWith a manyparticle reading particle reading of the particle of the particle equation: equation: FUNDAMENTAL ONTOLOGY Straightforward reading: The wavefunction The wavefunction The wavefunction + many particles The wavefunction + one “world” particle SPATIAL STRUCTURE Straightforward reading: Configuration space Configuration space Configuration space + ordinary threedimensional space Configuration space The Status of our Ordinary Three Dimensions in a Quantum Universe 535 One might wonder then, whether we can find the three-dimensional world of our ordinary experience in quantum mechanics if Bohmian mechanics is correct and that theory is given the first, straightforward reading. The idea then would be that the theory posits the existence of two physical spaces: a high-dimensional configuration space occupied by the wavefunction, and a separate, three-dimensional physical space occupied by the many particles that make up our tables, chairs, and the other objects of our manifest image of the world. Call this the ‘two-space reading’ of Bohmian mechanics. On the two-space reading of Bohmian mechanics, it looks like the central problem of this paper doesn’t arise. So perhaps all we need to worry about in the following sections is how to find the ordinary threedimensional space of our manifest image if GRW or Everettian theories are correct, or if the second, “one-space reading” of Bohmian mechanics is adopted. Perhaps. But it is worth taking a moment to see why many have not thought that this two-space reading of Bohmian mechanics really gives them what they were looking for.22 While it is undoubtedly clear that there exists a three-dimensional space according to the two-space reading of Bohmian mechanics, it is also unfortunately unclear whether or not this three-dimensional space is the space of our manifest image. For what fundamentally inhabits this three-dimensional space on the two-space reading? Particles. But what are these particles like? In particular, are these the particles that compose you or I, or the tables and chairs of our acquaintance, so that we may be said to also inhabit this three-dimensional space of Bohmian mechanics? The answer is not so clear. For even on this two-space reading of Bohmian mechanics, where the particles are ontologically fundamental, it is still true that it is states of the wavefunction that determine these particles’ behavior over time. The particles themselves only have positions. And what they do at later times according to this theory, i.e. what positions they move to over time, depends on the wavefunction. The properties (i.e. positions) of these particles at one time, t 1 , are not sufficient according to this theory to determine their features at any later time, t 2 , nor are facts about these particles even sufficient to determine the chances of the particle’s having certain features at any later time, t 2 . So, if we tried to view ourselves as constituted ultimately out of these Bohmian particles, we would be forced to conclude that all causal features we have constituting our ability to affect the world, that is, do what it is that we do, are ultimately grounded in features of the wavefunction, not in any features of ourselves or our constituent particles. It then seems like the three-dimensional space of two-space Bohmian mechanics is not the space in which we live. If you believe you are the type of thing that has not just a position, but the ability to affect the world around you, then you must deny that you inhabit the three-dimensional world of Bohmian mechanics.23 All that inhabits this space is a collection of inert particles. This is one line of reasoning that has convinced many philosophers 536 NOÛS and physicists that the two-space reading of Bohmian mechanics does not solve the problem of locating the three-dimensional space of our manifest image within the ontology of quantum mechanics.24 How to find the threedimensional space of our manifest image within the physical ontology is still a live issue for the two-space reading of Bohmian mechanics, as it is for the one-space reading of Bohmian mechanics, GRW, and Everettian quantum mechanics. Whether the preceding discussion was convincing or not, my suggestion is that we set aside this two-space reading of Bohmian mechanics in what follows. This will allow us to narrow our discussion onto the question of how to locate the three-dimensional space of our manifest image within a science whose most straightforward ontological reading posits only the high-dimensional configuration space as its physical space. 3. How to Think about the Configuration Space of Quantum Mechanics Let’s return now to discussing the nature of the configuration space of quantum mechanics. For it has yet to be shown why it is not a simple task to locate the three dimensions of our manifest image within this configuration space. To get a grip on the concept of configuration space, it is useful to ask: how many dimensions are there in configuration space? The correct answer to this question depends on contingent features of the world: the number of independent variables needed to specify the state of the world’s wavefunction () as a whole. One less correct, but more ubiquitous answer, an answer that we’ll find to be heuristically useful in a moment, is that the dimensionality of configuration space depends on the total number of particles in the system under consideration. The idea is that if the world contains a total of N particles, then the configuration space of the world is 3N-dimensional. The 3N dimensions of the space are understood in the following way.25 The first, second, and third dimensions of configuration space correspond to the three dimensions of the first particle; the fourth, fifth, and sixth dimensions to the three dimensions of the second particle; the seventh, eighth, and ninth dimensions to the three dimensions of the third particle; and so on. Then say, if the total number of particles in the universe is 1080 (near Arthur Eddington’s estimate), the wavefunction inhabits a space of 3 × 1080 dimensions. If this is right, then configuration space is very high-dimensioned indeed. And then, each point in the configuration space can be understood to correspond to a state that specifies all of the particles’ three-dimensional locations. As Jeffrey Barrett says: One can think of the positions of an N-particle system as being represented by a single point in 3N-dimensional configuration space (since there are N particles and three position coordinates for each particle). One might then picture The Status of our Ordinary Three Dimensions in a Quantum Universe 537 the motions of the particles by considering how that point would be pushed around . . . as the wavefunction evolves. (1999, p. 61) The preceding is a convenient way to come to grips with the very large dimensionality of configuration space, but it is ultimately not helpful for a couple of reasons. First, as has already been noted, the fundamental ontologies of the realist versions of quantum mechanics under consideration are either the wavefunction and the single world particle, or the wavefunction simpliciter. So, even if there are such things as particles with three-dimensional locations, they are at best derivative entities on these theories constituted in some way or other out of this more fundamental wavefunction (and possibly a single particle), and so they do not seem to be the sorts of things that would constrain the fundamental dimensionality of the configuration space. Even in two-space Bohmian mechanics, where one is able to read the theory as describing a world containing particles, the dimensionality of the configuration space will not be dependent on the existence of the particles. Both the configuration space and the particles in their three-dimensional space exist in their own right as fundamental entities. The fact that this “particle characterization” of configuration space is not entirely correct is often noted but ignored. Nevertheless, it is important to emphasize, as in the following textbook presentation: We may say, if we like, that [the wavefunction] is spread out over a 3ndimensional “coordinate” or “configuration” space in which each point represents a possible configuration of the system as a whole. But the use of such geometrical language is not essential and means merely that, since depends on 3n independent variables, it could be laid out as a “point” function only in a space having the corresponding number of dimensions. (Kemble 1958, pp. 21–2) As this explanation of configuration space demonstrates, the dimensionality of configuration space depends in a basic way only on the number of independent variables required to completely specify the state of the wavefunction at a time, not on anything having to do with particles in a three-dimensional space. This particle characterization of configuration space is useful as a heuristic, as it will allow us to hone in on approximately the right number of dimensions in our universe’s configuration space, but it is not strictly speaking correct. The concept of configuration space is the historical descendent of a related concept of the same name from classical mechanics. The “configuration space” of classical mechanics is typically not taken to be a genuine, physical space, but rather a purely mathematical representation that is used to conveniently summarize the positions of an entire system at a time. Instead 538 NOÛS of representing the locations of N particles in three-dimensional space by N points in a three-dimensional coordinate representation, classical physicists will often use one point in a 3N-dimensional coordinate representation to capture the same information. The concept of configuration space in quantum mechanics is no doubt derived from this other, highly useful concept of classical mechanics, but it is important to be clear that these are distinct concepts aimed at distinct entities. The dimensionality of the fundamental space of quantum mechanics is not determined by the number of particles in the universe. It is not a purely mathematical space invoked merely to summarize the locations of particles in some other, real three-dimensional space. What fundamentally exists in this theory is the wavefunction, and the dimensionality of the space the wavefunction inhabits is not determined by anything more fundamental.26 Before going any further, it is worth making it explicit that the configuration space of quantum mechanics is not to be confused with Hilbert space – the abstract, mathematical structure that is deployed in most textbook presentations of quantum mechanics. Hilbert space is used to represent states of the wavefunction as vectors. It is a convenient representation that allows various measurements we perform on states to be viewed mathematically as cases of vector operations. The Hilbert space does not have the status in quantum mechanics of a genuine physical space, inhabited by a world of vectors, but it is far more typical to see it regarded as a mathematical convenience.27 We now have the tools to understand why the case of quantum mechanics is so different from the case of relativity and the other theories we discussed above. In those theories, the three-dimensional space of our manifest image was contained within the space of the theories in the sense that three of the dimensions of the theory’s space were identical to the three dimensions of our manifest image. Then we could tell a plausible story about how we earlier might not have noticed the other dimensions of the theory’s space. However, the configuration space of quantum mechanics isn’t a space constituted by the three dimensions of our manifest image plus one, seven, or [(3 × 1080 ) − 3] more. No three of the dimensions of configuration space correspond to the three dimensions of our manifest image. Let’s see why. Imagine we are trying to describe three of the very many woodchips that make up my desk. We may start by coordinatizing the space in which these chips are located using three dimensions: x, y, and z. We may do this in such a way that x, y, and z correspond to the length, width, and height of our manifest image of the desk. And using this coordinate system, it seems we can give a complete description of the locations of these chips. The space in which the chips appear to live is three-dimensional because to specify each of their locations, we only need to specify these three values (their x, y, and z values). The Status of our Ordinary Three Dimensions in a Quantum Universe 539 For example: Chip 1: x=1 y=0 z=2 Chip 2: x=1 y=2 z=2 Chip 3: x = 10 y=2 z = 2. 3 of the desk’s wood chips Alternatively, we may represent the locations of the chips using the configuration space of quantum mechanics. To do this, we may exploit the particle characterization of configuration space discussed above. To see how to do this, it is possible to start by representing this system as a wavefunction spread out in a 3N-dimensional space, where N is the number of woodchips. Since the number of chips in this case is three, this wavefunction will be represented as inhabiting a nine-dimensional space. We can then represent the state of the whole system of chips as a point in nine-dimensional space.28 We will let the first three coordinates correspond to the x, y, and z dimensions of chip 1, the next three correspond to the x, y, and z dimensions of chip 2, and the last three correspond to the x, y, and z dimensions of chip 3. Then we can sketch a nine-dimensional configuration space representation of this system that partially makes up my desk: System: o=1 p=0 q=2 r=1 s=2 t=2 u = 10 v=2 w= 2 this system’s wavefunction Which of these nine coordinates corresponds to the x-coordinate of our threedimensional space? The correct answer is none of them. The o-coordinate corresponds to the x-coordinate-for-chip-1, the r-coordinate corresponds to the x-coordinate-for-chip-2, and so on. But no one of o, r, or u just is the x-dimension. To put this another way, we might ask: which coordinate in the nine-dimensional configuration space corresponds to the height of the desk? Again, it looks like no dimension of the configuration space is the height-dimension of the desk. The second, fifth, and eighth dimensions of the space look like they may in some sense correspond to the height of the 540 NOÛS desk, but none just is the height. The p-dimension may correspond to the height of chip 1, the s-dimension to the height of chip 2, and so on, but none corresponds to a common height that we may ascribe to the desk. Another way of recognizing the absence of our familiar three-dimensions from this picture is to note that if there is such a thing as height, then there is an object in the world that is extended in this dimension. This is captured in the three-dimensional graph by the representation of two occupied points at two different locations in the y-direction. It is in virtue of this that it appears that something is extended in this dimension. But as can be seen by the second figure, nothing is represented as being extended in any of the dimensions of the configuration space. So none of the nine dimensions of the configuration space correspond to our ordinary dimension of the height, nor to any of the other two dimensions of our manifest image. Although it not the case that any three of the dimensions of configuration space are the three dimensions of our manifest image, authors have tried to find some other way of locating the ordinary three-dimensional space of our acquaintance within the world we learn about from quantum mechanics. Two clearly divergent strategies have emerged. It is the goal of the next two sections to evaluate them. 4. Reconstruing the Status of the Wavefunction So, let us recap what is at issue. There are several viable, realist versions of quantum mechanics currently on offer. However, on the most straightforward, ontological readings of these theories, they show us that the space of the world we inhabit is not the familiar three-dimensional space we thought. In each case, we have a fundamental physical theory that enjoys an extremely high level of empirical support. Yet on the straightforward readings of these theories, there is no accompanying story (as there was for relativity theory, Kaluza-Klein and string theories) about how our familiar three dimensions are contained within the theory’s extremely high-dimensional world. No three dimensions of the configuration space of quantum theory correspond to the three dimensions of our manifest image. One response that has occurred to a number of authors is, on this basis, to simply reject what I have been calling the straightforward ontological readings of these theories. Perhaps one of Bohmian mechanics, Everettian or GRW quantum mechanics is correct. But we should not read an ontology off of these theories in the simple way I charted out above in Section 2. We need to be clear here: no parties to the present debate deny the reality of the wavefunction.29 Yet these authors insist that the Schrödinger equation is not about the evolution of the wavefunction over time. The theories are about something else. And this something else is something that exists in the three-dimensional space of our manifest image, and so there never really was a tension between the ontology of quantum mechanics and our manifest The Status of our Ordinary Three Dimensions in a Quantum Universe 541 image in this way in the first place. This something else is the collection of fundamental physical objects that constitute us and the ordinary material objects of our acquaintance. So, the straightforward readings of GRW, Everettian, and Bohmian quantum mechanics say the theories are about the evolution of the wavefunction over time. This alternative proposed reading says that these theories are about something else. The first thing one may wonder about this proposal is how this could be when there is a central dynamical law of all of these theories, the Schrödinger equation, which certainly appears to be about the evolution of the wavefunction over time. Moreover, we’ve already seen that if one is to give a complete description of the entangled states pervasive at our world, this requires a characterization in a space of a very high number of dimensions; that is, a characterization in configuration space. To make this proposal clearer, Valia Allori and her collaborators (Shelly Goldstein, Roderich Tumulka, and Nino Zanghı̀), following earlier work by Detlef Dürr, Goldstein, and Zanghı̀ (1992), invoke a distinction between what they call the ‘primitive ontology’ of a scientific theory (PO): what that theory is about, “the basic kinds of entities that are to be the building blocks of everything else [in that theory]”; and the nonprimitive ontology of that theory: phenomena, like laws, to which the theory appeals in order to (but only in order to) explain how the primitive ontology behaves (2008, pp. 363–5). Allori et. al. claim that for quantum mechanics, a variety of primitive ontologies are possible30 , but all of these include objects spread out in the ordinary three-dimensional space of our manifest image (or the four-dimensional Minkowski space-time that includes this space). The wavefunction is not part of the primitive ontology of quantum mechanics. They claim: Each of these [realist versions of quantum mechanics] is about matter in spacetime, what might be called a decoration of space-time. Each involves a dual structure (χ , ψ): the PO χ providing the decoration, and the wavefunction ψ governing the PO. The wavefunction in each of these theories, which has the role of generating the dynamics for the PO, has a nomological character utterly absent in the PO. (2008, p. 363) The idea is that although the Schrödinger equation may describe the evolution of an entity, the wavefunction, this entity should not be accorded the status of what quantum mechanics is ultimately about. According to Allori and her collaborators, we can think of the wavefunction as just something that has to be invoked by the theory in order to give a complete account of quantum systems, like a law. It thus follows on this view that the configuration space is not a genuine, physical space. The wavefunction is not what quantum mechanics is about, so the space it is supposed to inhabit is not a genuine space. Indeed as Dürr, Goldstein, and Zanghı̀ put it in 1992: 542 NOÛS . . . insofar as it is a field on configuration space rather than on physical space, the wave function is an abstraction of even higher order than the electromagnetic field. (1992, p. 850) For these authors, only the space of the primitive ontology is a genuine, physical space. Although Allori and her collaborators do not uniformly want to claim that the wavefunction just is a law, it is much like a law in that even if we are realists about laws, we do not take them to inhabit space, in the way we and other material objects inhabit space.31 This is part of what it means for them to accord to the wavefunction the status of nonprimitive ontology, saying it has a “nomological character.”32 Tim Maudlin (2007) has made closely related points regarding the ontology of realist versions of quantum mechanics. Although Maudlin does not make any specific claims about the wavefunction having the sort of nomological status ascribed to it in Allori et al. (2008), like these authors, Maudlin insists that any realist version of the theory must include in its ontology the kinds of things Bell called ‘local beables’. By ‘beable’, Bell just meant existent or entity.33 Local beables are those objects “which are definitely associated with particular space-time regions” (1987, p. 234). As Maudlin puts it, “local beables do not merely exist: they exist somewhere” (2007, p. 3157). As Bell is commonly interpreted, local beables must exist somewhere in ordinary threedimensional space, or at least four-dimensional space-time. Why is quantum mechanics obviously a theory of local beables, and so obviously a theory whose primitive ontology consists solely of local beables? One answer that can suffice for now is that quantum mechanics, like any candidate fundamental physical theory, is a theory that is intended to explain the behavior of material objects and the particles that make these things up – and these are local beables. The wavefunction by contrast is not a local beable. As we saw in the last section, it does not have any location in three-dimensional space; even according to the understanding of it we considered in the previous sections, it exists at best in configuration space. Therefore, the wavefunction is not part of the primitive ontology of quantum mechanics, and so its space is not a physical space. The argument may be summarized in the following way: (1) Quantum mechanics is a theory about local beables, i.e. objects with locations in the ordinary three dimensions of our manifest image. (2) Therefore, the primitive ontology of quantum mechanics must include local beables. (3) If quantum mechanics is correct, then objects can only have locations in the ordinary three dimensions of our manifest image if the space inhabited by the fundamental ontology of quantum mechanics includes these three dimensions. The Status of our Ordinary Three Dimensions in a Quantum Universe 543 Therefore, (4) The space inhabited by the fundamental ontology of quantum mechanics includes these three dimensions. The suggestion is then that the space of the world described by quantum mechanics does and must after all include the three dimensions of our ordinary experience. Since the wavefunction does not inhabit any such space, it is taken to be an object with a nonprimitive status – invoked by the theory to help (in some way) describe the local beables. Is this then the correct way to think about the ontological implications of realist versions of quantum mechanics? The main question that must be addressed if one is to adequately evaluate this proposal is: how plausible is it to take the wavefunction to have this nonprimitive status, as something that exists but nevertheless is not something that inhabits a physical space? John Bell seems to have thought that this is not plausible. He once said, speaking of a Bohmian version of quantum mechanics: No one can understand this theory until he is willing to think of [the wavefunction] as a real objective field . . . Even though it propagates not in 3-space but in 3N-space. (1987, p. 128) And, as David Albert puts it: The sorts of physical objects that wave functions are . . . are (plainly) fields – which is to say that they are the sorts of objects whose states one specifies by specifying the values of some set of numbers at every point in the space where they live, the sorts of objects whose states one specified (in this case) by specifying the values of two numbers (one of which is usually referred to as an amplitude, and the other as a phase) at every point in the universe’s so-called configuration space. (1996, p. 278) In quantum mechanics, states of the wavefunction appear very much like states of real things like electromagnetic fields. The wavefunction is characterized as spread out in configuration space with phase and amplitudes at each point in that space. Maudlin, Allori, and her collaborators want to argue that the wavefunction is not like a field, and thus does not have amplitudes at locations in a genuine physical space. We need an account however of why we are justified in thinking electromagnetic fields are the sorts of things that inhabit a genuine space, but the wavefunction is not. Prima facie, the wavefunction would not appear to have the abstract or nomological character Allori and her collaborators wish to ascribe it. It is natural to think of laws as entities that describe or govern the causal 544 NOÛS efficacy of other things. They are not themselves the things that have the causal efficacy.34 But certainly at least on the Bohmian version of quantum mechanics the wavefunction appears to have at least some efficacy, guiding the particle or particles into specific states. And the wavefunction has more analogies with electromagnetic fields as well. Just as there is a set of physical laws, Maxwell’s equations, that describe the evolution of electromagnetic fields over time, there exists a law, the Schrödinger equation, that describes the evolution of the wavefunction over time. One might think that this doesn’t establish that either electromagnetic fields or the wavefunction exist. But note that the question isn’t whether or not the wavefunction exists. All sides to the present debate agree it exists. The wavefunction must exist in order to capture the facts of entanglement. The question is whether it is part of quantum mechanics’ primitive ontology or nonprimitive ontology. Since, in general, we do not think that elements of a theory’s nonprimitive ontology, i.e. its laws, change, it would seem that the wavefunction belongs to quantum mechanics’ primitive ontology.35 After all, it is the type of entity whose state evolves over time, in conformity with the Schrödinger equation. One might object that physicists have recently raised the possibility that the fundamental laws of physics might themselves be evolving over time. Perhaps there is a meta-law that would govern this evolution in the other laws. Then we could view the Schrödinger equation as having a status like such a meta-law. It doesn’t describe how the primitive ontology of the theory evolves, but rather how the nonprimitive ontology that governs the primitive ontology evolves. When physicists talk about the laws of physics evolving over the history of our universe, however, what they usually have in mind is the possibility that what we thought were constants in nature (e.g. the speed of light or the charge of the electron) are really variables whose values change over time.36 What does this mean? It doesn’t seem to really mean that the fundamental laws of the universe are changing, but rather that the fundamental laws of the universe are different than we thought they were. For example, one of Einstein’s postulates in his special theory of relativity was that the speed of light in a vacuum (c) is constant. Now, if c has actually changed its value since the beginning of time, then this postulate is incorrect. My point is: it is not really so clear that there exist precedents for thinking that elements of a theory’s nonprimitive ontology are changing. Thus it seems that perhaps the wavefunction is more like a field (part of a primitive ontology) than a law (part of a nonprimitive ontology). It is not good reason to deny the wavefunction primitive status simply because it fails to be matter in space-time. Such a principle would have us similarly rejecting the primitive status of electromagnetic fields.37 In any event, these authors certainly have a reason to want to reject these claims of analogies with electromagnetism and deny that the wavefunction is The Status of our Ordinary Three Dimensions in a Quantum Universe 545 part of the primitive ontology of quantum mechanics. For they think that in order to account for the appearance of a three-dimensional world, the fundamental physical ontology must be three-dimensional. What we therefore need to explore is whether it is possible to capture the appearances – our perception of tables, chairs, measurement apparatus, and so on – in a world whose fundamental space is configuration space. We’ll be discussing such work in the next section. The upshot of this for the present section will be to remove the motivation for the proposal of Maudlin, Allori, and her collaborators. We may be able to account for the appearances in a fashion that is more scientifically conservative. Instead of rejecting the straightforward ontology read off of realist versions of quantum mechanics and replacing it with something closer to our manifest image of the world, we may be able to accept the straightforward ontology of quantum theory – the wavefunction in configuration space – and capture the appearances using those resources alone.38 5. Three-dimensional Space as a Non-Fundamental, Enacted Space An alternative, less revisionary strategy for recovering parts of our manifest image from the ontology of quantum mechanics has been proposed and developed in work by David Albert and Barry Loewer in discussion of GRW quantum mechanics and David Wallace on Everettian quantum mechanics. The idea is this. Accept the straightforward, ontological reading of these realist versions of quantum mechanics.39 In other words, accept that all there is fundamentally is a wavefunction in configuration space. Then the claim is that in the actual world, the behavior of the wavefunction over time is such that it is able to play the functional role we ordinarily associate with material objects in a three-dimensional space. In Albert’s terminology, the wavefunction is thereby able to “enact” the existence of material objects in a three-dimensional space. These authors thus accept the first premise of the preceding section’s argument: (1) Quantum mechanics is a theory about local beables, i.e. objects with locations in the ordinary three dimensions of our manifest image. However, they deny this argument’s third premise: (3) Objects can only have locations in the ordinary three dimensions of our manifest image if the space inhabited by the fundamental ontology of quantum mechanics includes these three dimensions. How does this work? We begin by being functionalists about the material objects of our manifest image – all that is required for there to be a chair is for there to be something that can play the functional role of a chair. 546 NOÛS For there to be a person, there just must be something that can play the functional role of a person. Albert suggests that any physics that is going to have a chance at describing our world as we experience it is going to have to describe a wavefunction that evolves in such a way that it is able to play the functional role of a universe with tables and chairs and people in it (1996, pp. 279–280). If this is right, then we may avoid the need to move to the less straightforward reading of quantum mechanics offered by Maudlin, Allori, and her collaborators in order to account for the appearances. We can allow that the fundamental space of quantum mechanics is the high-dimensional configuration space, but also claim that there is a derivative, functionallyenacted three-dimensional space occupied by tables, chairs, and people. This derivative space arises by the behavior over time of the wavefunction in the configuration space.40 This suggestion raises two questions. First, are the ordinary material objects of our manifest image the sorts of things that can be functionally enacted by the wavefunction? And second, is the three-dimensional space of our manifest image itself the sort of thing that can be functionally enacted by the wavefunction? For the purposes of this paper let us concede the point that the movement of areas of high amplitude of the wavefunction through the configuration space can enact something like a person, or a table, or a measuring device. That is, let us concede that these are the sorts of things that are not fundamental entities, but rather are constituted out of more fundamental entities’ playing the right kind of causal roles.41 This concession has extremely interesting consequences for our theories of reduction, realization and constitution; our theories, that is, of what relation the medium-to-large-sized objects of our manifest image bear to what fundamentally exists according to our best physical theories. In particular, an influential picture of reduction advocated by Paul Oppenheim and Hilary Putnam (1958, p. 8) according to which material objects relate to the fundamental physical ontology by mereological relations of part and whole (the fundamental physical ontology being the parts of everything else), would seem to have things backwards. The objects of fundamental physics are not parts of tables and chairs, but rather tables and chairs would appear instead to be parts of the wavefunction.42 It is very hard to wrap our heads around the idea that we ourselves are ultimately constituted by the dynamical behavior of a wavefunction spread out in a very high-dimensional configuration space. The diagrams in Section 3 may help to make this point vivid. If we appear to see a system of many particles spread out to make up our world in three-dimensional space, then (assuming the wavefunction is in nearly an eigenstate of position43 ) this is realized by one nearly point-sized region of high wavefunction amplitude in configuration space. The entire world ultimately amounts to a near-point-sized speck in configuration space. More work needs to be done to determine whether our concepts of reduction, realization, and constitution can accommodate the The Status of our Ordinary Three Dimensions in a Quantum Universe 547 live epistemic possibility that material objects are realized by, constituted by, or reduce to mere specks in configuration space. Still, granting this functionalism about ordinary objects is coherent (you, me, tables, chairs, planets, and galaxies could all be enacted by the wavefunction), we now need to ask: could space, or space-time be like that? Could a genuine, physical three-dimensional space be a functionally-enacted object? Now, someone could think space was this sort of thing. This is similar to how Leibniz thought of space. His view was that there isn’t this fundamental, physical thing, space, that exists in its own right. Rather, space exists when there are other substances standing in certain relations to each other. It wasn’t Leibniz’s view that space existed insofar as there were specifically causal relations obtaining between objects, but the views are still similar in this way: they ground the existence of space in the obtaining of relations between something or things more fundamental. I do not want to deny that one could have this view about three-dimensional space. What I want to deny is that one ought to think this is a view granting genuine, physical status to three-dimensional space when one also believes that there is another space that exists in its own right as a fundamental, physical substance. The predicament for those who try to find three-dimensional space in the ontology of quantum mechanics by viewing it as functionally enacted out of the behavior of the wavefunction is that they want to hold that this three-dimensional enacted space is real (though not fundamental) to get local beables and at the same time accept that there is a separate nonenacted, fundamental configuration space. Here is how Albert understands the configuration space. It is: an arena within which the dynamics does its work, a stage on which whatever theory we happen to be entertaining at the moment depicts the world as unfolding: a space (that is) in which a specification of the local conditions at every address at some particular time . . . amounts to a complete specification of the physical situation of the world. (1996, pp. 282–3) Configuration space is the genuine physical space here. It is not something that arises from the playing out of dynamical roles; it is the fundamental structure required for the dynamical roles to be played in the first place. By contrast, the three dimensions of our manifest image on this view are akin to the images that may seem to be appear when we are viewing a movie with 3D glasses. I am claiming that while functionally-enacted chairs are chairs, and functionally-enacted people are people, for a substantivalist, functionallyenacted space is nothing more than a simulation. There are several reasons to think this. Following John Earman, we may consider “two time-honored tests for substance” (Earman 1989, p. 111). These are not plausible accounts for existence simpliciter (for many objects of our manifest image, like tables 548 NOÛS and chairs may fail to satisfy them), but certainly seem to be features commonly associated with the nature of substantival space. First, substantivalist space is supposed to exist independently of other objects.44 But if threedimensional space is functionally enacted, it depends for its existence on other objects implementing causal roles, and thus would seem not to be a genuine, substantival space. In addition, according to Earman, substances are supposed to be themselves causally active. Whether or not functionallyenacted objects can themselves be coherently taken to be causally active has been a matter of heated debate in the philosophy of mind. A common verdict one finds, e.g. in Kim (1998) is that if what it is for something x to exist is for there to be something else y playing a certain causal role, then it seems that unless the x and the y are identified, there is no causal work left for the x to do. So, it is at least controversial that a functionallyenacted three-dimensional space could be causally active in the sense Earman appears to have in mind. We may also add to Earman’s list a third test to see if a space is legitimately viewed as the substantivalist would have it: substantival space is usually taken to be a background against which other events in our universe play out. For example, for Newton, space is the background required for objects to have absolute accelerations.45 But if three-dimensional space is functionally enacted, then it is not the background for fundamental interactions, but is instead constructed out of these interactions. If one wants to claim the three-dimensional space that is functionally enacted out of the behavior of the wavefunction is a genuine physical space, this would appear to threaten the substantivalism about configuration space Albert and others seem to endorse. If one takes a substantivalist attitude toward configuration space, then the approach of Albert and Loewer fails to provide a plausible way of finding three-dimensional space as (genuinely) existing within the ontology of quantum mechanics. The kind of functionalism that is plausible for ordinary, material objects fails according to substantivalists about space or space-time structures. The three-dimensional space of Albert and Loewer is nothing more than a simulation of space, a mirage.46 Albert himself gestures at the idea that perhaps this three-dimensional space is just space in a different sense. He distinguishes the physical space that is “the arena within which the dynamics does its work,” the configuration space of quantum mechanics, from the physical space that is “the space of possible interactive distances,” the three-dimensional space of our manifest image (1996, p. 282). He calls the former “the more fundamental one”. What I wish to argue here is that it is the only genuine concept of space, at least for a substantivalist. I conclude that neither account we have considered gives us a plausible way of understanding three-dimensional space as (genuinely) existing within the ontology of quantum mechanics. In this last section, I’d like to return to the argument of Section 4 and investigate whether there is an overwhelming The Status of our Ordinary Three Dimensions in a Quantum Universe 549 reason to recover the three-dimensional space of our manifest image from the ontology of quantum mechanics. 6. Must We Recover Three-Dimensional Space? Those authors we discussed in Section 4 were motivated to rethink the ontological status of the wavefunction (and its space) by their conviction that quantum mechanics is a theory about local beables, and the fact that the wavefunction is not a local beable. Recall the argument: (1) Quantum mechanics is a theory about local beables, i.e. objects with locations in the ordinary three dimensions of our manifest image. (2) Therefore, the primitive ontology of quantum mechanics must include local beables. (3) If quantum mechanics is correct, then objects can only have locations in the ordinary three dimensions of our manifest image if the space inhabited by the fundamental ontology of quantum mechanics includes these three dimensions. Therefore, (4) The space inhabited by the fundamental ontology of quantum mechanics includes these three dimensions. Local beables are believed to be the subject of the theory because they provide the simplest explanation of the following facts. First, we think quantum mechanics is about a world containing things like tables and chairs, objects that are something like the way we ordinarily take them to be. Another reason we have not up to this point considered is that the theory is justified by a wealth of data involving the status of material objects like pointers of measuring devices.47 The simplest explanation of both facts is that there really are such material objects inhabiting the kind of space we think they do. Now perhaps these points can be explained construing material objects functionally in the way Wallace, Albert, and Loewer suggest. For there to be a measuring device, there just needs to be something that can play the functional role of a measuring device. For there to be a table, there just needs to something that can play the functional role of a table. What is not required, I would like to suggest however, is the existence of a real, three-dimensional space. For there to be something that plays the causal roles of tables, chairs, people, and pointer readings, to ground the appearances and the confirmation of the theory, it does seem that there needs to be something located in some physical space or other. We need an arena in which the dynamics of our theory can unfold. And indeed we can concede that this be the kind of thing that can simulate heights, widths, and depths. However, there seems to be no reason why the physical ontology should entail that the appearance of three-dimensional space (or four-dimensional 550 NOÛS space-time) be anything more than a simulation. My suggestion, in other words, is that we should reject the first premise of the preceding argument: (1) Quantum mechanics is a theory about local beables, i.e. objects with locations in the ordinary three dimensions of our manifest image. Why would we need anything more than a simulation of a threedimensional space? According to Maudlin: The contact between theory and evidence is made exactly at the point of some local beables: beables that are predictable according to the theory and intuitively observable as well . . . Collections of atoms or regions of strong field . . . because they are local beables, can unproblematically be rock-shaped and move in reasonably precise trajectories. If the theory says that this is what rocks really are, then we know how to translate the observable phenomena into the language of theory, and so make contact with the theoretical predictions. (2007, p. 3159) I am not arguing that on a straightforward, ontological reading of realist versions of quantum mechanics that there are not rocks. I agree with Albert, Loewer, and Wallace that there are. However, I claim that this does not imply that there is anything that is genuinely rock-shaped (if this implies occupying locations in a physical, three-dimensional space) or that moves in precise trajectories through such a space. Since we can allow that the wavefunction simulates the behavior of something in a three-dimensional space, we have a way of making sense of our observations in terms of the language of the theory. Even if this isn’t the simplest account of our observations, it is the most scientifically conservative. Most of the authors writing on this topic have been inspired by Bell to assume that there must be local beables in any fundamental, physical ontology. Bell defined ‘local beable’ in the following way: We will be particularly concerned with local beables, those which . . . can be assigned to some bounded space-time region . . . It is in terms of local beables that we can hope to formulate some notion of local causality. (1987, p. 53) According to Bell, it is desirable to have objects with precise locations and trajectories in the physical space of a theory to be able to make sense of local causality. For there to be causal interactions at precise locations in a theory seemed to him (plausibly) to require the existence of entities with definite locations at which such causal interactions take place. The straightforward reading of realist versions of quantum mechanics gives us such objects: parts of the wavefunction with high amplitude that move and accelerate through the configuration space over time with precise locations and trajectories. So in the sense that really matters, we get local The Status of our Ordinary Three Dimensions in a Quantum Universe 551 beables: real, physical entities with precise locations in space. It is likely that Bell wanted more, that he meant by ‘local beables’: real, physical entities with precise locations in the space of our manifest image.48 But this doesn’t seem required by the physics, nor the grounding of our manifest image. Moreover, this seems unmotivated by Bell’s own emphasis on the issue of local causality. If we want a physical account that includes local physical interactions, there ought to be locations in some space or other where these interactions are actually taking place. But it isn’t clear why Bell should require these to be locations in a three-dimensional space. Without question, we need to be able to make sense of the fact that it seems to us as if these interactions are taking place in a three-dimensional space. But as has already been stated, this can be accomplished without there actually being such a space. Hans Reichenbach also argued that as a matter of empirical fact, to ground local causality, the space of our physical ontology must be threedimensional: The three-dimensionality of space has often been looked upon as a function of the human perceptual apparatus, which can visualize spatial relations only in this fashion. Poincaré tried to find a physiological foundation for this number . . . Even if this physiological explanation were tenable, it completely overlooks the fact that the number 3 of dimensions represents primarily a fact concerning the objective world and that the function of the visual apparatus is due to a developmental adaptation to the physical environment . . . It is the characteristic of three-dimensionality that it and only it leads to continuous causal laws for physical reality. (Reichenbach 1928, p. 274) Reichenbach’s argument for this claim proceeded as follows. First, he claimed that any tenable physical theory must be a causally local theory in the sense that it obeys the following principle. “Causal effects cannot reach distant points of space without having previously passed through intermediate points” (Reichenbach 1928, p. 275). The current physics of his day was, according to him, a theory that is causally local in that sense. Moreover, Reichenbach argued that any transformation of one physical theory positing a space of n dimensions into another with a space of n = n dimensions will disrupt the original theory’s preservation of local causality. This was an instance of a more general thesis of Reichenbach’s that whenever one transformed one geometry into another with distinct topological features, at least some causal relationships that turned out to satisfy a locality principle when viewed in the initial geometry would no longer be local in the geometry that resulted from the transformation. The dimensionality of a space is one of its topological features. We have a physical theory that satisfies local causality and posits a three-dimensional physical space. Therefore, it is not possible for there to be any other theory that preserves this theory’s physical content while positing a distinct number of spatial dimensions and is similarly tenable 552 NOÛS (in the sense of preserving local causality). Therefore, Reichenbach claimed, no tenable physical theory can posit anything other than a three-dimensional physical space. There is a lot to say about this argument. The most relevant point for our purposes however is that although it was reasonable when Reichenbach wrote this to argue that the current physics preserved local causality within a three-dimensional spatial geometry, when it comes to quantum mechanics and what we have learned about it up to now, we know that at least when formulated as a theory about objects in a three-dimensional physical space, this theory does not obey Reichenbach’s principle of local causality. This is a result that was shown conclusively by Bell (see his 1987, and for further discussion Maudlin 1994). Interestingly enough, local causality does appear to obtain according to one version of quantum mechanics we have discussed: the Everettian account. And if this is correct, then given that the most sophisticated and well-worked-out understanding of Everettian quantum mechanics, that of Wallace and his colleagues, posits a physical space of extremely high dimensionality, we look to get, using Reichenbach’s own reasoning, the conclusion that any tenable physical theory cannot posit a physical geometry with only three dimensions. So, there appear to be two reasons for insisting that we must somehow include in our physical ontology the existence of a genuine, three-dimensional space. First: our perceptual interaction with the world and justification of our scientific theories proceeds through interaction with material objects that appear three-dimensional. In response to this, I have pointed out that Albert, Loewer, and Wallace have already argued convincingly that a highdimensional quantum ontology can ground these appearances. There really are material objects, even if their three-dimensionality is a mirage, and they are ultimately grounded in the behavior of the wavefunction in configuration space. The second defense of local beables derives from the claim that local causality requires grounding in a genuine three-dimensional space. I have argued that Reichenbach’s reason for this looks to fail for reasons he could not have anticipated, and anyway, local causality seems only to require that one’s physical ontology include objects with precise locations in some space or other. This space need not be the three-dimensional one of our acquaintance. In the beginning of this paper, I described a satisfying process by which we are sometimes able to adjust our picture of the world in light of scientific evidence to see it as more highly dimensioned than we may have earlier thought. This required seeing our familiar three dimensions as three of the many dimensions of this new, physical orthodoxy. I have argued that this cannot be accomplished for realist versions of quantum mechanics, and this is part of what makes quantum mechanics such a puzzling theory for those of us who want to understand the fundamental ontology it presents. This forces us to reject our earlier notions of reduction and realization, and The Status of our Ordinary Three Dimensions in a Quantum Universe 553 challenges our intuitive sense of being extended in three dimensions. But just because we cannot find our ordinary three dimensions in the world of quantum mechanics, this does not mean that we may not be able to locate ourselves in a quantum world. If we are functionalists about material objects and ourselves, perhaps this is enough to see ourselves as inhabiting such a high-dimensional world as quantum mechanics demands. Appendix: From Entangled States to the Existence of a Wavefunction in Configuration Space The quantum state of a system is the state of the system’s wavefunction (). We will use the standard Dirac notation to express quantum states. For example, the quantum state of a system of just one particle that is at location (4, 0, 0) will be represented by: 1 = |(4, 0, 0) > The most interesting fact about quantum mechanics is that it allows for systems to evolve into states that are superpositions of a feature; for example, the state in which there is a single particle in a superposition of being at location (4, 0, 0) and being at location (7, 0, 0). (When a state is not in a superposition of position, it is said to be in an eigenstate of position.) This superposition will be represented in the following way: 1 1 | (4, 0, 0) > + | (7, 0, 0) > 2 = 2 2 To say that a system is in this quantum state is not to say that the particle is at location (4, 0, 0), nor is it to say that it is at location (7, 0, 0). It is also not to say that the particle is at both location (4, 0, 0) and at location (7, 0, 0); nor is it to say that the particle is at neither location (4, 0, 0) nor location (7, 0, 0). One thing that is true of a particle that is in this quantum state is that if the particle’s location is measured, then there is a 0.5 chance that it will be found at location (4, 0, 0). And if the particle’s position is measured, there is a 0.5 chance that it will be found at location (7, 0, 0). The probability is 0 that the particle will be found at any other location at that time. These probabilities are given by the square of the coefficients in the representation of the quantum state. Entangled states may be described as superpositions involving multiparticle systems. For example, we might consider a system of two particles that are entangled with respect to their position: 1 1 | (4, 0, 0) >1 | (7, 0, 0) >2 + | (7, 0, 0) >1 | (4, 0, 0) >2 3 = 2 2 554 NOÛS What is true of particles in such a state? On the assumption that the quantum state is ontologically complete, it is not true that either particle has a determinate location, either of being at (4, 0, 0) or of being at (7, 0, 0). However, it is true that if one were to successfully measure the positions of the particles, they would be found in one of the following two states. There is a 0.5 chance that particle 1 would be found at (4, 0, 0) and particle 2 would be found at (7, 0, 0). And there is a 0.5 chance that particle 1 would be found at (7, 0, 0) and particle 2 would be found at (4, 0, 0). And these are the only possibilities. It is natural to conclude from this fact that although neither particle in 3 has a determinate position, the relation between the positions of the two particles is determinate. For, there is a probability 1 that the particles will be found at a distance of 3 from each other in the x-dimension. How should one describe this state as being instantiated in a threedimensional space? One way is by representing at each point in the space the chances that a particle is at that location by a peak with a certain amplitude. Then to represent 3 , we will have two peaks of amplitude 0.5 at each of the two locations in three-dimensional space. In the figure below, we have marked the chances that each particle will be found at each location. Peaks for particle 1 are gray. Particle 2’s are white. y x z To see why this does not give us an adequate characterization of 3 , consider the distinct quantum state, 4 : 4 = √1 √ /2 | (4, 0, 0) >1 | (4, 0, 0) >2 + 1/2 | (7, 0, 0) >1 | (7, 0, 0) >2 4 is clearly a distinct quantum state than 3 . For if a system is in state 3 , then the particles may be correctly described as being at different locations a distance of 3 apart in the x-dimension (even though it is not necessarily true that either particle is at either location). However, if a system is in 4 , this is not the case. Here, the particles would not be truly said to be at different locations. For there is a probability of 1 that both particles will be found upon measurement to be in the same location. Nevertheless, despite these quantum states being clearly distinct and indeed empirically distinguishable, the three-dimensional representation of 4 will be the same as what was given above for 3 . The reader may check this against the above diagram. If The Status of our Ordinary Three Dimensions in a Quantum Universe 555 we want an adequate characterization of either state, one that distinguishes 3 from 4 , we will need to move to a higher dimensional configuration space. The configuration space that is used to represent quantum states in general has 3N dimensions, where N is the number of particles in the system to be characterized. Since the present examples each involve two entangled particles, the configuration space we will need to use in order to represent these states will be 6-dimensional. The first three coordinates in the configuration space correspond to x, y, and z coordinates for particle 1, and the second three coordinates in the configuration space correspond to x, y, and z coordinates for particle 2. 3 can be represented by peaks at locations (4, 0, 0, 7, 0, 0) and (7, 0, 0, 4, 0, 0) in the configuration space and the following diagram: x2 y2 y1 z1 x1 z2 Here, the peaks are used to represent locations where there is some nonzero chance that the entire system will be found. The two peaks correspond to the two terms in the superposition. In configuration space, 4 constitutes a completely different state x2 y2 x1 y1 z1 z2 556 NOÛS For a system in 4 , the peaks in the configuration space representation are at locations (4, 0, 0, 4, 0, 0) and (7, 0, 0, 7, 0, 0). As has been shown, entangled states can only be distinguished, and hence completely characterized in a higher-than-3-dimensional configuration space. They are states of something that can only be adequately characterized as inhabiting this higher-dimensional space. This is the quantum wavefunction. As I discuss in the third section above, none of the dimensions of configuration space are dimensions of the three-dimensional space of our manifest image. Notes 1 I am deeply indebted to David Albert, Valia Allori, John Bennett, Barry Loewer, and Jill North for extensive comments on an earlier draft, as well as to audiences at the University of Rochester, Vassar College, and Rutgers University for valuable discussion. 2 Following Wilfrid Sellars, I use ‘manifest image’ to describe our way of understanding our place in the world that relies upon what we learn from our perceptual interaction with it and introspection. This is intended as a contrast to the ‘scientific image’ (Sellars 1962). 3 As will become clearer I hope later on, this is the question of whether we need a physical ontology that includes local beables. 4 The formalism that results if one assumes such an anti-realism is different than the kind of formalism one finds in realist versions, so I call these different ‘versions’ of quantum mechanics, not different ‘interpretations’. 5 There are several places to look for a more comprehensive introduction to these theories. The best place to begin is probably Bell’s (1987) collected papers. He discusses Bohmian mechanics in ch. 17, the spontaneous collapse theory of Ghirardi, Rimini, and Weber in ch. 22, and the Everettian view in ch. 11. More references may be found in the footnotes below. 6 Von Neumann himself was not Danish, but Hungarian. Still, it is the version of quantum mechanics formulated by von Neumann, after being first suggested by Niels Bohr and his students, that is now commonly referred to as ‘Copenhagen’ quantum mechanics. 7 What is this thing, the system’s wavefunction? For some, a sufficient answer is this: it is the object whose evolution over time the fundamental laws of quantum mechanics describe. More will be said about this below. 8 See Bell 1987, ch. 23 for a more extensive discussion of the problem with a fundamental physical theory’s invoking concepts like measurement. 9 This is often called the ‘Everettian view’ because of its association with Hugh Everett III, who proposed confining the dynamics of quantum mechanics solely to the Schrödinger equation in his 1957 paper. It has since been developed into a clear and promising position in recent work by David Wallace (2002) and Simon Saunders (1995) among others. 10 An early variant on this version of quantum mechanics was proposed by David Bohm (1952), hence its current moniker. It has since been elaborated and clarified by Shelly Goldstein and his collaborators. See, for example, his (2006). 11 This is usually called ‘the guidance equation’, the thought being that this law describes how the wavefunction “guides” the motion of the particle or particles the theory posits. 12 The details of this debate are beyond the scope of this paper. For more, see Albert (1996), Monton (2000), and Maudlin (2007). 13 It is also the case that (arguably) the particle equation can itself be derived from the Schrödinger equation plus some natural symmetry considerations. 14 Philip Pearle developed a similar approach earlier (see Pearle (1976)). 15 The precise nature of the collapse isn’t important for the purposes of this paper. However, it is worth emphasizing that the theory doesn’t say that the system will undergo random behavior The Status of our Ordinary Three Dimensions in a Quantum Universe 557 at the time of the collapse. Rather the theory specifies a very specific kind of change in the system’s wavefunction at times of collapse. 16 Sometimes this version of quantum mechanics is presented not as one involving two laws, Schrödinger’s equation and another higher-level law saying when the Schrödinger equation fails to obtain, but rather as involving just one law that is a probabilistic revision of Schrödinger dynamics. I am here following Bell’s presentation of the theory: “The idea is that while a wavefunction . . . normally evolves according to the Schrödinger equation, from time to time it makes a jump” (1987, p. 202). 17 We know this, because anything with mass (m) is a material object according to Newton. In his Principia Mathematica, mass is defined as quantity of matter. 18 I say this view is untenable; this does not mean it is not still advocated in the physics community. This sort of eliminativism was defended most recently in the pages of Physics Today by David Mermin (2009), and raised a quite lively debate. Mermin argued that reifying the wavefunction was the manifestation of a more general bad habit that “makes life harder than it needs to be” of taking abstractions useful as calculational devices to be “real”. Reifying the wavefunction, according to Mermin, makes life harder by inducing people to write books and organize conferences on the topic of the measurement problem, or worry about faster-than-light signals produced as a result of quantum non-locality. 19 See also my (2010) for a more comprehensive discussion. 20 This much is true according to all of the realist versions of quantum mechanics we have considered (aside from certain Bohr-inspired variants of the problematic Copenhagen account) and anti-realist versions of quantum mechanics. On these accounts, it is not material objects that enter entangled states. It is the wavefunction that does so. This is something on which all parties to the present debate can agree, including my opponents in the following sections of this paper. 21 As I’ll discuss a bit more below, in the classical mechanics that followed Newton’s early presentation, use is also made of a higher-dimensional space. However, that space is not indispensable in order to completely describe the states of classical systems. This higherdimensional classical space instead has the status of an alternative representational framework that is used for convenience. This is why, unlike the configuration space of quantum mechanics, this classical high-dimensional space does not have the place in the theory of a genuine, physical space – the space anything actually inhabits. 22 This doubt is shared by all of the physicists and philosophers I discuss in the following sections of this paper, with the exception perhaps of Bell. (It is not clear to me from his work whether Bell understood Bohmian mechanics in this way or not). 23 It might help some philosophers to think of occasionalism here. The two-space picture is much like this view of causal interaction advocated by Malebranche except this time the role of God is played by the wavefunction. 24 There is also a more basic point that it is not clear even how to write down a law that expresses how the wavefunction may direct the behavior of the particles if one adopts this understanding of Bohmian mechanics. In general, physical theories describe interactions by inter alia specifying the geometrical relations that must obtain between the two entities for an interaction to take place. In this case, where the wavefunction and particles are supposed to exist in altogether distinct spaces, one cannot do this, for there are no geometrical relations obtaining between such objects in two separate spaces. I am grateful to David Albert for this point. 25 This way of specifying the dimensionality of configuration space is given by Bell (1987), Albert (1996), Lewis (2004), and many others. 26 Ideally, those endorsing this sort of realism about the wavefunction would introduce a new, distinct name for the space it is supposed to inhabit, but the name ‘configuration space’ is entrenched. 27 The question of the status of such vector spaces gets raised most explicitly when one starts to investigate the matter of describing quantum systems with various properties such as 558 NOÛS spin, charge, and the like. Since it is peripheral to the present topic and the issue of configuration space, I will set it aside here. 28 The reason why the chips may be represented by a single point, as opposed to a wavefunction smeared out over more of the configuration space, is that I am assuming our system is not in an entangled state. See the appendix for more discussion of these concepts and my (2010) for why this is not a generally accurate assumption for quantum systems. 29 So this proposal is not anything like what was considered briefly in the middle of Section 2, p. 9. 30 For Bohmian mechanics, the primitive ontology might consist of particles. For GRW and Everettian quantum mechanics, primitive ontologies of mass densities have been explored among other things. 31 As Zanghı̀ and Roderich Tumulka have emphasized in conversation, the precise status of the wavefunction really depends on which version of quantum mechanics is under consideration. For example, in Goldstein and Stefan Teufel’s (2001) it is claimed that in Bohmian mechanics, the wavefunction ought to be understood as a “field on the abstract space of all possible [particle] configurations.” In GRW quantum mechanics (which itself may be formulated in several versions), its status may be different, as the theory does not fundamentally involve particle configurations. What is important for our discussion is not what according to this view the precise status of the wavefunction is, but rather what its status is not – namely an entity like a field spread out over a genuine, physical space. 32 I have heard this position often confused with the two-space reading of Bohmian mechanics described at the end of Section 2 above. However, note that this is a distinct interpretation of quantum mechanics. The suggestion Allori and her collaborators wish to make is not that there are two physical spaces – ordinary three-dimensional space and configuration space. Their view is that there is only one space: the space inhabited by a material primitive ontology that does not include the wavefunction. 33 In introducing this concept, Bell was primarily interested in moving away from talk of observables in discussions of quantum mechanics, i.e. what can be observed according to the theory, as Heisenberg commonly insisted, and toward a discussion of beables, i.e. what exists according to the theory. 34 I am grateful to John Bennett for helpful comments on this point. 35 This worry might be removed if certain theories of quantum gravity turn out to be correct. As Goldstein and Teufel (2001) have argued, on a theory of quantum gravity incorporating the Wheeler-Dewitt equation, the wavefunction of the universe would actually be static, making it more akin to what we normally think of as a law. 36 This seems to be a common view in the physics community, for example, in Uzan (2003), p. 403: “Indeed, it is difficult to imagine a change of the form of physical laws (e.g. a Newtonian gravitation force behaving as the inverse of the square of the distance on Earth and as another power somewhere else) but a smooth change in the physical constants is much easier to conceive.” 37 Though perhaps this is what some of these authors intend. See especially Dürr et. al. 1992. 38 As Allori has pointed out to me in comments to this paper, what is scientifically conservative may be open to interpretation. For although the approach she and her collaborators favor revises quantum ontology as it may be straightforwardly read off the formalism, it is scientifically conservative in another sense, namely in remaining closer to the ontology that one finds naturally in classical mechanics. 39 Let’s set aside Bohmian mechanics for the moment. 40 Albert has a very nice story about how the wavefunction of our world is able to play this role. I won’t reproduce it in detail here, but see his (1996, pp. 280–281). The crux of the story is that, in order to support our quasi-Newtonian manifest image, it must be the case that when the amplitude of the wavefunction gets high at certain locations in configuration space, the system begins to accelerate more quickly than it does at other kinds of locations. These locations are The Status of our Ordinary Three Dimensions in a Quantum Universe 559 ones that should be occupied by regions of the wavefunction with high amplitude in order to ground the existence of material objects coming into contact. The diagrams in Section 3 can help to illustrate the idea. According to Albert, when the wavefunction moves into a state with high amplitude around the location (1, 2, 3, 1, 2, 3, 1, 2, 3) for example, we should expect the movement of regions of high amplitude to accelerate quite rapidly through configuration space. This makes sense since this corresponds roughly in the three-dimensional representation to a system of three particles with locations (1, 2, 3), (1, 2, 3), and (1, 2, 3), i.e. particles that are moving close enough to impact each other. We would expect this to create an acceleration in the movement of the particles in three-dimensional space. This corresponds to an expectation that the regions of high-amplitude wavefunction should accelerate in configuration space around locations like (1, 2, 3, 1, 2, 3, 1, 2, 3). 41 This is something very compellingly defended by David Wallace in his (2003), inspired by the work of Daniel Dennett, especially Dennett’s (1991). A similar view is endorsed by Albert and Loewer (1996), though they disagree with Wallace that the wavefunction could play such a role in Everettian quantum mechanics, as opposed to GRW quantum mechanics. There is disagreement between these authors also on the issue of whether the wavefunction can play the functional role to enact the material objects of our manifest image in Bohmian mechanics. Wallace argues (in collaboration with Harvey Brown in their 2005) that it can, and so the particle(s) of Bohmian mechanics are superfluous. 42 Jonathan Schaffer (2008) defends a version of ontological monism partly based on this point. 43 See the appendix for more clarification on the meaning of ‘eigenstate’. 44 Although Newton was not a substantivalist, he also viewed physical space in this way. Thus he claimed in the Scholium to his Principia that “absolute space, in its own nature, without relation to anything external, always remains similar and immovable”. 45 This is also suggested by the above passage from the Scholium to Newton’s Principia. 46 I am trying to find an analogy that will more clearly illustrate the problem. The best I can do is the following. Many philosophers believe that qualia – the qualitative features that make for phenomenal consciousness are basic, intrinsic features (e.g. Chalmers 1996). 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