### Introduction

There has been much discussion in the philosophical literature of the past few decades about the compatibility of Humean supervenience and quantum mechanics. In this paper, after discussing arguments from Tim Maudlin, Barry Loewer, Michael Esfeld, and others, I will argue that Humean supervenience is compatible with primitive ontology interpretations of quantum mechanics, favoring what I will call “BSA” approaches over “higher dimensional” approaches.### Humean Supervenience

Before assessing its compatibility with quantum mechanics, it is appropriate to specify exactly what Humean Supervenience is. As it was initially developed by David Lewis, Humean supervenience is the theory that reality consists of the spatio-temporal arrangement of point-sized quantities. All else, including laws of nature and modal facts, supervene on this “mosaic” of local matters of fact. However, quantum mechanics may pose a problem for viewing the fundamental constituents of the world as local, as entanglement seems to provide reason to reject locality.### Maudlin’s Perspective

Tim Maudlin has influentially argued that Humean supervenience is incompatible with certain empirical facts about quantum entanglement. He begins his argument by considering two central components of Humeanism. The first is separability, which states that components of space-time can be divided into arbitrarily small sections, each of which has its own physical state independently of any other section. The second component is physical statism, that all facts about the world, including modal and nomological facts, are determined by its total physical state. These components are fundamental to humeanism because if they are true it would follow that 1) the world is fundamentally non-modal and modal facts supervene on the spacetime points, and 2) there are no fundamental necessary connections in the Humean sense because each spacetime point is separated from the others.Potential problems for Humean Supervenience arise because entangled states in quantum mechanics prima facie seem to contradict separability. The example he gives in (2007) is as follows. Consider a singlet state with entangled particles p and q. We know from this state that upon measuring the spin of either particle in the z direction, there is a 50% chance of measuring the spin of z as up and a 50% chance of measuring the spin as down. However, because this system is entangled, there is the additional fact that upon measuring the system the resultant spins will always be correlated. So it seems that in this case there are facts about the spin states in the system that are over and above the combination of any intrinsic spin properties of p and q. This seems to contradict separability because part of the combined system state is not determined by the local states at each space-time point.

### Reconciliation

There are a number of attempts to reconcile Humean Supervenience and entanglement, several of which I will evaluate in the paragraphs that follow. First though, it is worth clarifying what exactly someone attempting a reconciliation must show. This is going to depend on what the correct interpretation of quantum mechanics is, so I’ll consider three below.The interpretation that is perhaps least in tension with Humean Supervenience is Bohmian mechanics. In order to show that some system of Bohmian particles is compatible with Humean Supervenience, one would need to show that all the system properties can be reduced to intrinsic properties of spacetime point, with everything else supervening on the spatiotemporal relations between those points. Whether or not Humean supervenience can be reconciled with GRW is going to depend on which formulation of GRW is in question. There are at least three common ways to understand the ontology of GRW theories. The first way is that the wave function is fundamental and particle talk supervenes on that. This first way is clearly a violation of Humean Supervenience because the wave function is both non-local and fundamental, which contradicts separability. A second way, usually associated with Ghirardi, is to take the wave function to describe how a mass distribution evolves. I will later defend the compatibility of this view Humean supervenience. The third, often associated with Bell (1987), is that the fundamental ontology consists of “flashes” occurring at space-time points. I will also defend the compatibility of this view with Humean supervenience.

There are perhaps insurmountable problems for reconciling the many world interpretation and Humean Supervenience similar to the issues faced by GRW the first formulation of GRW. If the many world interpretation takes the wave function to be something physically real and fundamental, then insofar as the wave function is non-local, points in the 4D block are not going to be separable.

### Lower/Albert View

One influential effort to reconcile entanglement and Humean Supervenience is due to Loewer (1996). While he agrees that non local states pose a problem for the Humean in the traditional 4D block picture, he believes that the Bohmian, for one, can claim that fundamentally there is a 3n dimensional configuration space, and any system of particles in the 4D block actually reduces to a field in this higher dimensional space. This means that the manifest world should be understood as something like a projection or shadow of a world particle in configuration space, structured in such a way that it appears three dimensional to observers.The tenability of this view depends on several factors, the most important of which is whether it’s possible for a world like ours to emerge from this higher dimensional space. There are clear instances where configuration space should be understood as a mathematical tool rather than as a description of physical space. To take an example from classical mechanics, consider a rigid body with its location defined relative to an origin and it’s orientation defined relative to some frame. The configuration for this rigid body is six dimensional because it has a 6 degree of freedom, but no one would talk this to imply that this physical phenomenon represented by this configuration space must necessarily be an actual point particle in higher dimensional space.

There are some reasons for thinking that the configuration space in Bohmian mechanics should be taken more seriously as a representation of the world than in classical mechanics. One difference is that the higher dimensional space seems indispensable to the quantum mechanical picture in a way it isn’t in classical mechanics. In the rigid body example above, the six dimensions can easily be separated into two three dimensional spaces, one of which describes the location of the object and the second of which describes the orientation of the object in motion. If we generalize this strategy, we might claim that a dimensional configuration space should be taken as fundamental when it is no longer separable into multiple lower dimensional spaces. This example, if correct, shows why this higher-dimensional space is indispensable in quantum mechanics. If we consider again a singlet state and the configuration space associated with it, attempting to separate this space into individual subspaces describing each particle cannot capture information about the correlations between the particles. Another important point is that in the classical mechanical case, because of the way velocity will determine position, the total possible states of the system given some initial state is really only a subspace of the total configuration space, while in the quantum mechanical case, the whole space physically possible. This is further reason that the configuration space is indispensable in the quantum mechanical case but not the classical mechanical one.

How might it be possible for the manifestly three dimensional world to emerge from this higher dimensional space? There have been several attempts to explain this, one of which is due to Bell. Because in Bohmian mechanics there is, in addition to the wave function, a hidden variable that determines the results of our measurements, as long as the state of any Bohmian particle can be interpreted in a three-dimensional space, we can expect that the entirety of our experience can be interpreted three-dimensionally. Bell points out that it is the “hidden variables” we experience, while the wave function remains hidden. There is, however, a problem with this view. For any given coordinate in this space, there is nothing in the coordinate itself that specifies how the parameters should be mapped onto the three-dimensional space. This could pose a problem for interpreting configuration space as something real, as I will attempt to show below.

Consider again, a rigid body in four dimensional space. Clearly the orientation of the axis relative to which the coordinates specifying the state of the rigid body are a matter of definition. So if the position of the rigid body is represented by the coordinate (1, 1, 1, 1) relative to some axis, there may be some other axis relative to which the coordinate is (2, 2, 2, 2). And the same way that the axis position can be arbitrarily defined, the rotation of these axes can be arbitrarily defined, meaning that there’s some axis relative to which the coordinate is (2, 1, 2, 1). Clearly in all these situations the actual state of the object is the same, which means that the state of the object is invariant under transformations of the coordinate system. This rigid body is analogous to the world particle in configuration space.

But if I now say this 4 dimensional coordinate actually “projects” down to two particles in two different locations, much the same way that the world particle in configuration space is supposed to “project” to our three dimensional world, we find that the coordinates are no longer invariant over transformations of the axis. Combined with the fact that there is no preferred way to orient the axis, it would follow that the single point in higher dimensional space fails to fully determine the locations of the two particles. If this analogy also holds in the Bohmian case, it would follow that the world particle in configuration space doesn’t fully specify the state of the manifest three dimensional world, which provides evidence that the configuration space should be understood as a mathematical construct rather than something physically fundamental.

There is, however, another suggestion due to David Albert (1996) that may resist this objection. One could claim that the wavefunction state does not need to determine a three dimensional state at each moment of time, but rather that it is the effect of the evolution over time that creates our experience of a three dimensional world. Albert’s claim is that the laws that govern the evolution of the system are what creates the impression that the world is three dimensional. This means that because the dynamics of the world particle are restricted, a specific grouping of the coordinates, while artificial, will seem more compatible with the physical facts than others. Peter Lewis (2004), claims that for any two particles approaching each other, there is a grouping of coordinates into N groups of 3 so that when these groups approach each other, there are higher velocities of approach relative to the other groupings. This allows us to only consider a subset of the total coordinates, because these subspaces are more approximate representations of the total space.

Because these 3 groupings are more natural, one might suppose that evolution has equipped us psychologically such that the world appears three dimensional, even though the wave function exists fundamentally and in higher dimensional space. Here I think Albert has made a serious proposal for how we could come to interpret a higher dimensional space in lower dimensions. While I think this proposal could explain how we could come to interpret a higher dimensional space in lower dimensions, I don’t think it solves the problem that a particle in higher dimensional configuration space underdetermines the state of the 3 dimensional world that should emerge from it. As with the coordinate transformation example I gave previously, since how the 3d emergent world emerges from the world particle isn’t invariant under coordinate transformations, additional information must be given for how to map the dimensions of the configuration space onto features of the actual world. But it isn’t clear from the world particle alone how to do this, so the world particle as a physical phenomenon would underdetermine the state of the actual world.

While, if it could avoid the objection I have above, this view is an effective way of reconciling entanglement and separability (not the separability of space-time points, but fundamental state separability), reducing the physical world to a particle in high dimensional configuration space is such a radical shift in ontology that one could reasonably question what would remain of Humean Supervenience in this picture. In the 4D view, Lewis takes causes to supervene on the counterfactual relations between points in spacetime. So even though causation isn’t fundamental, it supervenes on the relations between the fundamental entities. If we wanted our notion of cause to supervene on the fundamental entities of the higher dimensional view, it would supervene on the relations between the single world particle that exists at each moment of time. A full discussion of how the higher dimensional picture could change our views on causation would require another paper, but I’ll make a few observations here. First off, it seems that overdetermination wouldn’t pose a problem for counterfactual theories of causation amongst others in fundamental physics (though one still might want to deal with overdetermination to make sense of ordinary language talk about causation), because only a single world particle exists at each moment in time. For the same reason it doesn’t seem that preemption cases apply to fundamental relations. On the other hand, there seems to be some intuitive appeal for taking a causal process approach.

From these arguments, I conclude that the manifest world would fail to emerge from a physically real configuration space.

### Bhogal/Perry

Loewers approach can be difficult to evaluate. While the approach itself is well defined, the resultant picture is so opposed to our everyday experience that it is difficult to intuitively grasp. An alternative approach has been suggested by Bhogal and Perry. They describe their approach as follows: “The move, put simply, is this: A pair of particles being in the singlet state is not determined by the intrinsic physical states of those two particles; rather, it’s determined by the states of the pair together with the intrinsic physical states at other points in the mosaic.”The central idea is that the humean mosaic is populated with point particles that are arranged in such a way that the best system account (BSA) of laws will involve the Bohmian guidance equation. So on this view, the wave function is not fundamental, but rather supervenes on the arrangement of Bohmian corpuscles. By taking this view, the Humean relinquishes the idea that the universal wave-function actually pilotes the Bohmian corpuscles. While this may seem to be giving up the key insight of Bohmian mechanics, there may be further justification for giving up the pilot wave, as suggested by Esfeld (2014). Since the universal wave-function exists in configuration space, it cannot also be a wave of field in physical space, which means it could not guide or pilot the development of particles in physical space.

One could object to the above strategy by claiming that the Bohmian guidance equation won’t arise from a best system account or that there will be many other theories that compete with it, as acknowledged by Lewis (1994). For example, Neil Dewar (2016) argues that the standards of strength and simplicity are inherently vague, which makes the laws that would emerge from a best system account of some set of events is highly dependent on the specific definitions of strength and simplicity one applies. First of all, it’s worth mentioning that if it follows from Humeanism that it will be difficult to agree on a best system account of laws than this quite nicely reflects the current state of debates about the foundations of quantum mechanics there are large numbers of plausible interpretations of the data with little consensus and little empirical way to distinguish between them. However, these kinds of objections, while interesting, I think are tangential to the central project of Humean Supervenience because they say little about the fundamental ontology. Furthermore, even if it turned about to be true that normal quantum mechanics didn’t emerge from the best system account of the Bohmian particles, I would take this as reason to re-evaluate the best system account of laws rather than the larger project of Humean Supervenience. While of course the BSA and HS have traditionally been intertwined, I don’t think it matters much for the overall project of Humeanism whether the account of laws is the best system account or some other as long as the laws end up supervening on the patterns in the mosaic.

Another interesting objection was raised in Dewar (2018). This objection has been coined the problem of determinate quantities. The essential idea is that the Bohmian picture the wave function is not a determined quantity at each spacetime point. Since there is such a massive amount of information in the total wave function in configuration space, it would be highly unlikely that the best system account of the particles would contain so much more information than is in the configuration of the particles themselves. I think it’s worth taking a step back and considering what this argument, if valid, would actually show. First, it does not show an incompatibility between Bohmian mechanics as a description of the relations between spacetime points and Humean Supervenience, it would just show that such an account does not properly balance strength and simplicity. For this reason, I think at most this should make the Humean reconsider his/her position on the best systems account of laws, not any more central tenets of separability and physical statism. We also should not take this argument as relevant to our fundamental ontology.

### GRW

There have been more recent attempts to extend the compatibility of quantum mechanics from Bohmian mechanics to primitive ontology theories in general. Esfeld (2014) goes so far as to say quantum physics strengthens Humeanism rather than refuting it. Consider, for example, the GRW flash theory (GRWf). In contrast to Bohmian Mechanics, GRW theories modify the Schrodinger equation to include the collapse of the wave function. In GRWf, this law describes the following primitive ontology: whenever the wave function collapses in configuration space, this represents an event occurring at that point. These events are called flashes, and they are all that there is in spacetime. This means that apart from the collapses, the evolution of the wave function does not say anything about the distribution of matter in space, but only reflects the probabilities that a flash will occur. This view is similar to Bohmian mechanics in that there aren’t any physically instantiated superpositions, but it differs because while in Bohmian mechanics there will be a continuous distribution of matter in space, in GRWf the flashes are more sparsely distributed so there aren’t word lines. Since the GRWf flashes are essentially a subset of the Bohmian corpuscles, it would seem that any strategy that reconciles Bohmian mechanics with Humean Supervenience will extend to GRWf also.Though I think this strategy is a valid way to reconcile GRWf and Humean supervenience, the flash ontology creates problems for the BSA similar to the Bohmian view. To reiterate, the problem is that since there is such a massive amount of information in the total wave function in configuration space, it would be highly unlikely that the best system account would contain so much more information than is in the states of the particles themselves. However, the issue is much stronger in the case of GRWf than Bohmian mechanics, because the Bohmian trajectories provide much more information from which to derive the BSA than the flashes. In other words, the BSA wave function is significantly more underdetermined on the GRWf view than on the Bohmian view. As before, I don’t see the BSA as essential to Humean supervenience, and so I don’t think these kinds of objections as a reason to reject it. Rather I see this at most as a reason to consider other ways laws of nature might supervene on the mosaic. In addition, to make the derivation of the BSA on the GRWf picture seem more plausible, one could point out that since all measurements are of flashes on this view, scientists have already undertaken the work of deriving the equations of quantum mechanics from the GRW flashes.

### Comparing Bhogal and Loewer

There are advantages and disadvantages to the BSA and higher dimensions strategies. I agree with Bhogal and Perry that their account is much more elegant in the Bohmian case. Little needs to be changed about the traditional formulation of Humean Supervenience to reconcile it with entanglement, and the solution is intuitive and well defined. The disadvantage to this view is that it seems, in principle, to be only relevant to Bohmian mechanics and a Bohmian-like way of viewing GRW. In (2015) they explain their account by “applying it to the example of Bohmian Mechanics”. Though it’s unclear if they implied otherwise, I think there is little hope that there will be any other examples where this account is applicable, primarily because it relies on the unique way Bohmian Mechanics solves the measurement problem. To reconcile this with quantum mechanics it must be interpreted in such a way that 1) there are hidden variables for particle locations 2) these particles are separable and local and 3) the wave function is not a fundamental part of the world. It seems that any interpretation compatible with these three thesis will be Bohmian in nature.The “higher dimensions” view has the advantage that it is potentially generalizable to other interpretations, though I have tentatively argued that this view fails for other reasons. Though this view can be more difficult to intuitively grasp, it also has the slight advantage of being closer to how scientists normally treat quantum mechanics. There are, however, problems with understanding whether the manifest three dimensional world would emerge from a physically real world particle in configuration space. Because of this issue and the overall simplicity of the alternative, I see the Bhogal/Perry view as the most plausible fundamental ontology view. For anyone who sees Bohmian mechanics as an untenable solution to the measurement problem, however, the Loewer/Albert view would be preferable.

### Conclusion

I have argued that Humean supervenience can be reconciled with Bohmian-like primitive ontology theories of quantum mechanics. I have also argued that the BSA view is preferable to a “higher dimensions” view under Bohmian mechanics, due to difficulties in how the world we experience could emerge from a physically real configuration space.### Citations

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Bhogal, H., & Perry, Z. (2015). What the Humean Should Say About Entanglement. Noûs, 51(1), 74–94. doi: 10.1111/nous.12095

Dewar, Neil. (2016). What the Humean cannot say about entanglement. Retrieved from http://philsci-archive.pitt.edu/12046/.

Dewar, Neil (2018). La Bohume. Synthese. doi: 10.1007/s11229-018-1800-1

Esfeld, M. (2014). Quantum Humeanism, or: Physicalism without Properties. The Philosophical Quarterly, 64(256), 453–470. doi: 10.1093/pq/pqu030

Maudlin, T. (2007). Why Be Humean? The Metaphysics Within Physics, 50–77. doi: 10.1093/acprof:oso/9780199218219.003.0003

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Loewer, B. (1996). Humean Supervenience. Philosophical Topics, 24(1), 101–127. doi: 10.5840/philtopics199624112