No More Humean Supervenience about Laws of Nature

1_gammaIf you are a Humean, you may believe that laws of nature do not govern, but rather neatly fit the overall pattern of everything that happens in the universe. A law then would not have much governing to do beyond offer an explanation. Unfortunately, there are empirical reasons to believe that this may not be so. Here I try to make the case for some of these. After constructing Humean Supervenience as conjunction of Modal Combinatorialism, Supervenience and Nomological Reduction, I consider quantum-entanglement to show how Modal Combinatorialism is violated leading to a downfall of HS in general. I will consider a response to this by Loewer and conclude that on empirical grounds HS under Lewis’ formulation fails, although its basic intuition of non-governing laws persists.


HS is the thesis that all facts about the world supervene on facts relating to individual instantiations of fundamental properties and facts (see opening of Lewis, 1994; Weatherson, 2012: 1). The idea is that the conjunction of fundamental properties alone can give a complete picture of the world. There are point-sized properties distributed in space-time. Them, together with their relations are the base onto which nomic properties supervene, which means that there cannot be a change in nomic properties without a change in fundamental properties. The sentiment here is that nomic properties are nothing above non-nomic properties and explanatorily reducible to them (Lewis, 1983: 358). A law of nature is hence only a generalisation that picks out regularities. If God gave us a big book of all particular facts and none happen to reside outside f=ma, then it may appear as a generalisation in the preface to save space, but only describes the distribution of particulars instead of governing their behaviour (Beebee, 2000: 575). The Humean component of the thesis hence postulates that there are no necessary connections between particulars (ibid: 571).

One of the core premises of HS is that of Spacio-Temporalism. It posits that all fundamental relations are spacio-temporal in nature and all fundamental properties are those of occupants of points in space-time (Lewis, 1994: 474). For Lewis, fundamental properties have essences, which leads to a neat picture wherein a world is completely characterised by a “spacio-temporal distribution of intrinsic properties” (Weatherson, 2012: 2). These properties are Humean, since their instantiation is not contingent on anything beyond having a point to occupy and has no bearing on the instantiation of fundamental properties elsewhere (Loewer, 1996: 177). This concept of independence is expressed in another premise, namely of Modal Combinatorialism. It states that: “anything can co-exist with anything else” (Lewis, 1986b: 181, in Weatherson, 2012: 5). This seeks to summarise how point sized spacio-temporal properties are independent of one another and their existence not contingent on a connection with other points. It posits that the world is made of discrete events, like pieces of a mosaic (Karakostas, 2009: 17). Let me illustrate with an example. A digital display is composed of millions of pixels and whatever image they jointly instantiate will supervene on facts about the individual pixels, since there cannot be a change in the overall image without a change in the pixels. Those individual facts are the intrinsic properties of a pixel, as for instance brightness and colour plus its spacial relation to the other pixels. For a Humean World, the image is more complex in more dimensions, but the sentiment is the same: There is a vast mosaic composed of particular point-sized properties (Lewis, 1986: ix) and nomic facts about our world arise from those individual instantiations.

Having discussed how nomic properties result from fundamental ones, let us now turn to how we pick out laws of nature from nomic facts. After all, if nomic properties are mere expressions of regularities, how can HS make a distinction between accidental and nomological generalisations? What counts as a law for Lewis are those theorems and axioms of the best system of scientific analysis (Lewis, 1994: 478). Such a system represents some Kuhnian scientific paradigm where many competing world views may persist and the best paradigm is that which finds a balance between simplicity and explanatory power. This naturally entails epistemic issues associated with what should count as simple and powerful, since simplicity will be language and mind-dependent, all of which are beyond the scope of this post (Loewer argues that mathematics can do this independently, see 1996: 192; in case of ties between systems, Lewis hopes that “if fortune favours us, there will be a unique best system” (Lewis, 1994: 479). This is really an epistemic concern though and not for me). Let me give an example. Reichenbach formulated two true generalisations, that 1) There is no ton sphere of uranium, and 2) There is no ton sphere of gold. While 1 is nomologically and necessarily true because of the volatile nature of uranium, 2) is accidentally true but could become false (1959; in van Fraassen, 1989: 27). HS distinguishes 1) and 2) by testing whether they are axioms in the best system, which 1) is in nuclear physics, while 2) is not (Loewer, 1996: 188; I cannot help though but point out that 2) is actually false in nature since there are ton spheres of gold at the centre of massive stars produced during solar fusion and novas). The mentioned idea of necessity is troublesome for HS since the law that “There must be Z” should entail that it is necessary that Z. According to Beebee this is the case in Lewis since this necessity is just a way of saying that an axiom is part of our best system, hence something is physically necessary iff it is a law (2000: 577). Here we must understand necessity as an honorary title with little ontological grounding.

Let us take stock. After characterising HS as thesis of laws of nature wherein a law is picked out by scientists as describing a nomological generalisation that supervenes on and is reducible to independent particular facts, allow me to now turn more closely to what it is for a nomic property to supervene and reduce. Surprisingly, HS turns out to be rather hard to phrase as an actual supervenience thesis. Beebee opts for the modally strong necessary supervenience across possible worlds (2000: 572), where “Two worlds identical in their spacio-temporal distribution of fundamental properties are identical in their contingent facts.” This however turns out to be too strong since HS does not postulate that there cannot be non-spacio-temporal properties, just that they are not instantiated in a Humean world (Weatherson, 2012: 2). Recognising this, Lewis reformulates a local supervenience claim: “For two worlds in which no alien properties are instantiated, identity in spacio-temporal distribution of fundamental properties entails identity of contingent facts” (paraphrased from Lewis, 1986a: 15), where alien properties are fundamental non-Humean. The discussion on whether this supervenience thesis can hold is beyond the scope of this post. Suffice it to say that the existence of non-instantiated alien properties entails counterexamples (Weatherson, 2012: 2). To overcome them, Lewis eventually moved to the more vague claim of Familiar Supervenience, where identity holds for “worlds like ours” (Lewis, 1994: 475), which begs to ask what a such a world like may be.

In essence, it is exceedingly difficult to state HS as supervenience claim. More interestingly, I find it unclear what exactly supervenience on its own can amount to. For Lewis, its failure is best characterised as leading to governing laws where two worlds identical in fundamental properties can differ in their nomic facts, or, the particulars do not determine the nomic facts if supervenience fails. Therefore supervenience is required to ensure that laws do not govern. I am not sure this simple relationship exists. To elaborate, let me quickly introduce the most prominent opponent of HS, namely Armstrong’s necessitarian view. Here, to say that nomologically, all Fs are Gs is true because of some variable X, which is the property of being a law. For Armstrong, X is primitive and somehow analysable in terms of necessitation (1983: 93; it is under dispute how necessity is understood, for Armstrong it is a causal relation). The differentiation between accidental and nomological generalisations is hence easy in looking whether Fs and Gs are related by X or not (see Beebee, 2000: 577 for more detail). Hence, Fs are Gs may be true in one world, but not another regardless of particulars. Still, suppose world1 and world2 are Armstrongian where relations possess X and govern. In world1 Fs are Gs holds and hence the particulars are distributed to accommodate the law. In world2, Fs are Hs and particulars differ from world1 since now they are distributed such that Fs are Hs holds. Supervenience would be true for world1 and world2, which is not problematic since all supervenience amounts to is an affirmation of co-variance between nomic and non-nomic facts. Except for the nomological nihilist (see Loewer, 1996: 198), it is not under dispute whether laws and fundamental properties are related. Therefore supervenience alone does not suffice to refute governing laws.

The more promising premise of HS is that of nomological reductionism. It claims that laws do not just covariate with particulars but are explicable in terms of them. Again it appears difficult to flesh out what this means. Ordinary conceptions of functional or bridge law reduction are aimed at deriving lower from higher level laws and thus not applicable. Rather, I believe Lewis had in mind that nomic facts inherit the ontology of their Humean instantiators. It is not merely the case that laws supervene on fundamental properties, but they attain their truth value from those fundamental properties and are laws in virtue of them. Hence we are not actually trying to reduce the ontology of laws to particulars. The great metaphysical difference between the Humean and her adversary is what preempts what. While for the Humean the local matters of particular fact determine whatever supervenes on them, the anti-Humean opts for the opposite, and nomic-features determine the supervenience base. This is the very core of the intuition of whether laws govern or not (Beebee, 2000: 573). What follows is that Beebee’s conception that “HS is reductionist because laws supervene” (ibid: 572, 577) is upside down, rather laws must supervene because they get their ontology from their supervenience base.

After having developed HS, let me now consider a counterexample from quantum theory. In its commitment to Modal Combinatorialism, HS appeals to the separability principle in classical physics, which seeks to express how properties in different space-time regions can act independently and completely determine an entire system (Karakostas, 2009: 6). This principle already entails supervenience since any change in a system is contingent on changes in local properties. Let me give an empirical example: In general relativity, the gravitational field of some region is entirely determined by the ten independent, point-sized components of the metric tensor within the given space-time manifold. By contrast, in a quantum system in an entangled state, the global properties of the system are not “dependent upon nor determined by any properties of its parts” (ibid: 11). Furthermore still, the constituents of the system do not have any intrinsic properties that could suffice to guarantee the entangled state of the system beyond relational ones. This entails that a system will not be derivable, or reducible to any combination of its constituents and the behaviour of the system is not supervenient on the properties of the individual particles. Worse than that, quantum theory does not allow to view system parts as individual and autonomous property instantiations as Lewis had in mind (ibid: 13).  Under HS, a quantum system should be like the pattern on the monitor, where the individual quantum events are causally inert, completely independent of each other, and hence exhibit no necessary connections to others or the system overall. With the violation of the separability principle in quantum theory falls Modal Combinatorialism. Therefore HS is false.

Modal Combinatorialism is of particular importance to HS since it seeks to show how nomic properties are less fundamental than spacio-temporal ones, since those are autonomous and causally static while laws are contingent on them. If Modal Combinatorialism is false, then supervenience amounts to nothing as laws could supervene on all kinds of things (Weatherson, 2012: 9). Lewis responds to the counterexample by claiming that he is “not ready to take lessons from quantum physics as it is now” (Lewis, 1986b: xi). I believe it is a terrible argument to dismiss physics a priori simply on the basis that Lewis does not like its philosophical implications. Still, one must contend that his protest was against the young quantum physics of the time. Today, quantum theory has reached the level of maturity Lewis demanded, yet the problem of non-locality persists (Karakostas, 2009: 19). A more modern defence against entanglement comes from Loewer, who builds his argument on the ontology of quantum theory formulated by Bohm, which contains a world particle. This fundamental particle he believes moves through some configurational space and entangled quantum states as all else supervene on it (Loewer, 1996: 180). It is unclear whether Lewis‘ sentiment of particulars in space-time can be rescued when moving into configurational space. This view is very abstract and only based on a particular reading of Bohm, hence scientifically dubious. I am with Karakostas, who asserts that “any consideration about the fundamental space of the world should be grounded in physical science, not a priori philosophical considerations” (2009: 29), which is why I am unconvinced by Loewer and the counterexample from quantum entanglement stands.

Despite the largely conclusive evidence against Lewis’ formulation of HS, I believe we do not necessarily have to become supporters of governing Armstrongian laws. The intuition of the regularity theorist persists. That nomic facts somehow reduce to the distribution of fundamental properties is a philosophically tempting theory, even if these turn out not to be spacio-temporal in nature. Likewise, reduction of causation to counterfactual dependence or that of special sciences to physics are not immediately refuted by quantum entanglement being true. What we do require is an entirely new framework for HS with a new kind of fundamental property on which to supervene. Loewer has tried to do this, but he still rests on the idea that point sized entities are most fundamental in the form of his world particle in configurational space. I would speculate that perchance a new theory of HS could be formulated without Modal Combinatorialism and Spacio-Temporalism altogether, where entangled states or systems may be viewed as fundamental and the laws of a quantum system supervene on some emergent property of all particulars taken together. This would certainly go against Lewis’ intuition, but at this point it seems difficult to justify salvaging his HS instead of a new one with a new name. That task however must be left for another post.

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