A challenge for Deniers of Humean Supervenience about Laws of Nature Part I: Humean Supervenience

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{Forthcoming in Philosophy and Phenomenological Research}

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A Challenge for Deniers of Humean Supervenience about Laws of Nature

Part I: Humean Supervenience

John Earman

Department of History and Philosophy of Science

University of Pittsburgh

John T. Roberts

Department of Philosophy

University of North Carolina at Chapel Hill1
Abstract: This is the first part of a two-part article in which we defend the thesis of Humean Supervenience about Laws of Nature (HS). According to this thesis, two possible worlds cannot differ on what is a law of nature unless they also differ on the Humean base. The Humean base is easily to characterize intuitively, but there is no consensus on how, precisely, it should be defined. Here in Part I, we present and motivate a characterization of the Humean base that, we argue, enables HS to capture what is really stake in the debate, without taking on extraneous commitments.

“I tend to picture the [facts of the form “it is a law that s” and “it is not a law that s”] as having been sprinkled like powdered sugar over the doughy surface of the non-nomic facts.” -- Marc Lange2

“Avoid empty carbohydrates.” -- Runner’s World3
1. Introduction

Much of the contemporary debate concerning the nature of laws of nature has focused on the thesis of Humean supervenience about Laws of Nature (henceforth, HS):

HS: What is a law of nature, and what is not, supervenes on the Humean base.

The Humean base may be characterized – very roughly -- as the complete set of basic facts not offensive to those who are skeptical of non-logical, necessary connections in nature. The base includes particular facts about the existence of physical objects and their occurrent properties and spatiotemporal relations to one another. Excluded from the base are irreducibly general facts (e.g. that all electrons are negatively charged, that there exists at least one electron), and facts that involve laws of nature or other non-logical, natural modalities (e.g., facts about causal relations, counterfactuals, and irreducible dispositions).

HS will be a precise thesis only when two questions are answered: “What sense of ‘supervenes’ is intended?” and “What, exactly, is the criterion for deciding whether something belongs to the Humean base?” These questions are answered differently by different philosophers who have written on this topic.4 Hence, “HS” is a name shared by many different theses, differing from one another in subtle ways, though they are all intended to capture the same general view of the world.

That general view is the one held by those who take seriously the claims of science to discover laws of nature, but who see at best a metaphor in the idea that nature is “governed” by laws. It is the view that the laws of nature are not an independent metaphysical ingredient of the world, standing over and above the totality of more humble facts that they are supposed to govern; on the contrary, the more humble facts exhaust what there is of the world, and a complete specification of them would settle everything there is to settle about what is a law of nature and what is not.5

We think that HS, suitably formulated, is true. It expresses a necessary condition for laws of nature, as such, to be within our scientific reach. That is to say: the possibility of empirically justified belief, of a law of nature, that it is a law of nature, depends on HS. 6 Our argument for this claim will depend on a certain view of what it is for something to be within our scientific reach. It is possible to reject this view without lapsing into inconsistency, so our argument will not be conclusive. But we will try to show that rejecting this view is a most unattractive option.

However, we think that HS is defensible only if it is formulated in the right way, and we do not think that either the friends or the foes of HS have so far managed to formulate it as a thesis both precise enough and plausible enough to be given a serious and successful defense. Our task in Part I is to articulate and motivate what we take to be the optimal formulation of HS. In Part II, we will present our epistemological argument for it.

It might be objected that HS has already been formulated – by David Lewis7 – and any “new formulation of HS” is really just a new thesis with a stolen name. But HS has taken on a life of its own since it was introduced by Lewis. Many philosophers have argued for and against something called “Humean Supervenience,” giving non-equivalent formulations of it.8 Still others have argued for or against similar yet subtly different theses with other names9. For the most part, these authors have successfully joined the issue with one another. They have not seemed to talk past one another, but rather to be conducting a coherent debate over a single issue, even though they have not all formulated that issue in the same way. Thus, HS has acquired a status like the one that doctrines like materialism, dualism, and empiricism often appear to have: There seems to be an idea there, that one can be determinately for or against, that considerations can speak clearly for or against, even while it remains an open question exactly how the idea should be formulated.

We will begin our discussion with a brief review of some of the most persuasive arguments against HS. We will do this for two reasons. The first is that it will help to clarify the burden that must be carried by anyone attempting to argue for HS: It is necessary to produce reasons capable of outweighing the powerful intuitions marshaled by the foes of HS. The second is that doing so will help to make clear what constraints must be met by anyone who, like us, offers a new formulation of HS. Critics have managed to find some extremely counterintuitive consequences of HS. These counterintuitive consequences are not, we think, merely the products of clever logical trickery; they go straight to the heart of the matter, and anyone willing to stand up for HS ought to be willing to accept these consequences. Indeed, these counterintuitive consequences are intimately connected to, and perhaps partly constitutive of, what is exciting and distinctive about HS. So, as hard to swallow as they are, these consequences are not to be evaded by new and clever formulations of HS; any so-called “reformulation of HS” on which it did not have these counterintuitive consequences would be a bland and pale substitute for the real thing. Conversely, any supervenience thesis about laws of nature that shares these counterintuitive consequences shares the distinctive metaphysical flavor of the standard formulations of HS, and is prima facie a good candidate for the name “HS.” What we hope to show here is that there is a a good candidate which is also plausibly true.

2. Some Arguments Against HS
There are certain pairs of apparent possibilities that would, if genuinely possible, be counterexamples to HS. So an advocate of HS must deny these appearances of possibility. This denial, according to some philosophers, is too implausible to sustain; hence HS must be rejected. We don’t accept these arguments, since we think there are reasons to accept HS that outweigh the intuitions they appeal to. We will review these arguments here, not to offer a refutation of them, but to bring out forcefully what any friend of HS must accept, and what sort of burden she must carry.

One straightforward attempt to present a counterexample to HS10 goes as follows:

Consider a possible world in which there is nothing but a Newtonian spacetime and a single material particle, traveling with some constant velocity for all of time. What goes on in this possible world is consistent with the laws of Newtonian physics. But it is also consistent with the hypothesis that there is exactly one law of nature, namely the uniform-velocity law, which says that all material objects always travel with uniform velocity. So there are at least two lonesome-particle worlds: one where the laws are Newtonian, and one where the only law is the uniform-velocity law. Hence, HS is false.

There is a quick reply: Conceivability or imaginability is not always a good guide to possibility, and whatever reasons we may have for believing HS would be reasons for rejecting the genuine possibility of one or both of the lonesome-particle worlds, despite their apparent conceivability. Such a quick reply doesn’t settle the matter, of course. The two lonesome-particle worlds really seem to represent genuinely possible ways that our world could have been, and many philosophers would find it counterintuitive to deny that this is what they are. Intuitions on this particular case are varied, though; other philosophers do not feel any strong intuitive compulsion to grant that a world that is so simple could be one in which there could be a set of laws as complicated as those of Newtonian mechanics.

However, the literature contains other putative counterexamples that are more persuasive. Michael Tooley (1977, pp. 669-72) describes a possible world in which there exist elementary particles of ten kinds. Hence, two-particle collisions come in 55 different kinds. Suppose a complete history of the motions of these particles to be specified. In this history, 54 of these 55 kinds of collision are exemplified, perhaps in numerous cases. But the 55th kind – collisions of X-particles with Y-particles – never takes place. This isn’t because there is a law forbidding such collisions; it’s just that none ever occur, due to physically contingent conditions. Further, there are laws governing free particle motion, and laws that govern collisions of the 54 exemplified kinds. It seems possible that there is also a law governing X-Y collisions. But nothing said so far settles what that law is. So it seems that there are multiple possible worlds, each answering to the description given so far, each of which has a different law for X-Y collisions. But all such worlds clearly agree with respect to the Humean base. So these worlds provide a counterexample to HS, if they are indeed genuine possible worlds. Here again, the defender of HS can simply deny that these possible worlds all exist. But in this case, the intuition that the worlds are genuinely possible seems stronger than it did in the case of the two lonesome-particle worlds. Tooley’s worlds are neither so simple, so barren, nor so obviously unlike our own world as the lonesome-particle worlds. So there seems to be less prima facie reason to doubt their existence. Again, the issue comes down to the question of which is more plausible: the intuitions that speak in favor of the existence of Tooley’s possible worlds, or HS itself. But the bar for arguments intended to establish that HS is the more plausible option seems to have been raised.

That bar is raised even higher by Carroll (1994, pp. 60-68) with his Mirror Argument. Carroll begins by describing two possible worlds, U1 and U2, that even a philosopher committed to HS has no apparent reason to deny. He then proceeds to show that if certain apparently plausible premises are accepted, then the existence of the two unobjectionable worlds entails the existence of two further possible worlds, U1* and U2*, which constitute a counterexample to HS. U1 and U2 are worlds that contain X-particles and Y-fields. In U1, it is a law that X-particles always have spin up when they are within Y-fields; this law is dubbed “L1.” U2 agrees with U1 in every non-nomic detail except that in it, there is one X-particle, particle b, that has spin down while it traverses a Y-field; L1 thus has a counterexample in U2, so it is not a law there. In both U1 and U2, a slight change in circumstances, namely the position of a certain mirror, would have prevented particle b from entering the crucial Y-field. It seems that the total set of Humean-base facts that would have resulted in U1 if this change were made is the same that would have resulted in U2 if this same change were made. (For, if the change were made, then particle b would never traverse a Y-field; but the only Humean-base difference between U1 and U2 is in the spin of particle b as it passes through a particular Y-field.) In each of U1 and U2, it is apparently physically possible that this change be made. If we assume the standard possible-worlds semantics for counterfactuals, and we assume that in any possible world, a physically possible non-nomic change cannot result in a change in the laws, then it seems to follow that there exists a pair of worlds with the same Humean base but different laws. These are U1*, which is the closest possible world to U1 in which the mirror’s position has been changed, and U2*. which is the closest possible world to U2 in which the mirror’s position has been changed. In U1*, L1 is a law (because L1 is a law in U1, and changing the position of the mirror cannot change the laws), whereas in U2*, L1 is not a law (because it is not a law in U2, and again, changing the position of the mirror cannot change the laws).

In response to this argument, a defender of HS must either reject the existence of one of the two possible worlds Carroll starts with (which do not present a counterexample to HS, and do not appear to be objectionable in any way) or else reject one of the further premises Carroll relies on, for which he makes an impressive case.11 The cost of endorsing HS now appears to be even higher. If HS is now said to be plausible enough to prefer rejecting Carroll’s intuitive premises to rejecting HS, then a pretty convincing argument for HS is needed.12

As impressive as they are, the arguments of Tooley and Carroll are not conclusive refutations of HS. They depend on the presumption that the strength of the modal intuitions that stand behind them is sufficient to make them more plausible than HS itself. But how plausible HS is depends on how strong a case can be made for it, and as far as we know, a sustained positive argument for HS has yet to be given. If philosophical argument is at bottom a matter of weighing the costs and benefits of adopting various positions, then we need to see both sides of the ledger.

One of our main purposes in discussing these arguments against HS is to establish a standard for evaluating proposed formulations of HS. Suppose that someone were to propose a new formulation of HS (as we will do in the following section). One might well wonder whether the new thesis really deserves to be called a version of HS; perhaps it is really just a new thesis. Have we been given a refinement of HS, or has the subject been changed? The distinction here is no doubt a rather vague one. But it seems to us that there is one clear condition that any thesis must satisfy if it is to qualify as a version of HS: the putative counterexamples to HS discussed above must be putative counterexamples to it.

3. Refining HS

Though there are many definitions of HS about, it is uncontroversial to define the thesis as we have done above:

HS: What is, and what isn’t, a law of nature supervenes on the Humean base of facts.
That is to say, every pair of possible worlds13 that agree on the Humean base have the same laws of nature.14 What is controversial is how to define the Humean base.15
3.1. Lewis’s Humean Base

Lewis’s answer to this question has perhaps been the most influential. His Humean base is the spatiotemporal arrangement of local qualities, i.e. intrinsic properties that require no more than a spacetime point (or a point-sized object) for their instantiation. Lewis’s vision appears to be a qualitative version of classical (that is, non-quantum) local field theory. Indeed, he suggests that it is inspired by classical physics (Lewis (1986, pp. x-xi)). In a letter to Max Born, Einstein argues, in effect, for the doctrine that everything about the physical world supervenes on something like Lewis’s Humean base:

If one asks what ... is characteristic of the world of ideas of physics, one is first of all struck by the following: the concepts of physics relate to a real outside world, that is, ideas are established relating to things such as bodies, fields, etc., ... It is a further characteristic of these physical objects that they are thought of as arranged in a space-time continuum. An essential aspect of this arrangement of things is that they lay claim, at a certain time, to an existence independent of one another, provided these objects ‘are situated in different parts of space.’ Unless one makes this kind of assumption about the independence of the existence (the ‘being-thus’) of objects which are far apart from one another in space – which stems in the first place from everyday thinking – physical thinking in the familiar sense would not be possible. It is also hard to see any way of formulating and testing the laws of physics unless one makes a clear distinction of this kind. This principle has been carried to extremes in the field theory by localising the elementary objects on which it is based and which exist independently of each other, as well as the elementary laws which have been postulated for it, in the infinitely small (four-dimensional) elements of space. (Einstein (1971, pp.170-171))

Einstein thus held not only that the world consists of a distribution of objects all of whose properties are local (i.e., independent of what properties may be instantiated in other places), but also that this is a necessary condition of the very possibility of physical theorizing. Further, in the development of field theory (of which Einstein clearly approves) this idea is taken to an extreme by making the basic objects themselves point-like (i.e. “in the infinitely small (four-dimensional) elements of space”). This view is very much like Lewis’s version of HS.16

Its debt to (late) classical physics is the undoing of Lewis’s thesis, in our view. The consensus among contemporary physicists is that Einstein was just wrong to say that the very possibility of physical thinking depends on the supervenience thesis in question.17 Quantum mechanics appears to be flatly inconsistent with the supervenience of everything on “point-sized” instantiation of properties. This is because, according to quantum mechanics, there exist entangled states of composite physical systems in which multiple, space-like separated sub-systems have a joint state, though none has its own state characterizable in terms that refer only to its own spatiotemporal location. Hence, the quantum state of a composite system does not, in general, supervene on states of its separate, “point-like” parts. To be sure, there are many peculiarities of quantum mechanics, and it is certainly not the last word. But one of the lessons of Bell’s theorem and the Aspect experiments seems to be that the non-locality predicted by quantum mechanics is probably a real phenomenon. As Maudlin (unpublished B) argues, its commitment to locality makes Lewis’s formulation of HS empirically implausible.

3.2. Loewer’s Humean Base

Another possible view of the Humean base is motivated by a consideration of the kinds of particular facts that supporters of HS are inclined to see as paradigm members of the base. These tend to be things like values of the fundamental physical quantities, such as mass, charge, position, and velocity of particular objects at particular times. This suggests the following general characterization: the Humean base consists of all the values of the fundamental physical magnitudes, for all particular physical systems at all times. This is closely related to the proposal of Loewer (1997), who suggests characterizing the Humean base in terms of the “fundamental spaces” of a physical system according to current physical theory. The fundamental space of (non-relativistic) quantum mechanics is not ordinary three-dimensional physical space, or even four-dimensional space-time, but an abstract Hilbert space, whose points are vectors representing physically possible quantum states. On Loewer’s view, the Humean base consists of the locations of physical systems in some such “fundamental space.”

The worry here is that any such proposal takes for granted the notion of a fundamental physical magnitude, or alternatively, the notion of a fundamental space. But we do not yet know which magnitudes are the fundamental physical ones, nor what sort of space can qualify as a fundamental space. Moreover, there is no uncontroversial way of explaining what a magnitude would have to be like in order to qualify as a fundamental physical magnitude; similarly, there is no uncontroversial way of characterizing what a mathematical space would have to be like in order to be a fundamental physical space. One could try defining fundamental physical magnitudes as the magnitudes picked out by the most basic terms of some physical theory, or fundamental spaces as the mathematical spaces employed in some physical theory. But different physical theories refer to different fundamental magnitudes and use different mathematical spaces. Which physical theory should we use?

If we specify that the Humean base consists of the values of the basic magnitudes of our current physical theory, we would tie the fortunes of HS to those of current physical theory, which seems a bad move. Should we specify that the Humean base consists of the values of the basic magnitudes of some future physical theory, perhaps the one that physicists will accept at the Peircean limit of enquiry? If we do that, then we risk making HS vacuous. For, suppose that in a century or so, scientists come to accept some new theory quite unlike any current physical theory. Should it count as a “physical theory” or not? Without some account of what makes a theory a “physical” theory, the resulting version of HS is without content. (The worry here is similar to one attending some attempts to define “physicalism” and “materialism”; see van Fraassen (2002, pp. 49-61).)

To bring out the difficulty here, suppose that some future theory is formulated in terms of a new kind abstract space, different positions in which correspond to systems governed by different laws of nature, or to different second-order connections among basic physical properties, somewhat like the second-order relation of nomic necessitation posited by Armstrong (1983), Dretske (1977) and Tooley (1977). Would we want to allow the positions of systems like that to count as elements of the Humean base? Surely not; a physical theory that turns out to have been anticipated by the Armstrong-Dretske-Tooley theory of laws would be a physical theory incompatible with HS. We dare not predict with certainty that scientists will never formulate and accept such a theory. But if they ever do, then they will thereby adopt a theory with a fundamental space, locations in which cannot plausibly be included in the Humean base.

A more attractive alternative is to define the Humean base as the facts about the fundamental physical magnitudes, and locations within fundamental spaces, postulated by the true fundamental physical theory, whatever that is. But this proposal really has no advantage over the previous one: We do not know whether there is a true fundamental theory that is anything like current physics; for all we know, the best candidate for the title “true fundamental theory” may include fundamental spaces like the one just described.

A promising move here would be to try to formulate some general constraints that any scientific theory of the fundamental nature of the world would have to meet in order to count as a physical theory, and in order for its fundamental magnitudes and fundamental space to count as the kinds of magnitudes and spaces that belong in the Humean base. This would allow one to use Loewer’s way of characterizing the Humean base without either giving too many hostages to fortune or rendering HS vacuous. We have no objection to this strategy. However, as will emerge below, we think the job can be done more directly, by giving a different characterization of the Humean base that does not appeal to the notions of fundamental physical theory, fundamental physical magnitude, or fundamental space.

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