Productive Laws in Relativistic Spacetimes

1. Introduction

In debates about the metaphysics of laws of nature, one of the most plausible and formidable positions is occupied by proponents of productive accounts of laws. Foremost among these is Maudlin’s (2007), which posits Fundamental Laws of Temporal Evolution (FLOTEs) that guide the development of the universe by producing later states from earlier states. On this view, the productive activity of the laws underwrites their explanatory power: laws help to explain events by providing the history of their production.

Maudlin’s account is deeply intuitive. If philosophical theories of lawhood were judged solely by their intuitive force, then I think most parties to these debates would admit that this is the clear winner. No doubt this is partly because of the account’s natural coherence with the physical theory in which most of us first acquire the concept of law, namely Newtonian mechanics. Knowingly or not, that theory probably provides many of us with our paradigm case of lawhood in $F=ma$. With this dynamical law at its center, Newtonian mechanics does indeed allow us to speak sensibly of the laws evolving earlier states of the world into later states.

What I want to argue here, however, is that the notion of productive laws is seriously problematized by the shift from Newtonian mechanics to relativity. Once we move from the Galilean spacetime of Newtonian mechanics to the sorts of spacetimes required by special and general relativity, productive laws lose most if not all of their intuitive appeal. In particular, by focusing on the simpler Minkowski spacetime of special relativity, I’ll argue that there are two significant threats to productive accounts of laws like FLOTEs. First, Minkowski spacetime makes it unclear how we are supposed to understand the relata of the nomic production relation. And second, as Putnam (1967) and others have suggested, it motivates a “block universe” view of time that, I will argue, obviates the need for productive laws in the first place.

This agenda raises three dialectical points that ought to be addressed before proceeding. First, Maudlin is sympathetic with Bohmian mechanics, which itself is inconsistent with special relativity because it requires a privileged foliation of spacetime. Thus one might suggest that it is unfair to attack Maudlin on the grounds that his view of laws is inconsistent with special relativity, since he has antecedent commitments that are likewise inconsistent. I agree with this suggestion, and it is not my goal to attack Maudlin in this paper. Rather, my goal is to evaluate the prospects for a particular theory of lawhood that Maudlin has compellingly and persuasively articulated. That view is proposed not as a theory of the laws of Bohmian mechanics, but as a theory of lawhood simpliciter. Some people, including Maudlin himself, have thought that “The idea of a law governing temporal evolution can be extended to special relativistic, and (I think) even general relativistic, contexts” (2007: 48). My aim here is to show that these extensions are untenable. If this is right, then that theory of lawhood is seriously impugned.

This raises the second dialectical point. Given that special relativity is not the final word, how serious, in the end, are my objections? The answer will depend on our final physics, specifically on why special relativity is not the final word. Special relativity does not accommodate gravity; to do that we move to a general relativistic framework. If special relativity is merely rejected in favor of general relativity, then as Savitt (2000) points out, every general relativistic spacetime locally has the structure of Minkowski spacetime. In that case, the arguments I raise here would, I think, still apply.¹ However, general relativity itself should eventually be supplanted by a theory of quantum gravity. I won’t speculate here about the prospects of such a theory maintaining a genuinely relativistic spacetime structure²—if it does, then my arguments should still apply; if not, not. Such is life for the non-omniscient.³

The third dialectical point is that you might think that it’s just obvious that FLOTEs will not work in relativistic spacetimes. After all, the problems I’ll be pointing out are based on relatively straightforward issues that arise as soon as one moves to a relativistic context, and they have analogs in discussions about the metaphysics of time. But I hope to show that the issues here are actually fairly subtle: there are moves open to a proponent of FLOTEs that strictly speaking would render them compatible with relativity. It’s just that by employing these moves, the view loses most or all of its theoretical advantages over other theories of lawhood.

So much for the dialectical points. Here’s how this paper will proceed. In §2 I’ll review Maudlin’s account of FLOTEs and what he says about their relationship with the passage of time. In §§3–4, I’ll explicate the dual threats to FLOTEs posed by relativistic spacetimes and consider how a proponent of FLOTEs might respond. In §5, as a way of avoiding these threats, I’ll consider the merits of positing a privileged foliation along which the productive activity of the FLOTEs proceeds. I’ll conclude in §6.

2. Maudlin’s Non-Humean Package

Maudlin’s FLOTEs are fundamental entities of the universe that occupy a distinct ontological category from the constituents of spacetime and the spatiotemporal arena itself. Their job, as he puts it, is to “specify how the state of the universe will, or might, evolve from a given initial state” (2007: 172). Of course, a great deal hangs on what Maudlin means by “specify” here. One reading would be that the laws merely entail what happens at later times given what happens at an initial time. This is not what Maudlin intends. Rather, “specification” has metaphysical import: “the laws of temporal evolution operate, whether deterministically or stochastically, from the initial state to generate or produce later states” (ibid.: 174, my emphasis). The basic picture is thus that the laws have a sort of input/output form, where the input is an earlier state and the output is a later state, and their role is to produce the later state from the earlier one.

According to Maudlin, this nomic production relation underwrites the laws’ ability to explain natural phenomena: “This sort of explanation takes the term initial state quite seriously: the initial state temporally precedes the explananda, which can be seen to arise from it (by means of the operation of the law)” (ibid.: 176). So if you want to know why such-and-such an event occurred at time $t_2$, we can appeal to the state at $t_1$ as well as the operation of the laws to show that the state at $t_1$ evolves, according to the laws, into the state at $t_2$.

Superficially, this looks like the old-fashioned Deductive Nomological model of explanation (Hempel and Oppenheim, 1948). One of the familiar problems with that model was that it allowed explanations in the wrong directions. Not only can one derive the length of the shadow from the angle of the sun above the horizon, the height of the flagpole, and the laws of optics, but one can also derive the height of the flagpole from the length of the shadow, the angle of the sun above the horizon, and the laws of optics. Only one of these derivations should count as an explanation, but both satisfy the DN model’s criteria. So there’s an explanatory asymmetry that the DN model fails to capture.

Maudlin’s view captures this asymmetry of explanation by appeal to the asymmetry of production. The laws produce the later state from the earlier state, but not vice versa. Thus even if the laws allow us to infer the earlier state from the later state, this would not give us an explanation of how the earlier state arose in the first place.

As all of this suggests, Maudlin also has a robust conception of the passage of time. Part of what allows us to speak sensibly about the laws producing later states from earlier states, he thinks, is that there is an inherent anisotropy in the temporal dimension. That is, given an event, there is an objective fact about which direction is forward in time from that event. And unlike others who have sought to reduce the temporal asymmetry to other asymmetries in the total distribution of events (e.g. Albert (2000), Loewer (2007, 2012)), Maudlin regards the temporal asymmetry as fundamental, maintaining that it both supports talk of the passage of time, and is required for the very notion of nomic production to be coherent in the first place; without it there would be no sense to talk of “production.”⁴

The passage of time, however, is meant to be more than just a temporal asymmetry:

The passage of time connotes more than just an intrinsic asymmetry…[it] underwrites claims about one state “coming out of” or “being produced from” another, while a generic spatial (or temporal) asymmetry would not underwrite such locutions. (2007: 109–110)

So talk of the passage of time is supported by an intrinsic asymmetry in the temporal dimension, but it also requires some extra ingredient, because not just any asymmetry would support talk of production. What is this extra ingredient? According to Loewer’s (2012: 119) interpretation of Maudlin, the passage of time “consists in the fundamental arrows of time [i.e. the fundamental temporal asymmetry] determining the direction in which the laws operate.” So it is the conjunction of the temporal direction and the operation of the FLOTEs along that direction that makes it sensible to talk about genuine temporal passage, and the genuine production of later states from earlier states, in the way that a generic asymmetry by itself would not.

Maudlin’s view is clearly at home in a theory like Newtonian mechanics. Not only did Newton espouse something very much like Maudlin’s notion of temporal passage⁵, but the Newtonian picture also allows us to make good sense of the operation of the FLOTEs. As we saw, the form of the FLOTEs requires us to identify both an input state and an output state to serve as the relata of the production relation. In the Newtonian picture, there are clear and unambiguous choices for these relata: the present state of the world that pervades all of space provides the input on which the Newtonian laws act to produce subsequent states of the world as output. These global states at moments of time are natural relata for the production relation.

Moreover, there really are no other candidate relata to be found in Newtonian mechanics. One might try to specify relata that are less spatially expansive, but no matter where one draws the boundary of such regions, the Newtonian laws will allow that events outside of that region may influence the events within that region at subsequent moments. As Maudlin notes (ibid.: 178), this forces us to make the input relatum a global state.

Once we have expanded the input relatum to encompass all of space, there is no reason not to do the same with respect to the output relatum. I suppose that in principle one might take the output relatum to be a local state or even a single event, and allow that the laws produce each of the events at $t_n$ from the preceding global state at $t_{n–1}$ on a piecemeal, albeit simultaneous, basis.⁶ But it is hard to see what motivation there could be for fragmenting the productive activity of the laws like this, and this does not seem to be Maudlin’s view.

Note that locality, on this view, is a rather superficial phenomenon. If the relata of FLOTEs are global states of the world, then while it may indeed turn out that there are stronger correlations between spatially local variables than there are between spatially distant ones, fundamentally speaking the entire earlier state is nomically tied to the entire later state. Nor would genuine locality be saved if we followed the suggestion in the previous paragraph and made the output relatum a single event, since the earlier relatum from which the event is produced is still a global state, and is not confined to the region in the spatial vicinity of the event being produced.

Despite this result, on the whole the view is incredibly intuitive. This fact is not lost on Maudlin; indeed he takes it to be one of the primary attractions of the view, especially over Humean alternatives that regard facts about the temporal asymmetry and laws as reducible to facts about the entire manifold of events. Dubbing his theories of FLOTEs and temporal passage as “Maudlin’s Non-Humean Package,” he says that his motivation for accepting it over the Humean alternative is that it is “much closer to the intuitive picture of the world that we begin our investigations with” (ibid.: 182).

I agree. In many ways, Maudlin’s Non-Humean Package could be thought of as a sort of default position, a natural starting point that we should abandon only if we find good reasons to do so. Of course, many philosophers claim to have found such reasons. Given that FLOTEs fail to supervene on the particular matters of fact, Humeans have sometimes raised worries about our epistemic access to them (Loewer 2012). Others have objected that taking the laws as primitive makes it unclear how they are supposed to play their explanatory role (Roberts 2016). And still others have objected that positing a primitive anisotropy in the temporal dimension wouldn’t help to explain our experience of time’s passage (Price 2011).

Without taking a stand on any of these claims, in the following I want to suggest a different reason that should motivate us to consider abandoning the default position of Maudlin’s Non-Humean Package: it is problematized by relativity. The issue, in short, is that while the Package itself is highly intuitive, relativity is not. Insofar as we have good reason to think that our world is relativistic, we also have good reason not to trust the very intuitions about laws and temporal passage that motivate acceptance of the Package in the first place.

In the following, I’ll focus for simplicity on special relativity. The two threats I want to consider still have force in general relativity (though see footnote 1), but they are easier to discuss in the flat Minkowski spacetime of SR than in the variably curved spacetimes of GR.

3. What Produces What?

If we want to make sense of the productive activity of the FLOTEs in SR, we need to characterize the kinds of states that serve as the input and output relata. Unlike in Newtonian mechanics, this is far from straightforward in SR.

Maudlin gives us the following advice about how to conceptualize FLOTEs in a special relativistic setting:

[T]ake a deterministic FLOTE and adjunct principles that operate in a special relativistic spacetime. Take a surface that cuts every maximal timelike trajectory in the spacetime exactly once (a Cauchy surface). Specifying a Cauchy surface is the analog to choosing a moment of time in a Newtonian regime; roughly one can think of it as a surface that cuts the spacetime in two horizontally…Boundary values can be specified on this surface, such as the distribution of particles, intensities of fields, etc…The FLOTE now specifies how those values will evolve through time. (ibid.: 18)

Maudlin thus appears to be suggesting that we regard the relata of the production relation the same way we did in Newtonian mechanics, namely as global states of the entire world.

The problem is that specifying a Cauchy surface is not an exact analog of choosing a moment of time in Newtonian mechanics. Given an event in the Galilean spacetime of Newtonian mechanics, there is exactly one moment of time that it lies on. But given an event in the Minkowski spacetime of special relativity, there are an infinite number of Cauchy surfaces that intersect it. So instead of having a single natural candidate for the input relatum, we have a plurality.

Part of what conceals this problem in Maudlin’s discussion is that much of it is framed in terms of the evaluation of counterfactuals. His aim is to show that one can use FLOTEs as part of a recipe that delivers intuitively correct results about the truth values of a wide variety of counterfactual conditionals. Given a counterfactual, the core steps of the recipe are: (1) select a Cauchy surface as directed by the antecedent, (2) modify boundary values on that surface as minimally as possible so as to make the antecedent true, and (3) evolve that modified state forward (or backward) according to the laws to see whether the consequent comes out true.⁷

In this context, the productive role of the FLOTEs takes a backseat to their inferential role. To see this, it suffices to note that the recipe directs us to evolve the modified state forward or backward according to the laws to infer what would have happened if the scenario specified by the antecedent had been true. So it is not the productive activity of the FLOTEs that allows them to play their role in counterfactual reasoning, but rather the inferences they enable us to make about some parts of the manifold given information about other parts. They can play this inferential role without our having to worry about which Cauchy surface we apply them to, as long as we are able to make the changes specified by the antecedent of the counterfactual. All such surfaces will give us consistent results about which events occur, so there is no worry about whether the laws really act on all of them or one of them. The laws don’t really act on any of them—they’re counterfactual, after all.

But in the actual world, the question of which Cauchy surface(s) the FLOTEs act on becomes more pressing: we need an answer in order to understand how the laws are supposed to exert their productive power. So how should we interpret Maudlin’s suggestion?

The only structure that is intrinsic to a special relativistic spacetime is given by the Minkowski metric, which determines the distances between events, and thereby also defines the lightcone structure (see Figure 1). Relative to a particular event e, all other events are divided into one of three sets, those that are timelike, lightlike, or spacelike separated from e. Events at timelike separation are in e’s future or past; events at lightlike separation lie on the forward or backward lightcones emanating from e; and events at spacelike separation bear no temporal relation to e—i.e. they are neither before, after, nor simultaneous with it. One can impose a coordinate system on the spacetime and thereby induce a temporal ordering for all events, even ones that are spacelike separated. But there is no uniquely correct way to do so supported by the underlying geometry; the coordinate system is a human projection, not part of the intrinsic structure of the spacetime.

Figure 1

Lightcone structure of Minkowski spacetime. Events at timelike, lightlike, and spacelike separation from e are indicated, as are three Cauchy surfaces that intersect e, represented by dotted lines; note they may be either straight or curved. The Cauchy surfaces have an extra dimension suppressed for ease of representation.

If we are going to respect the resulting democracy of inertial frames, then it seems we cannot privilege a particular Cauchy surface as the one on which the laws act, i.e. the one in which dynamical evolution “really” occurs.⁸ So we will have to interpret Maudlin’s suggestion as meaning that the FLOTEs act on every Cauchy surface. Even still, this is compatible with at least two different ways of construing the input and output relata of the production relation.

4. The Block Universe

The second major relativistic threat to FLOTEs traces back to an argument originally advanced by Putnam (1967) to the effect that special relativity requires a block universe (or “eternalist”) conception of time.¹³ Here I will briefly review that argument and then explain why it is problematic for Maudlin’s FLOTEs.

The argument is designed as a refutation of the presentist (or growing-block-ist) idea that a present moment sweeps through the temporal dimension, separating the real events of the past/present from the as-of-yet unreal events of the future. Consider two reference frames defined via two inertial observers, A and B, in relative motion with respect to each other (see Figure 2). At any given point, each observer will regard certain events spacelike separated from themselves as occurring simultaneously. What will these surfaces of simultaneity look like? The diagram indicates a few of them with dashed lines: in A’s frame, the surfaces of simultaneity radiate outward on straight spacelike lines perpendicular to A’s worldline. But B’s surfaces of simultaneity tilt upward with respect to A’s, so B regards events as simultaneous that A does not. Of course, all of this is perfectly symmetric; from B’s frame, it’s A’s simultaneity surfaces that appear tilted.¹⁴

Figure 2

Spacetime diagram of inertial observers A and B in relative motion with cross-cutting simultaneity surfaces. A’s surfaces are horizontal in the diagram. In this frame, if the slope of B’s trajectory is s, then the slope of B’s simultaneity surfaces is $1/s$.

Note that A and B’s worldlines cross at the origin o. Now take an event p on some faraway alien planet that lies on that simultaneity surface of B’s that also intersects o. If we regard the simultaneity surfaces as present moments separating the real past/present events from the unreal future events, then at o, p is real for B but unreal for A. If this is nonsense (and surely it is), then it must be wrong to regard these simultaneity surfaces as distinguishing the real from the unreal. Worse yet, consider some inertial observer at p, who has a simultaneity surface that intersects B’s worldline at point q, which happens to be B’s death. If reality transfers along these simultaneity surfaces (and assuming that no observer is singled out as special, then it’s hard to see why it wouldn’t), then B’s death at q is just as real as the event at p, which is just as real as B’s presence at o.¹⁵

The upshot is that special relativity appears to refute the presentist (or growing-block-ist) view of time in favor of a block universe (“eternalist”) view. As Sider (2001: 42) puts it, the argument constitutes “the fatal blow to presentism.” In common terms, it forces us to accept that the future is just as real as the present.

This conclusion is widely (though not universally) endorsed by philosophers of physics. In addition to the authors already cited in footnote 13, others who accept the argument include Gibson and Pooley (2008), Hoefer (2011), Monton (2006), Saunders (2002), and Savitt (2000) (though Savitt appears less sanguine about it in his (2011)). Callender calls it “utterly convincing” (2017: 53). Maudlin (2002: 260; 2007: 109, 115) also appears to endorse the block universe view, though he doesn’t say whether Putnam’s argument played a role in that endorsement. A notable exception is Zimmerman’s (2011) tour de force, which gives a compelling defense of presentism in the context of special relativity. Zimmerman argues in favor of privileging a particular foliation, and tries to minimize the conflict with special relativity that appears to result.¹⁶ We’ll return to this strategy in the next section, but for now, I want to consider the implications for FLOTEs if the conventional wisdom about Putnam’s argument is correct.

Those implications are straightforwardly problematic. If eternalism is the right view of time, then every event that will ever occur is “there already” in the spatiotemporal manifold. Fundamentally, then, there is nothing for the FLOTEs to produce; all of it already exists. The FLOTEs are just idle wheels.¹⁷

To be clear, I am not objecting to Maudlin’s taking the laws as ontologically primitive, nor to the idea that one can simply stipulate that these primitive entities play certain roles (cf. Schaffer, 2016). What I’m objecting to is the idea that the role that’s been stipulated for the FLOTEs is there to be played in the first place.

I know of nowhere that Maudlin explicitly addresses this worry. The closest he gets is in his rebuttal to Gödel’s (1949) argument that relativity undermines the idea of an “objective lapse of time.” Gödel claimed that such a lapse implies the existence of a privileged foliation of “nows,” and that this is incompatible with relativity. Maudlin (2002: 265–266) objects that a foliation is irrelevant to the lapse of time, and that what’s really needed is an orientation.¹⁸ However, elsewhere Maudlin seems to be committed to the idea that the passage of time requires more than just an orientation. Recall his claim in §2 that a generic temporal asymmetry would not underwrite talk of temporal passage, and that we needed to add the productive activity of the FLOTEs along that asymmetry for such talk to make sense. But if the entire manifold is already there, what could the productive activity of the FLOTEs possibly amount to? Even if we allow that an orientation by itself would be sufficient for temporal passage (and that a foliation would not), the crucial point is that it is not sufficient to legitimize talk of production. A complete manifold equipped with an orientation is no more hospitable to FLOTEs than is a complete manifold equipped with a preferred foliation. As long as the manifold is complete, there is nothing for the FLOTEs to do.

It is somewhat surprising that no one has yet made note of this objection. My own diagnosis is sociological: aside from some noteworthy exceptions¹⁹, philosophical work on the metaphysics of laws and the metaphysics of time seems to proceed relatively independently of one another. It may be that disciplinary boundaries are the culprit, as philosophy of time is typically a topic in metaphysics proper, whereas the metaphysics of laws is more frequently discussed in philosophy of science. This may contribute to a sort of compartmentalization, where questions about time and questions about laws tend to get addressed in relative isolation from each other.²⁰

An alternative explanation is that I have misunderstood Maudlin’s position: perhaps he is really an advocate of the growing block view of time. This is a subtle interpretive issue, and the best I can do is point the reader to quotations from Maudlin, many of which are cited above. Maudlin states his position most explicitly on page 109 of The Metaphysics within Physics (2007): “I am one of those unusual defenders of the block universe who does not deny that there is any objective flow of time. The four-dimensional universe is a single entity of which the passage of time, in one particular direction, is an ingredient.” Given that by “temporal passage” he means a fundamental asymmetry in the temporal dimension, (ibid.: 108) the picture one gets here is of a four-dimensional block with fundamental “arrows” pointing toward the future in the temporal dimension (this is Loewer’s interpretation of Maudlin in his (2012)). But one might argue that this should really be interpreted as a commitment to a four-dimensional growing block where the temporal arrows indicate the direction in which the FLOTEs produce growth. In that case, the leading edge of the growing manifold would play the role of the present moment, so we would again be faced with the task of identifying a relativistically kosher present. The prospects of a relativistic growing manifold are considered in §5.

Given a commitment to a (non-growing) block universe, I am not sure exactly how a proponent of the Package might try to respond to the objection that the FLOTEs have no role to play. One possibility would be to try to shift to a more deflationary conception of production to accommodate the fact that the output relata already exist. Callender (2017: 157), for example, talks of production occurring whenever the laws license inferences about one hypersurface given information about another: “[T]o the extent that one can speak of the physical state in a region of spacetime ‘determining’ another, one can often show that the state of the universe at one of these ‘times’ produces the next.” If Maudlin were to avail himself of this sense of production, presumably he would want to add that it only occurs along the future direction in the manifold, to avoid countenancing production (and hence explanation) in both temporal directions. We might call this the “oriented inferential conception” sense of production:

Oriented Inferential Production: A set of laws is productive in the oriented inferential sense if they permit nontrivial inferences about the content of later hypersurfaces given information about the content of earlier hypersurfaces.

This is a thin and yawn-inducing sense of production²¹ that could be adopted even by a Humean (as Callender himself is). All it requires of the laws is that they can amplify information in a particular direction, and this is a matter of the laws’ formal structure, not their metaphysical status.²² But if so, then it becomes unclear what is gained by Maudlin’s primitivism. FLOTEs were taken as primitive because they function not just to specify but to produce—in a robust, unanalyzable, yet intuitive sense—later states on the basis of earlier states. This robust kind of production was also supposed to underwrite the ability of the FLOTEs to furnish genuine explanations of natural phenomena. Maudlin explicitly takes it to be a serious advantage of his view that it can provide an explanation of how the manifold (“Humean mosaic”) comes to be.²³ Now, under pressure from the block universe of relativity, we are trying to say a bit more to clarify the nature of the production relation. If it really just amounts to a sort of directed implication relation, then what has ultimately been bought by primitivism about the laws? Humean laws can likewise “produce” states in this sense, so if that relation allows FLOTEs to explain the mosaic, it also allow Humean laws to explain the mosaic. Not only is the resulting account not doing justice to the intuitions that motivate it, but everything it can do can also be done by Humean laws.

It might be objected that Humean laws couldn’t play this explanatory role because of circularity worries. If the laws themselves are determined by the mosaic, then how can they turn around and explain it? This worry has been pressed by a number of authors, including Maudlin himself (2007: 172).²⁴ If it were right, then primitivism about FLOTEs would still have an explanatory advantage over Humeanism.

But two points deserve mention about this idea. First, this would be a rather hollow victory. It maintains that FLOTEs can explain the mosaic because they have a primitive status that endows them with such explanatory power. It is hard to see that much has been accomplished here.

Second, and more significantly, if the relevant kind of explanation is backed by production, and the production relation is to be understood in this oriented inferential sense, then the circularity worry appears to lose traction. What the laws are really doing here is licensing the inference from a given hypersurface to a later hypersurface, but they arguably aren’t directly involved in the explanation. Hicks (2021) argues that it is indeed a mistake to think of laws as figuring directly into explanations of their instances; rather, one can instead conceive of them as backing the explanation of the explanandum state on the basis of the explanans state.²⁵ On this view, there is no explanatory circularity for Humeanism, since the laws are not directly part of the explanans. Hicks’s conception of the explanatory role of laws squares quite well with a conception of explanation according to which it is ultimately a matter of Oriented Inferential Production—i.e. a matter of what can be inferred, in a temporally oriented manner, on the basis of what. If this is right, why should it matter what the metaphysical status is of the entity licensing that inference?

None of this is meant to endorse Oriented Inferential Production. I only mean to point out that if a proponent of FLOTEs were to adopt this conception in response to the worries about eternalism, their account would lose its explanatory advantage over Humeanism.

In short, Maudlin’s Non-Humean Package faces a dilemma: either the productive activity of the FLOTEs is ontologically robust, in which case it looks incompatible with eternalism (and, via Putnam’s argument, with relativity), or it is ontologically innocent, in which case the view has no clear advantage over Humeanism. Perhaps there is some other notion of nomic production that we can use to navigate this strait, but if so I am not aware of it, nor do I see how it could maintain the intuitive advantages of the view.

By now this is a recurring theme. In modifying our conception of FLOTEs to be compatible with relativity, in one way or another we forgo what was most compelling about them. There is an alternative, however. Instead of modifying FLOTEs to be compatible with the traditional understanding of relativity, we might instead modify our understanding of relativity to be compatible with FLOTEs. The next section considers the merits of this strategy: can it be done consistent with the letter of special relativity, and how scientifically revisionary would it be?

5. Privileging a Foliation

Essentially all of the problems that we’ve encountered to this point could be solved by positing a privileged foliation along which FLOTEs operate. We would then have natural candidates for the relata of the production relation, namely the successive leaves of the privileged foliation, giving us the following conception of production:

Privileged Hypersurface Production: Production occurs (only) along a single foliation of the manifold into spacelike hypersurfaces.

With Privileged Hypersurface Production, we have a notion that is objective (i.e. not hypersurface-relative) and yet it does not result in radical overdetermination of the produced events. Essentially it is the intuitive notion of production from Newtonian Mechanics.

With respect to Putnam’s argument, positing a privileged foliation along which FLOTEs operate would allow us to maintain that the future is not (yet) real, and that the FLOTEs are literally producing the manifold along an empirically inaccessible hypersurface, in accord with a growing block view of time. The privileged foliation might coincide with some particular inertial frame’s simultaneity surfaces, though in principle it need not (if the privileged hypersurface is curved). The empirical inaccessibility of which hypersurface is privileged would also allow us to explain why we might have found Putnam’s argument compelling.

The main issue here is that the existence of a preferred foliation is typically regarded as inconsistent with the content of special relativity. As Maudlin himself puts it: “If a preferred family of hyperplanes is part of the intrinsic structure of spacetime then the fundamental postulate of the Theory of Relativity is false” (2011: 185). But as he also points out, it is a subtle issue identifying exactly what counts as part of the intrinsic structure of spacetime, and there are ways of privileging a particular foliation without appealing to anything in spacetime’s intrinsic structure.

For example, Zimmerman (2011: 209) imagines a physical theory according to which a particular kind of particle is of singular importance, and then a manifold in which it just so happens that all such particles are inertially co-moving. Since the particles are privileged, there is some sense in which the frame in which they are all stationary is also privileged. But its privileged status has nothing to do with the intrinsic structure of spacetime; it is, rather, a matter of the contingent initial conditions of the universe. There is no inconsistency with special relativity here.

So some ways of privileging a foliation are relativistically kosher, while others are not. To decide whether a privileged foliation along which FLOTEs exert their productive power is inconsistent with special relativity, we need some criterion for when a foliation is privileged in a way that depends on the intrinsic structure of spacetime. Zimmerman (2011) seeks the same sort of criterion, though with slightly different goals in mind. He is trying to argue that presentism is consistent with special relativity, and wants to maintain that there are ways of privileging a foliation that are consistent both with the content of SR and with varieties of presentism. Drawing on Maudlin’s (2011) discussion of which versions of quantum mechanics are inconsistent with special relativity, the criterion Zimmerman ultimately arrives at is that a theory is inconsistent with SR if it postulates laws that directly appeal to intrinsic spacetime structure beyond the Minkowski metric (ibid.: 216). He elaborates the idea as follows:

What is inconsistent with merely Minkowskian intrinsic structure is to explain some fact about the contents of spacetime as being due to the special nature of one foliation, and then not be able to appeal to any deeper laws that fail to mention that foliation. If the laws of a theory merely pick out the relevant frame of reference in terms of contingent material contents, and the contents merely happen to pick out that frame; then it is the material contents that are doing the work. But if a theory’s most basic laws (whether they govern physical or metaphysical features of the manifold) must invoke one inertial frame of reference or foliation “by name”, as it were; then there is something special about the frame or foliation itself, quite apart from the manifold’s content. The law is an indication that the manifold includes built-in “rails”, directing things in a certain way: some structure that is part of spacetime itself is doing the work. (ibid.: 213).

Zimmerman’s idea is that if the fundamental laws directly reference the material contents of spacetime—and only pick out a particular frame indirectly by reference to those contents—that foliation may be privileged, but not in a way that is inconsistent with SR. So if we return to his special particle theory where they are all stationary in an inertial frame, whether or not we have an inconsistency with SR will depend on whether the laws refer to the material contents such as the particles, or whether they refer to successive leaves of the foliation themselves.

What is it for the laws to reference material constituents of spacetime as opposed to spacetime structure itself? Zimmerman acknowledges that this idea is somewhat vague, but presumably it cannot merely be a matter of how the laws are commonly expressed in our physical theories. As Hicks and Schaffer (2017) point out, physicists often use context sensitive pragmatic criteria to decide how to write down equations, so it would seem unwise to try to read off deep metaphysical structure directly from the equations found in physics texts and papers.

What the laws “refer to” is ultimately going to depend not just on the physical theory in question, but also on the metaphysics of laws. In particular, we need to ask what the role of the laws is according to the given philosophical theory of lawhood. If the role of the laws is to relate properties—as it would be according to necessitarians like Armstrong (1983), Dretske (1977), and Tooley (1977), or (rather differently) according to dispositional essentialists like Bird (2007)—then we need to know the nature of the properties that the laws relate: are they best understood as properties of occupants of spacetime, or of spacetime itself? If the role of the laws is to summarize particular matters of fact—as on Humean Best System views—then we need to know the nature of the particular facts they are summarizing. In other words, we need to look to the metaphysical form of the laws, not merely the form of their equations, to decide whether they are referencing intrinsic spacetime structure.²⁶

On some views of laws, it may require quite subtle analysis—of both the physical theory in question and the metaphysical account of laws on offer—to determine whether or not the laws reference intrinsic spacetime structure. Fortunately, the analysis is rather straightforward when it comes to the conception of FLOTEs currently under consideration. The relata of FLOTEs, on this view, are particular spacelike hypersurfaces of the spacetime manifold; the FLOTEs function to produce later such hypersurfaces from earlier ones. It seems clear here that the laws “reference” intrinsic spacetime structure, namely the privileged hypersurfaces of the manifold upon which they operate. Remember how Loewer suggested we understand this: the fundamental arrows of time—which we are now positing to exist along a privileged hypersurface—determine the direction in which the laws operate. The job of the FLOTEs is to look for the leading edge of the manifold and evolve it in the direction indicated by the arrows thereon. If so, the manifold has “built-in ‘rails’, directing things in a certain way,” as Zimmerman puts it. The verdict, then, is straightforward: if our criterion of inconsistency with SR is that the laws refer to intrinsic spacetime structure beyond the Minkowski metric, then the current conception of FLOTEs is inconsistent with SR.

How damning is this result? One option for a proponent of FLOTEs would be to search for an alternative criterion of inconsistency with SR on which there is none here. Surely there are candidate criteria that would render a friendlier verdict about the present proposal (namely, a FLOTE-directed privileged foliation), though admittedly the verdict of inconsistency seems quite plausible here—if this isn’t a case of inconsistency, then what is?—so any alternative criterion that reaches the opposite verdict would have to explain away that plausibility.

Probably a better option would just be to own up to the inconsistency. Though inconsistency with a physical theory as well-confirmed as relativity is a serious worry for a philosophical account of laws, as I mentioned in the introduction, there already exist first order physical theories that are likewise inconsistent with it, e.g. Bohmian mechanics.²⁷ Moreover, this inconsistency might go away in some general relativistic spacetimes, where the spacetime geometry itself could privilege a particular foliation.²⁸ And lastly, we know that general relativity is not the final word: someday it will probably be replaced by a more successful theory of quantum gravity, and at this point it’s anyone’s guess whether the spacetime structure of that theory will turn out to be as inhospitable to FLOTEs as those of SR and GR.

So the proponent of FLOTEs can take some solace in the fact that the spacetime structure of relativity may yet be replaced by better physics. But we ought not overstate this. Conflict with relativity is conflict with one of the most highly confirmed physical theories ever. Insofar as there are reasons to think that the world is relativistic—and there are plenty—there are reasons to think that FLOTEs are an inaccurate picture of the nature of laws.

6. Conclusion

When Pierre-Simon Laplace famously articulated the idea of causal determinism, the picture was at once metaphysical and epistemological:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. (1814 [1902])

What starts here as a metaphysical picture about the present state of the universe being the cause of its future becomes an epistemological picture about the intelligence’s ability to predict the future on the basis of the present. Maudlin’s FLOTEs are essentially a precisification of the vague metaphysical picture articulated by Laplace. I have argued that relativity undermines that metaphysical picture, though it’s worth noting that it does not touch the epistemological one. Given complete information about a Cauchy surface, a Laplacean intelligence could still predict the entire future, and what relativity teaches us is that it doesn’t matter which Cauchy surface the intelligence knows about; complete information about any one of them would allow the intelligence to infer the rest of the manifold.²⁹

It is rare that a philosophical view of laws sticks its neck out far enough to be so seriously tested by actual physics. Some philosophers, e.g. Lange (2012: 185), argue that philosophical accounts of laws ought to be general enough that they are compatible with all theories that physicists (rightly) take seriously. But Maudlin is explicit about his disagreement with this position. As just one example, he observes that FLOTEs cannot abide closed timelike curves, for if they existed then the productive activity of FLOTEs would operate in a circle around them—and production cannot operate in a circle.³⁰ In Maudlin’s words, “the existence of closed timelike curves would imply the non-existence of this sort of productive explanation, and might suggest that a Humean account is the strongest that can be had” (2007: 175). My basic suggestion has been that even without CTCs, relativity already implies the non-existence of the sort of productive explanations that Maudlin seeks.

In a sense, my objection here is an instance of a broader dialectical point. Non-Humeanism is sometimes characterized, rather generally, by its commitment to the idea that laws “govern” the particular matters of fact. If you spend enough time with Humeans, you will hear the complaint that this notion of governance is opaque: it is unclear just what is being asserted by the claim that laws govern—unclear what role is thereby being ascribed to the laws—and correspondingly it is unclear how we are to evaluate it.³¹

I think this complaint is fairer against some accounts than others, and at first glance Maudlin’s account simply avoids it. In contrast with the obscurity of “governance,” Maudlin’s notion of production seems relatively straightforward: here we can see what role is being ascribed to the laws, and correspondingly we can see what it would take for the view to be true or false.

But clarity is both a virtue and a risk. Insofar as I can see what it might mean to say that the laws produce later states of the world from earlier states, the truth of that claim would require the falsity of relativity. At the very least, if one wants to maintain that the idea of FLOTEs still makes sense in relativistic spacetimes, then it’s no longer clear what the production relation is supposed to be. Presumably it’s something extra, something that Humean laws don’t do. But what that extra thing is, and why it is aptly called “production,” I do not know.³²