the maximal sub-nomically stable set at $w$, contradicting the requirement
that $\Lambda$ be nonmaximal. Hence, defining $\Lambda$ to be nonmaximal rules
out the metaphysical possibility of strong determinism.

I regard that as a significant and under-appreciated cost of Lange’s account.
Strong determinism is logically consistent and conceptually coherent. On what
grounds are we entitled to dismiss it as metaphysically impossible? Moreover,
there can be scientifically motivated, empirically adequate, and theoretically
virtuous candidates for laws that are strongly deterministic and provide new
insights into the foundations of physics (§2.4; §3.3). Ruling out strong
determinism by fiat seems contrary to the methodology of naturalistic
metaphysics. I think the lesson here is that we should rethink the original
motivation for a non-circular distinction between laws and accidents and
modify Lange’s account in light of the possibility of strong determinism.

In contrast, strong determinism is not only allowed on MinP but serves as a
good example of how laws constrain the universe—strongly deterministic laws
rules out every world except the actual world. If such laws are simple, they
provide strong and compelling explanation for every event in spacetime.

###  4.5 Maudlinian Primitivism

All previous accounts surveyed in this section attempt to reduce laws to
something else, such as the Humean mosaic, universals, dispositions, and
counterfactuals. An alternative view is simply to take laws as ontological
primitives. An influential primitivist account is developed by Maudlin (2007).
His version of primitivism, which in one aspect is similar to MinP, comes with
more metaphysical commitments than MinP.

As a primitivist about laws, he suggests that we should not analyze or reduce
laws into anything else. Laws are metaphysically fundamental; they are
primitive entities that do not supervene on other entities. To have a
sufficiently explanatory metaphysical theory, our fundamental ontology needs
to include not only spatiotemporal objects but also laws that govern them.
Maudlin rejects any reduction or deeper analysis of laws.

Maudlin is also committed to primitivism about the direction of time: that the
distinction between past and future is metaphysically fundamental and not
reducible to anything else. There is in effect a fundamental arrow or
orientation at every spacetime point that points to the future. Maudlin
combines the two commitments into a metaphysical package (p.182).

The two posits are closely connected on his account. For Maudlin, laws produce
or generate later states of the world from earlier ones. In this way, via the
productive power of the laws, subsequent states of the world (and its parts)
are explained by earlier ones and ultimately by the initial state of the
universe. It is this productive explanation that is central to his account.
Dynamic production is closely related to causation, and just like (paradigm
cases of) causation it is time asymmetric. Future states are dynamically
produced from earlier states but not vice versa. This, for example, allows
Maudlin’s account to vindicate a widespread intuition about Bromberger’s
flagpole. The shadow is produced by the circumstances and the length of the
pole (together with the laws). Although we can deduce from the laws the pole
length based on the circumstances and the shadow length, the pole length is
not produced by them. Hence, given the laws, the pole length and the
circumstances explain, but are not explained by, the shadow length.

Maudlin suggests that his package yields an attractive picture by being closer
to our initial conception of the world. He is committed to all three theses
discussed in §2.4: Only FLOTEs, Dynamic Production, and Temporal Direction
Primitivism. We may summarize Maudlin’s metaphysical package as follows:

Maudlinian Primitivism

Fundamental laws are certain ontological primitives in the world. Only
dynamical laws (in particular, laws of temporal evolution) can be fundamental
laws. They operate on the universe by producing later states of the universe
from earlier ones, in accord with the fundamental direction of time.

Maudlin allows there to be primitive stochastic dynamical laws—those laws that
involve objective probability such as the GRW collapse laws. Hence, dynamic
production need not be deterministic. An initial state can be compatible with
multiple later states, determining only an objective probability distribution
over those states. Perhaps the objective probability can be understood as
propensity, with stochastic production implying variable propensities of
producing various states, in proportion to their objective probabilities and
in accord with the direction of time. However, even if deterministic
production is an intelligible notion, it is not clear that stochastic
production or propensity is as intelligible. (Recall the earlier point about
“probabilistic necessitation” in Platonic Reductionism.)

Comparisons with MinP. Maudlinian Primitivism and MinP agree that fundamental
laws are metaphysically fundamental and that they govern. They disagree about
how they do it.

For Maudlin, dynamic production is essential, and every fundamental law needs
to have the form of a dynamical law (in the narrow sense of a FLOTE) that can
be interpreted as evolving later states of the universe from earlier ones. For
laws to produce, they operate according to the fundamental direction of time,
providing an intuitive picture close to our pre-theoretic conception of the
world: “the universe is generated from a beginning and guided towards its
future by physical law” (p.182). MinP is not committed to a fundamental
direction of time; nor is it committed to dynamic production as how laws
govern or explain. On MinP, explanation by simple constraint is good enough.
Many candidate fundamental laws such as the Einstein equation are not (in and
of themselves) FLOTEs that produce later states of the universe from earlier
ones. For the same reason, PH, WCH, IPH, and NBWF cannot be Maudlinian laws.
And neither can a purely spatial constraint such as Gauss’s law212121In
personal communication, Maudlin suggests that he now regards (4) as expressing
a metaphysical analysis or a definition of $\rho$ in terms of the divergence
of E.  or the simple rule responsible for the Mandelbrot world. On MinP, they
can all be understood as fundamental laws that express simple constraints.

The difficulty with dynamic production is not just that it precludes certain
candidate fundamental laws. It is also difficult to understand the notion
itself. What does dynamic production mean and what are its relata? Does it
relate instantaneous states or sets of instantaneous states of the universe?
If it relates instantaneous states, it is hard to understand dynamic
production even in paradigm examples of FLOTEs such as the one expressed by
Hamilton’s equations. (The initial data is not confined to a single moment in
time, if we understand momentum as partly reducible to variations in positions
over some time interval.) The notion becomes even less natural in relativistic
settings. Moreover, on a simple understanding of dynamic production, the
beginning of the universe does metaphysical work; it is what gets the entire
productive enterprise started. However, for spacetimes with no temporal
boundaries, it is unclear where to start the productive explanation. In
contrast, constraints operate on the entire spacetime, regardless of whether
there is an “initial” moment. Thus, MinP does not require a first moment in
time. (Perhaps a more sophisticated understanding of dynamic production does
not either.)

On MinP, even if the universe lacks a fundamental direction of time, we can
still recover a notion of productive explanation at a non-fundamental level.
For example, we can use PH to define a (non-fundamental) direction of time in
the usual way: earlier is defined as being closer to the time of PH, while
later is defined as being further away from that. We may regard FLOTEs as
evolving earlier states of the universe into later ones (with respect to PH).
In such a universe, dynamic production may be metaphysically derivative.
Still, we can contemplate a (non-fundamental) productive explanation of
Bromberger’s flagpole and vindicate the intuition that the pole length and the
circumstances explain, but are not explained by, the shadow length. Therefore,
the intuitive picture behind Maudlinian Primitivism can be preserved even if
there is no fundamental notion of dynamic production or a fundamental
direction of time.

###  4.6 Summary

We have looked at five accounts of laws that have been widely discussed in the
literature, and compared them with MinP. All four non-Humean accounts surveyed
here are more restrictive than MinP. Even though Humean Reductionism disagrees
with MinP regarding the metaphysical status of laws, they agree on many first-
order judgments about which equations may express laws. This suggests that the
fundamental disagreement between Humeanism and non-Humeanism is much more
subtle than is often recognized.

##  5 Simplicity

Physical laws are strikingly simple, although there is no a priori reason they
must be so. I propose that nomic realists of all types (Humeans and non-
Humeans) should accept that simplicity is a fundamental epistemic guide for
discovering and evaluating candidate laws. As it turns out, this
epistemological principle is independent of the metaphysical posits about laws
in both Humeanism and non-Humeanism.

###  5.1 Nomic Realism and the Epistemic Gap

Many physicists and philosophers are realists about physical laws. Call
realism about physical laws nomic realism. It contains two parts:

Metaphysical Realism:

Physical laws are objective and mind-independent; more precisely, which
propositions express physical laws are objective and mind-independent facts in
the world.222222A weaker version of metaphysical realism maintains that laws
are not entirely mind-dependent. That would accommodate more pragmatic
versions of the Humean best-system accounts including those of Hicks (2018),
Dorst (2019), Jaag and Loew (2020), and Callender (2023). The arguments below,
with suitable adaptations, still apply.

Epistemic Realism:

We have epistemic access to physical laws; more precisely, we can be
epistemically justified in believing which propositions express the physical
laws, given the evidence that we will in fact obtain.232323The terminology is
due to Earman and Roberts (2005). Here I’ve added the clause “given the
evidence that we will in fact obtain.” My version of epistemic realism is
logically stronger than theirs.

Nomic realism gives rise to an apparent epistemic gap: if laws are really
objective and mind-independent, it may be puzzling how we can have epistemic
access to them, since laws are not consequences of our observations. The
epistemic gap can be seen as an instance of a more general one regarding
theoretical statements on scientific realism (Chakravartty 2017).

The accounts surveyed in §4-5 all aspire to satisfy nomic realism. Let us
focus on BSA and MinP. Do they vindicate epistemic realism? Their metaphysical
posits, by themselves, do not guarantee epistemic realism. This should be
clear on MinP. Since there is no metaphysical restriction on the form of the
fundamental laws, if they are entirely mind-independent primitive facts about
the world, how do we know which propositions are the laws? However, an
analogous problem exists on BSA. This claim may surprise some philosophers, as
it is often thought that BSA has an epistemic advantage over non-Humean
accounts like MinP, precisely because BSA brings laws closer to us. BSA
defines laws in terms of the mosaic, and the mosaic is all we can empirically
access (Earman and Roberts 2005).

The problem is that we are not given the mosaic. Just like laws, the mosaic
entertained in modern physics is a theoretical entity that is not entailed by
our observations. Our beliefs about its precise nature, such as the global
structure of spacetime, its microscopic constituents, and the exact matter
distribution, are as theoretical and inferential as our beliefs about laws.
They are all parts of a theory about the physical world. Just as MinP requires
an extra epistemic principle to infer what the laws are, BSA requires a
similar principle to infer what the mosaic is like. The latter, on BSA, turns
out to be equivalent to a strong epistemic principle concerning what we should
expect about the best system given our limited evidence, which because of its
limitation pins down neither the mosaic nor the best system.

After all, on BSA laws are not summaries of our observations only, but of the
entire spacetime mosaic constituted by the totality of microphysical facts, a
small minority of which show up in our observations. The ultimate judge of
which system of propositions is the optimal true summary depends on the entire
mosaic, a theoretical entity. (For this reason, BSA should not be confused as
a version of strict empiricism.) And in current physics, our best guide to the
mosaic is our best guess about the laws. At the end of the day, MinP and BSA
turn out to require the same super-empirical epistemic principle concerning
laws. On neither account does the epistemic principle follow from the
metaphysical posits about what laws are.

To sharpen the discussion, let us suppose, granting Lewis’s assumption of the
kindness of nature (Lewis 1994, p.479), that given the mosaic $\xi$ there is a
unique best system whose axioms express the fundamental law $L$:

$L=BS(\xi)$ (11)

with $BS(\cdot)$ the function that maps a mosaic to its best-system
law.242424We might understand pragmatic Humeanism as recommending that we use
another best-system function $BS^{\prime}(\cdot)$ that is “best for us.”  Let
us stipulate that for both BSA and MinP, physical reality is described by a
pair $(L,\xi)$. For both, we must have that $\xi\in\Omega^{L}$, with
$\Omega^{L}$ the set of mosaics compatible with $L$. This means that $L$ is
true in $\xi$. On BSA, we also have that $L=BS(\xi)$. So in a sense, all we
need in BSA is $\xi$; $L$ is not ontologically extra. But it does not follow
that BSA and MinP are relevantly different when it comes to epistemic realism.

Let $E$ denote our empirical evidence consisting of our observational data
about physical reality. Let us be generous and allow $E$ to include not just
our current data but also all past and future data about the universe that we
in fact gather. There are two salient features of $E$:

  * •

$E$ does not pin down a unique $\xi$. There are different candidates of $\xi$
that yield the same $E$. (After all, $E$ is a spatiotemporally partial and
macroscopically coarse-grained description of $\xi$.)

  * •

$E$ does not pin down a unique $L$. There are different candidates of $L$ that
yield the same $E$. (On BSA, this is an instance of the previous point; on
MinP, this is easier to see since $L$ can vary independently of $\xi$, up to a
point.)

Hence, on BSA, just as on MinP, $E$ does not pin down $(L,\xi)$. There is a
gap between what our evidence entails and what the laws are. Ultimately, the
gap can be bridged by adopting simplicity (among other super-empirical
virtues) as an epistemic guide. Nevertheless, it helps to see how big the gap
is so that we can appreciate how much work needs to be done by simplicity and
other epistemic guides.252525It is worth contrasting the current setup with
the influential framework suggested by Hall (2009, 2015). In one way of
fleshing out the core idea of BSA, Hall imagines a Limited Oracular Perfect
Physicist (LOPP) who has as her evidence all of $\xi$ and nothing else. Her
evidence $E_{\text{LOPP}}$ contains vastly more information than $E$. On BSA,
$E_{\text{LOPP}}$ pins down $(L,\xi)$. Actually, this is still incorrect. On
BSA, we also need the assumption that $E_{\text{LOPP}}$ corresponds to the
entire spacetime, and the mosaic does not “continue” after $E_{\text{LOPP}}$.
This subtlety has not been sufficiently appreciated in the literature. Notice
that $E_{\text{LOPP}}$ is as theoretical for Humeans as for non-Humeans. The
Humean’s best guess about what is in $E_{\text{LOPP}}$ depends on her
expectation about what $L$ looks like given $E$.

The epistemic gaps can be illustrated by considering cases of empirical
equivalence. If different laws yield the same evidence, it is puzzling how we
can be epistemically justified in choosing one over its empirically equivalent
rivals, unless we rule them out by positing substantive assumptions that go
beyond the metaphysical posits of nomic realism. Here I briefly mention three
kinds of algorithms for generating empirical equivalents. For a more in-depth
discussion, see Chen (2023b).

Algorithm A: Moving parts of ontology (what there is in the mosaic) into the
nomology (the package of laws).

General strategy. This strategy works on both BSA and MinP. Given a theory of
physical reality $T_{1}=(L,\xi)$, if $\xi$ can be decomposed into two parts
$\xi_{1}\&\xi_{2}$, we can construct an empirically equivalent rival
$T_{2}=(L\&\xi_{1},\xi_{2})$, where $\xi_{1}$ is moved from ontology to
nomology.

Example. Consider the standard theory of Maxwellian electrodynamics, $T_{M1}$:

  * •

Nomology: Maxwell’s equations and Lorentz force law.

  * •

Ontology: a Minkowski spacetime occupied by charged particles with
trajectories $Q(t)$ and an electromagnetic field $F(x,t)$.

Here is an empirically equivalent rival, $T_{M2}$:

  * •

Nomology: Maxwell’s equations, Lorentz force law, and an enormously
complicated law specifying the exact functional form of $F(x,t)$ that appears
in the dynamical equations.

  * •

Ontology: a Minkowski spacetime occupied by charged particles with
trajectories $Q(t)$.

Our evidence $E$ is compatible with both $T_{M1}$ and $T_{M2}$. The outcome of
every experiment in the actual world will be consistent with $T_{M2}$, as long
as the outcome is registered as certain macroscopic configuration of particles
(Bell 2004). We can think of the new law in $T_{M2}$ as akin to the
Hamiltonian function in classical mechanics, which is interpreted as encoding
all the classical force laws, except that specifying $F(x,t)$ is much more
complicated than specifying a typical Hamiltonian. Both $F(x,t)$ and the
Hamiltonian are components of respective laws of nature that tell particles
how to move.262626Note that we can decompose the standard ontology in many
other dimensions, corresponding to more ways to generate empirically
equivalent laws for a Maxwellian world. This move is discussed at length by
Albert (2022). Similar strategies have been considered in the “quantum
Humeanism” literature. See Miller (2014), Esfeld (2014), Callender (2015),
Bhogal and Perry (2017), and Chen (2022a). Given metaphysical realism, at most
one of the two theories has the correct nomology.

Algorithm B: Changing the nomology directly.

General strategy. This strategy is designed for MinP. We can generate
empirical equivalence by directly changing the nomology. Suppose the actual
mosaic $\xi$ is governed by the law $L_{1}$. Consider $L_{2}$, where
$\Omega^{L_{1}}\neq\Omega^{L_{2}}$ and $\xi\in\Omega^{L_{2}}$. $L_{1}$ and
$L_{2}$ are distinct laws because they have distinct sets of models. Since $E$
(which can be regarded as a coarse-grained and partial description of $\xi$)
can arise from both, the two laws are empirically equivalent. There are
infinitely many such candidates for $\Omega^{L_{2}}$. For example,
$\Omega^{L_{2}}$ can be obtained by replacing one mosaic in $\Omega^{L_{1}}$
with something different and not already a member of $\Omega^{L_{1}}$, by
adding some mosaics to $\Omega^{L_{1}}$, or by removing some mosaics in
$\Omega^{L_{1}}$. $L_{2}$ is empirically equivalent with $L_{1}$ since $E$ is
compatible with both.272727See Manchak (2009, 2020) for more examples.

Example. Let $L_{1}$ be the Einstein equation of general relativity, with
$\Omega^{L_{1}}=\Omega^{GR}$, the set of general relativistic spacetimes.
Assume that the actual spacetime is governed by $L_{1}$, so that
$\xi\in\Omega^{L_{1}}$. Consider $L_{2}$, a law that permits only the actual
spacetime and completely specifies its microscopic detail, with
$\Omega^{L_{2}}=\\{\xi\\}.$ Since our evidence $E$ arises from $\xi$, it is
compatible with both $L_{1}$ and $L_{2}$. Since it needs to encode the exact
detail of $\xi$, in general $L_{2}$ is much more complicated than
$L_{1}$.282828$L_{2}$ is a case of strong determinism. See Adlam (2022a) and
Chen (2022c) for recent discussions. Given metaphysical realism, at most one
of $L_{1}$ and $L_{2}$ corresponds to the actual law.

Algorithm C: Changing the nomology by changing the ontology.

General strategy. This strategy is designed for BSA. On BSA, we can change the
nomology by making suitable changes in the ontology (mosaic), which will in
general change what the best system is. Suppose the actual mosaic $\xi$ is
optimally described by the actual best system $L_{1}=BS(\xi)$. We can consider
a slightly different mosaic $\xi^{\prime}$, such that it differs from $\xi$ in
some spatiotemporal region that is never observed and yet $E$ is compatible
with both $\xi$ and $\xi^{\prime}$. There are infinitely many such candidates
for $\xi^{\prime}$ whose best system $L_{2}=BS(\xi^{\prime})$ differs from
$L_{1}$. Alternatively, we can expand $\xi$ to $\xi^{\prime}\neq\xi$ such that
$\xi$ is a proper part of $\xi^{\prime}$. There are many such candidates for
$\xi^{\prime}$ whose best system $L_{2}=BS(\xi^{\prime})$ differs from
$L_{1}$, even though $E$ is compatible with all of them.

Example. Let $L_{1}$ be the Einstein equation of general relativity, with
$\Omega^{L_{1}}=\Omega^{GR}$, the set of general relativistic spacetimes.
Assume that the actual spacetime is globally hyperbolic and optimally
described by $L_{1}$, so that $L_{1}=BS(\xi)$. Consider $\xi^{\prime}$, which
differs from $\xi$ in only the number of particles in a small spacetime region
$R$ in a far away galaxy that no direct observation is ever made. Since the
number of particles is an invariant property of general relativity, it is left
unchanged after a “hole transformation” (Norton 2019). We can use determinism
to deduce that $\xi^{\prime}$ is incompatible with general relativity, so that
$L_{1}\neq BS(\xi^{\prime})$. Let $L_{2}$ denote $BS(\xi^{\prime})$.
$L_{1}\neq L_{2}$ and yet they are compatible with the same evidence we obtain
in $\xi$. Since $\xi^{\prime}$ violates the conservation of number of
particles, $L_{2}$ should be more complicated than $L_{1}$. Given metaphysical
realism, at most one of $L_{1}$ and $L_{2}$ corresponds to the actual law.

Algorithms A, B, and C can be combined to generate more sophisticated cases of
empirical equivalence. The question they raise is this: what breaks the tie
among empirical equivalents and epistemically justifies our belief in the
intuitively correct law, given the evidence that we will in fact obtain? In
other words, in cases of empirical equivalence, how can we hold on to
epistemic realism given our commitment to metaphysical realism? To do so, we
need a tie breaker.

###  5.2 Nomic Simplicity

I propose that simplicity is a fundamental epistemic guide to lawhood. Roughly
speaking, simpler candidates are more likely to be laws, all else being equal.
It secures epistemic realism in cases of empirical equivalence where
simplicity is the deciding factor. In particular, we should accept this
principle:

Principle of Nomic Simplicity (PNS)

Other things being equal, simpler propositions are more likely to be laws.
More precisely, other things being equal, for two propositions $L_{1}$ and
$L_{2}$, if $L_{1}>_{S}L_{2}$, then $L[L_{1}]>_{P}L[L_{2}]$, where $>_{S}$
represents the comparative simplicity relation, $>_{P}$ represents the
comparative probability relation, and $L[\cdot]$ denotes is a law, which is an
operator that maps a proposition to one about lawhood.292929For example,
$L[F=ma]$ expresses the proposition that F=ma is a law. The proposition F=ma
is what Lange (2009) calls a “sub-nomic proposition.”

As a fundamental principle, PNS is not justified by anything else.303030Saying
that simplicity is a fundamental epistemic guide to lawhood does not mean it
is the only such guide. Recall that PNS contains the proviso “other things
being equal.” But sometimes other factors are not held equal, and we need to
consider overall comparisons of theoretical virtues (epistemic guides) and
their balance. Other theoretical virtues can also serve as epistemic guides
for lawhood. For example, informativeness and naturalness are two such
virtues. A simple equation that does not describe much or describe things in
too gruesome manners is less likely to be a law. We can formulate a more
general principle:  Principle of Nomic Virtues (PNV) For two propositions
$L_{1}$ and $L_{2}$, if $L_{1}>_{O}L_{2}$, then $L[L_{1}]>_{P}L[L_{2}]$, where
$>_{O}$ represents the relation of overall comparison that takes into account
all the theoretical virtues and their tradeoffs, of which of which $>_{S}$ is
a contributing factor, $>_{P}$ represents the comparative probability
relation, and $L[\cdot]$ denotes is a law, which is an operator that maps a
proposition to one about lawhood. What is overall better is a holistic matter,
and it can involve trade-offs among the theoretical virtues such as
simplicity, informativeness, and naturalness. PNV should be thought of as the
more general epistemic principle than PNS. I shall mainly focus on PNS, but
what I say below should carry over to PNV. But what is special about PNS, and
why not use the oft-cited principle of simplicity as below?

Principle of Simplicity (PS)

Other things being equal, simpler propositions are more likely to be true.
More precisely, other things being equal, for two propositions $L_{1}$ and
$L_{2}$, if $L_{1}>_{S}L_{2}$, then $L_{1}>_{P}L_{2}$, where $>_{S}$
represents the comparative simplicity relation, $>_{P}$ represents the
comparative probability relation.313131It may be too demanding to require a
total order that induces a normalizable probability distribution over the
space of all possible laws. It is less demanding to formulate PS in terms of
comparative probability.

The reason is that PS leads to probabilistic incoherence while PNS
straightforwardly avoids it. Whenever two theories have nested sets of models,
say $\Omega^{L_{1}}\subset\Omega^{L_{2}}$, the probability that $L_{1}$ is
true cannot be higher than the probability that $L_{2}$ is true. For
concreteness, consider an example from spacetime physics. Let $\Omega^{GR}$
denote the set of models compatible with the fundamental law in general
relativity—the Einstein equation, and let $\Omega^{GR^{+}}$ denote the union
of $\Omega^{GR}$ and a few random spacetime models that do not satisfy the
Einstein equation (see Figure 7). Suppose there is no simple law that
generates $\Omega^{GR^{+}}$. While the law of $GR$ (the Einstein equation) is
presumably simpler than that of $GR^{+}$, the former cannot be more likely to
be true than the latter, since every model of $GR$ is a model of $GR^{+}$, and
not every model of $GR^{+}$ is a model of $GR$. This is an instance of the
problem of nested theories, as $\Omega^{GR}$ is a subclass of and nested
within $\Omega^{GR^{+}}$.

$\Omega^{GR}$$\Omega^{GR^{+}}$ Figure 7: $\Omega^{GR}$ is nested within
$\Omega^{GR^{+}}$.

On PS, in the case of nested theories, we have probabilistic incoherence. If
$L_{1}$ is simpler than $L_{2}$, applying the principle that simpler laws are
more likely to be true, we have $L_{1}>_{P}L_{2}$. However, if $L_{1}$ and
$L_{2}$ are nested with $\Omega^{L_{1}}\subset\Omega^{L_{2}}$, the axioms of
probability entail that $L_{1}\leq_{P}L_{2}$. Contradiction!

On PNS, we can avoid the problem. Even though we think that the Einstein
equation is more likely to be a law, it is less likely to be true than the law
of $GR^{+}$. According to PNS, what simplicity selects here is not truth in
general, but truth about lawhood, i.e. whether a certain proposition has the
property of being a fundamental law. Let us assume that fundamental lawhood is
factive, which is granted on both BSA and MinP. Hence, lawhood implies truth:
$L[p]\Rightarrow p$. However, truth does not imply lawhood: $p\nRightarrow
L[p]$. This shows that $L[p]$ is logically inequivalent to $p$. This is the
key to solve the problem of coherence.

On PNS, the contradiction that inflicts PS is removed, because more likely to
be a law does not entail more likely to be true. If $L_{1}$ and $L_{2}$ are
nested, where $L_{1}$ is simpler than $L_{2}$ but
$\Omega^{L_{1}}\subset\Omega^{L_{2}}$, then $L_{1}\leq_{P}L_{2}$. It is
compatible with the fact that $L[L_{1}]>_{P}L[L_{2}]$. What we have is an
inequality chain:

$L[L_{2}]<_{P}L[L_{1}]\leq_{P}L_{1}\leq_{P}L_{2}$ (12)

From the perspective of nomic realism, one can consistently endorse PNS
without endorsing PS. Some facts are laws, but not all facts are laws. Laws
correspond to a special set of facts. On BSA, they are the best-system axioms.
On MinP, they are the constraints on what is physically possible.

This is also a new and simple solution to the problem of nested theories /
problem of coherence. It is compatible with but less demanding and perhaps
more general than the recent proposal of Henderson (2022). Unlike Henderson’s
approach, my proposal works even when candidate laws are not structured in a
hierarchy.

###  5.3 Theoretical Benefits

We should accept PNS because of its theoretical benefits. Here I discuss six
attractive consequences of PNS.

Empirical equivalence. PNS is useful for resolving cases of empirical
equivalence constructed along Algorithms A-C. For Algorithm A, $T_{2}$ will in
general employ much more complicated laws than $T_{1}$. For example, the laws
of $T_{M2}$ specify $F(x,t)$ in its exact detail. For Algorithm B, $L2$ will
in general be more complicated than $L1$, if $\Omega^{L_{2}}$ is obtained from
$\Omega^{L_{1}}$ by adding or subtracting a few models. For example, a
strongly deterministic theory of some sufficiently complex general
relativistic spacetime, as described in the example, needs to specify the
exact detail of that spacetime, will employ laws much more complicated than
the Einstein equation. For Algorithm C, even though the mosaics of $L1$ and
$L2$ are not that different, if $L1$ is a simple system, then in general $L2$
will not be. In fact, given enough changes from the actual mosaic, there may
not be any optimal system that simplifies the altered mosaic to produce a good
system. With PNS, we are justified in choosing the theoretical package with
simpler laws, which agrees with standard theory choice.

Induction. We want to know what the physical reality $(L,\xi)$ is like. Given
our limited evidence about some part of $\xi$ and some aspect of $L$, what
justifies our inference to other parts of $\xi$ or other aspect of $L$ that
will be revealed in upcoming observations or in observations that could have
been performed? It does not follow logically. Without some a priori rational
guide to what $(L,\xi)$ is like or probably like, we have no rational
justification for favoring $(L,\xi)$ over any alternative compatible with our
limited evidence. On a given $L$ we know what kind of $\xi$ to expect. But we
are given neither $L$ or $\xi$. Without further inferential principles, it is
hard to make sense of the viability of induction.

In §2.6, we discussed the potential connection between laws and induction.
What does induction require from laws? One might demand that laws hold the
same way everywhere and everywhen, but that is either vacuous or too
inflexible. A better answer is the simplicity of laws, as required by PNS. A
simple law can give rise to a complicated mosaic with an intricate matter
distribution. The complexity is only apparent, because behind all the surface
phenomena is a simple law that is discoverable. By assuming that the universe
is governed by a simple law, one may make reasonable guesses about unobserved
parts of the universe, based on a simple rule compatible with observed data.
But nomic simplicity does not forbid laws about boundary conditions or about
particular individuals. Some simple laws may even have temporal variations,
such as a time-dependent function $F=\frac{1}{t+k}ma$ for some constant $k$.
As long as the temporal variation and spatial variation are not too extreme as
to require complicated laws, we can still inductively learn about physical
reality based on available evidence, even in a non-uniform spacetime with
dramatically different events in different regions.

Symmetry. Symmetry principles play important roles in theory construction and
discovery. Physicists routinely use symmetries to justify or guide their
physical postulates. However, whether symmetries hold is an empirical fact,
not guaranteed by the world a priori. So why should we regard symmetry
principles to be useful, and what are they targeting? I suggest that certain
applications of symmetry principles are defeasible guides for finding simple
laws. In such cases, their epistemic usefulness may come from the fact that
certain symmetries are defeasible indicators for simplicity.323232For a
related perspective, see North (2021). Consider the toy example

$F=ma\text{ for }(-\infty,t]\text{ and }F=(8m^{9}-\frac{1}{7}m^{5}+\pi
m^{3}+km^{2}+m)a\text{ for }(t,\infty)$ (13)

with $F$ given by Newtonian gravitation. This law violates time-translation
invariance and time-reversal invariance. In this case, we have a much better
law that is time-translation and time-reversal invariant:

$F=ma\text{ for all times}$ (14)

The presence of the two symmetries in (14) and the lack of them in (13),
indicate that all else being equal we should prefer (14) to (13). We can
explain this preference by appealing to their relative complexity. (14) is
much simpler than (13), and the existence of the symmetries are good
indicators of the relative simplicity. However, in this comparison, we are
assuming that both equations are valid for the relevant evidence (evidence
obtained so far or total evidence that will ever be obtained). The preference
is compatible with the fact that if empirical data is better captured by (13),
we should prefer (13) to (14).

In the relevant situations where symmetry principles are guides to simplicity,
they are only defeasible guides. Symmetry principles are not an end in itself
for theory choice. I shall provide two more examples to show that familiar
symmetry principles are not sacred, but rather defeasible indicators of
simplicity, that can be ultimately sacrificed if we already have a reasonably
simple theory that is better than the alternatives.

The first is the toy example of the Mandelbrot world (§3.2). The physical
reality consisting of $(L_{M},\xi_{M})$ is friendly to scientific discovery.
If we were inhabitants in that world, we can learn the structure of the whole
$\xi_{M}$ from the structure of its parts, by learning what $L_{M}$ is.
However, $L_{M}$ is not a law with any recognizable spatial or temporal
symmetries.333333There is, however, the reflection symmetry about the real
axis. But it does not play any useful role here, and we can just focus on the
upper half of the Mandelbrot world if needed. Nevertheless, the physical
reality described by $(L_{M},\xi_{M})$ is a perfect example of an ultimate
theory (though not of the actual world). It is an elegant and powerful
explanation for the patterns in the Mandelbrot world. What could be a better
explanation? I suggest that none would be, even if it had more symmetries. In
this case, we do not need symmetry principles to choose the right law, because
we already have a simple and good candidate law. The lack of symmetries is not
a regrettable feature of the world, but a consequence of its simple law.

The second and more realistic example is the Bohmian Wentaculus (Chen 2018,
2022a, 2023c). With the Initial Projection Hypothesis (§3.3.2), the initial
quantum state is as simple as the Past Hypothesis. This allows us to adopt the
nomic interpretation of the quantum state, and understand the mosaic $\xi_{B}$
as consisting of only particle trajectories in spacetime, with the fundamental
dynamical law $L_{B}$ as given by this differential equation:

$\frac{dQ_{i}}{dt}=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{q_{i}}W_{IPH}(q,q^{\prime},t)}{W_{IPH}(q,q^{\prime},t)}(Q)=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{q_{i}}\bra{q}e^{-i\hat{H}t/\hbar}\hat{W}_{IPH}(t_{0})e^{i\hat{H}t/\hbar}\ket{q^{\prime}}}{\bra{q}e^{-i\hat{H}t/\hbar}\hat{W}_{IPH}(t_{0})e^{i\hat{H}t/\hbar}\ket{q^{\prime}}}(q=q^{\prime}=Q)$
(15)

Since the quantum state is nomic, as specified by a law, the right hand side
should be the canonical formulation of the fundamental dynamical law for this
world. Notice that the right hand side of the equation is not time-translation
invariant, as at different times the expression

$\text{Im}\frac{\nabla_{q_{i}}\bra{q}e^{-i\hat{H}t/\hbar}\hat{W}_{IPH}(t_{0})e^{i\hat{H}t/\hbar}\ket{q^{\prime}}}{\bra{q}e^{-i\hat{H}t/\hbar}\hat{W}_{IPH}(t_{0})e^{i\hat{H}t/\hbar}\ket{q^{\prime}}}$

will in general take on different forms. However, the physical reality
described by the Bohmian Wentaculus may be our world, and the equation can be
discovered scientifically. The law is a version of the Bohmian guidance
equation that directly incorporates a version of the Past Hypothesis. Hence,
$(L_{B},\xi_{B})$ describes a physical reality that is friendly to scientific
discovery and yet does not validate time-translation invariance.

In the Bohmian Wentaculus world, symmetry principles can be applied, but the
fundamental dynamical law explicitly violates time-translation invariance. In
such cases, the lack of symmetries is not a problem, because we already have
found the simple candidate that has the desirable features. Again, the time-
translation non-invariance is a consequence of its simple law. PNS takes
precedence over symmetry principles and are the deeper justification for
theory choice.

Dynamics. We have good reasons to allow fundamental laws of boundary
conditions. However, many boundary conditions are not suitable candidates for
fundamental lawhood. Epistemic guides such as simplicity allow us to be
selective in postulating boundary condition laws, and to give more weight to
proposals that include dynamical laws.

The examples of good boundary condition laws have a common feature: they are
simple to specify. Many boundary conditions contain a great deal of
correlations, but only a select few are good candidates for fundamental laws,
namely those that are also sufficiently simple. One may wonder why we choose
the Past Hypothesis, a macroscopic description, over a precise microscopic
initial condition of the universe. The answer is that the former is much
simpler than the latter and is still sufficiently powerful to explain a
variety of temporally asymmetric regularities. The simplicity of the boundary
condition laws make it almost inevitable that we will have dynamical laws in
addition to boundary condition laws. The scientific explanations of natural
phenomena come from the combination of simple boundary conditions and
dynamical laws. As such, dynamical laws have to carry a lot of information by
themselves.

Determinism. Nomic realism is often accompanied with other reasonable
expectations about laws. On MinP, given any mosaic $\xi$, there are many
possible choices of $L$ such that $\xi\in\Omega^{L}$ and mosaics do not cross
in $\Omega^{L}$. Here is an algorithm to generate some of them: construct a
two-member set $\Omega^{L}=\\{\alpha,\beta\\}$ such that $\alpha$ and $\beta$
agree at no time (or any Cauchy surface). Any law with such a domain meets the
definition of determinism. As long as $\alpha$ is not a world where every
logically possible state of the universe happens some time in the universe, it
is plausible to think that there are many different choices of $\beta$ that
can ensure determinism. Without a further principle about what we should
expect of $L$, determinism is too easy and almost trivial on MinP. On BSA, the
problem is the opposite. It becomes too difficult and almost impossible for a
world to be deterministic. Given the evidence $E$ we have about the mosaic,
even though $E$ may be optimally summarized by a deterministic law $L$, it
does not guarantee (or make likely, without further assumptions) that the
entire mosaic is optimally summarized by a deterministic law $L$. Small
“perturbations” somewhere in the mosaic can easily make its best system fail
determinism.343434See Builes (2022) for a related argument.

Hence, there is a question of what nomic realists should say that constitutes
a principled reason to think that determinism is not completely trivial (on
MinP) and not epistemically inaccessible (on BSA). With PNS, determinism is no
longer trivial on MinP. Given any mosaic $\xi$, even though there are many
deterministic candidates compatible with and true at $\xi$. Not every mosaic
will be compatible with a relatively simple law that is deterministic. The
non-triviality of determinism on MinP is the fact that it is non-trivial to
find a law that is simple and deterministic, as that is not guaranteed for
every metaphysically possible mosaic. With PNS, determinism is no longer
epistemically inaccessible on BSA. This follows from the more general
principle that PNS gives us epistemic justification to hold beliefs about
parts of the mosaic that we have not observed and will never observe. We are
justified in believing that the best system of the actual mosaic is relatively
simple, even though the actual evidence does not entail that. If the actual
evidence can be optimally summarized by a deterministic law restricted to the
actual evidence, we have epistemic justification to make inferences about
regions that will not be observed – the entire mosaic, $\xi$, can be
summarized by a simple law that happens to be deterministic.

Explanation. There is a strong connection between nomic realism and scientific
explanation. The point of postulating laws, on BSA and on MinP, is to provide
scientific explanations. However, not all candidate laws provide the same
quality of explanation or same kind of explanation. Hence, on both versions of
nomic realism, we might wonder if there is a principled reason to think that
we will have a successful scientific explanation for all phenomena.

On MinP, explanations must relate to us (§3.2). Constraints, in and of
themselves, do not always provide satisfying explanations. Many constraints
are complicated and thus insufficient for understanding nature. For example,
the constraint given by just $\Omega^{L}=\\{\xi_{M}\\}$, which requires a
complete specification of the mosaic, is insufficient for understanding the
Mandelbrot world. Knowing why there is a pattern requires more than knowing
the exact distribution of matter. On MinP, many candidate laws can constrain
the mosaic. But not all have the level of simplicity to provide illumination
about the mosaic. With PNS, we expect the actual constraint to be relatively
simple. The constraint given by the Mandelbrot law should be preferred to that
given by $\Omega^{L}=\\{\xi_{M}\\}$. The simple law provides a successful
explanation while the more complicated one does not.

On BSA, it is built in the notion of laws that they systematize the mosaic.
However, whether there is a systematization that is simpler than the mosaic is
a contingent matter, depending on the detailed, microscopic, and global
structures of the mosaic. Not every mosaic supports a systematization that
provides illumination in the sense of unifying the diverse phenomena in the
mosaic. BSA only entails that the best system is no more complex than the
exact specification of the mosaic. For example, some mosaics may support no
better optimal summary than the exact specification of the mosaic itself.
Hence, on BSA, having successful explanations is not automatic. It requires
the mosaic to be favorable. On BSA, some mosaics are favorable: they support
optimal summaries that are simpler than themselves and provide “Humean
explanations” about the mosaic. In fact, in a sense, most mosaics are not
favorable (Lazarovici 2020). There exist mosaics underdetermined by actual
evidence that do not support any good summaries. Given the actual evidence,
with PNS, we are epistemically justified in inferring that the actual best
system is relatively simple such that it can provide a “Humean explanation”
about the actual mosaic. In effect, we are expecting that the actual Humean
mosaic is a favorable one that completely cooperates with our scientific
methodology and is such that it can be unified in a reasonably simple best
system.

###  5.4 Summary

Nomic realists should postulate the principle of nomic simplicity. It
vindicates epistemic realism when there is empirical equivalence (at least in
those cases discussed in the paper), avoids probabilistic incoherence when
there are nested theories, and supports realist commitments regarding
induction, symmetries, dynamics, determinism, and explanation. With many
theoretical benefits for only a small price, it is a great bargain.

##  6 Exactness

Another hallmark of laws is their exactness, in contrast to the pervasive
vagueness we find in ordinary language. A good way of understanding something
is to study its opposite. In this section, I discuss a model of nomic
vagueness and a case study.

###  6.1 Nomic Exactness

Many predicates we use in everyday contexts do not have determinate boundaries
of application. Is John bald when he has exactly 5250 hairs on his head? There
are determinate cases of “bald,” but there are also borderline cases of
“bald.” In other words, predicates such as “bald” are indeterminate: there are
individuals such that it is indeterminate whether they are bald.353535There
are subtleties about how best to characterize vagueness. For reviews on
vagueness and the sorites, see Keefe and Smith 1996, Sorensen 2018, and Hyde
and Raffman 2018.  Moreover, the boundaries between “bald” and “borderline
bald” are also indeterminate. There do not seem to be sharp boundaries
_anywhere_ , a phenomenon as higher-order vagueness. Vagueness gives rise to
many paradoxes (such as the sorites) and serious challenges to classical
logic.

We might expect that, at the level of fundamental physics, the kind of
vagueness that “infects” ordinary language should disappear. That is, the
fundamental laws of physics, the predicates they invoke, and the properties
they refer to should be exact. The expectation is supported by the history of
physics and the ideal that physics should deliver an objective and precise
description of nature. All the paradigm cases of candidate fundamental laws of
nature are not only simple and universal, but also _exact_ , in the sense
that, for every class of worlds (or class of solutions), fundamental laws
either determinately apply or determinately fail. Suppose the fundamental laws
are Newton’s equation of motion $F=ma$ and law of universal gravitation
$F=Gm_{1}m_{2}/r^{2}$: there is no vagueness about whether a physical history
(described in terms of trajectories of point particles with Newtonian masses)
satisfies the laws. In other words, nomologically possible worlds form a set.

Fundamental nomic exactness—the ideal (roughly) that fundamental laws are
exact—supports an important principle about the mathematical expressibility of
fundamental laws. If some fundamental laws are vague, it will be difficult to
describe them mathematically in a way that genuinely respects their vagueness
and does not impose sharp boundaries anywhere. The kind of mathematics we are
used to, built from a set-theoretic foundation, does not lend itself naturally
to model the genuine fuzziness of vagueness. One could go further: the
language of mathematics and the language of fundamental physics are supposed
to be exemplars for the “ideal language,” a language that is exact, suggested
in Frege’s _Begriffsschrift_ , Russell’s logical atomism, and Leibniz’s
_characteristica universalis_. The successful application of mathematical
equations in formulating laws seems to leave no room for vagueness to enter
into a fundamental physical theory. If there is fundamental nomic vagueness,
and if vagueness is not completely mathematically expressible, then the
fundamental physical theory is not completely mathematically expressible.

###  6.2 Nomic Vagueness

Figure 8: The motion of a projectile in Newtonian mechanics. Picture from
Zátonyi Sándor, (ifj.) Fizped, CC BY-SA 3.0
<https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons

How should we understand the exactness of paradigm fundamental laws of nature?
Let us start with the familiar case of Newtonian mechanics (with Newtonian
gravitation). Its laws can be expressed as a set of differential equations
that admit a determinate set of solutions. Those solutions specify all and
only the possible histories compatible with the laws; each solution represents
a nomologically possible world of the theory.

Consider the projectile motion illustrated in Figure 8. Suppose that the
projectile has unit mass $m$ and the gravitational acceleration is $g$. We can
specify the history of the projectile with the initial height, initial
velocity, maximum height, and distance traveled. There is a set of histories
compatible with the laws. For any history of the projectile, it is either
determinately compatible with the equations or determinately incompatible with
the equations.

If $W$ represents the space of all possible worlds, then the nomologically
possible worlds of Newtonian mechanics corresponds to $\Omega^{NM}$, a proper
subset in $W$ that has a determinate boundary, where the boundary is not in
spacetime but in modal space. For any possible world $w\in W$, either $w$ is
contained in $\Omega^{NM}$ or it is not. For example, in Figure 9, $w1$ is
inside but $w2$ is outside $\Omega^{N}$. In other words, $w1$ is nomologically
possible while $w2$ is nomologically impossible if Newtonian laws are true and
fundamental. Call $\Omega^{NM}$ the domain of Newtonian mechanics. We can
capture an aspect of fundamental nomic exactness as the exactness of the
domain:

Domain Exactness

A fundamental law $L$ is domain-exact if and only if, (a) for any world $w\in
W$, there is a determinate fact about whether $w$ is contained inside $L$’s
domain of worlds, i.e. $L$’s domain has no borderline worlds, (b) $L$’s
domain, which may also be called $L$’s extension, forms a set of worlds, (c)
$L$’s domain is not susceptible to sorites paradoxes, and (d) $L$’s domain has
no borderline borderline worlds, no borderline borderline borderline worlds,
and so on.

Figure 9: An exact fundamental law and a vague fundamental law represented in
modal space.

In contrast, a domain-vague law has none of (a)–(d). Intuitively, a domain-
vague law has a vague boundary in the following sense. In Figure 9, a domain-
vague law is pictured by a “collection” of worlds with a fuzzy boundary. Just
as a cloud does not have a clear starting point or a clear end point, the
fuzzy “collection” of worlds does not delineate the worlds into those that are
clearly compatible and those that are clearly incompatible with the law. To
borrow the words of Sainsbury (1990), a domain-vague law classifies worlds
“without setting boundaries” in modal space. For example, $w3$ is clearly
contained inside the domain of the vague law, since it is so far away from the
fuzzy boundary; but $w4$ is not clearly contained inside the domain of the
vague law, and neither is it clearly outside; $w2$ is clearly outside the
domain. More precisely, I propose that we understand domain vagueness as the
opposite of domain exactness:

Domain Vagueness

A fundamental law $L$ is domain-vague if and only if $L$ meets all four
conditions below.

  * (a’)

$L$ has borderline worlds that are not determinately compatible with it. For
some world $w\in W$, there fails to be a determinate fact about whether $w$ is
contained inside $L$’s domain of worlds.

  * (b’)

$L$ lacks a well-defined extension in terms of a set of models or a set of
nomological possibilities. Nomological necessities and possibilities turn out
to be vague.

  * (c’)

$L$ is susceptible to sorites paradoxes. We can start from a world that is
determinately lawful, proceed to gradually make small changes to the world
along some relevant dimension, and eventually arrive at a world that is
determinately unlawful. But no particular small change makes the difference
between determinately lawful and determinately unlawful.

  * (d’)

$L$ possesses higher-order domain-vagueness. Whenever there are borderline
lawful worlds, there are borderline borderline lawful worlds, and so on. It
seems inappropriate to draw a sharp line anywhere. This reflects the genuine
fuzziness of domain vagueness.

Domain vagueness has features similar to those of ordinary-language vagueness.
Domain exactness and domain vagueness capture the kind of fundamental nomic
exactness and fundamental nomic vagueness we are most interested in here.
(There is another kind of fundamental nomic vagueness that results from vague
objective probabilities or typicalities. See Goldstein (2012). Fenton-Glynn
(2019) offers an account of imprecise (but not vague) chances in the best-
system theory.)

###  6.3 A Case Study

The Past Hypothesis (PH) presents a case of fundamental nomic vagueness. To
begin, let us consider the following version of PH that is sometimes proposed:

Super Weak Past Hypothesis (SWPH)

At one temporal boundary of space-time, the universe has very low entropy.

SWPH is obviously vague. How low is low? The collection of worlds with “low-
entropy” initial conditions has fuzzy boundaries in the space of possible
worlds. Hence, if SWPH were a fundamental law, then we would have domain
vagueness.

However, SWPH may not be detailed enough to explain all the temporal
asymmetries. For example, in order to explain the temporal asymmetries of
records, intervention, and knowledge, Albert (2000) and Loewer (2020b) suggest
that we need a more specific condition that narrows down the initial
microstates to a particular macrostate. One way to specify the macrostate
invokes exact numeral values for the macroscopic variables of the early
universe. Let $S_{0},T_{0},V_{0},D_{0}$ represent the exact values (or exact
distributions) of (low) entropy, (high) temperature, (small) volume, and
(roughly uniform) density distribution of the initial state. Consider the
following version of PH:

Weak Past Hypothesis (WPH)

At one temporal boundary of space-time, the universe is in a particular
macrostate $M_{0}$, specified by the macroscopic variables
$S_{0},T_{0},V_{0}$, and $D_{0}$.

WPH is a stronger version of PH than SWPH. By picking out a particular (low-
entropy) macrostate $M_{0}$ from many macrostates, WPH more severely
constrains the initial state of the universe. WPH is also more precise than
SWPH. (Some may even complain that the WPH is too strong and too precise.)
Unfortunately, WPH is still vague. The collection of worlds compatible with
WPH has fuzzy boundaries. If WPH were a fundamental law, then we would still
have nomic (domain) vagueness: there are some worlds whose initial conditions
are borderline cases of being in the macrostate $M_{0}$, specified by the
macroscopic variables $S_{0},T_{0},V_{0}$, and $D_{0}$.

The vagueness of WPH is revealed when we connect the macroscopic variables to
the microscopic ones. Which set of microstates realizes the macrostate
$M_{0}$? There is hardly any sharp boundary between those that do and those
that do not realize the macrostate. A macrostate, after all, is a coarse-
grained description of the physical state. As with many cases of coarse-
graining, there can be borderline cases. (To connect to our discussion in
§2.1, the vagueness of macrostates is similar to the vagueness of “is bald”
and “is a table.”) In fact, a macrostate can be vague even when it is
specified with precise values of the macro-variables. This point should be
familiar to those working in the foundations of statistical
mechanics.363636Commenting on the vagueness of the macrostate boundaries,
Loewer (2007) writes, “Obviously, the notion of _macro state_ is vague and
there are many precisifications that would serve the purposes of statistical
mechanics.” Goldstein et al. (2020) write, “there is some arbitrariness in
where exactly to ‘draw the boundaries.”’ However, it is worth spelling out the
reasons to understand where and why such vagueness exists.

Figure 10: A diagram of phase space where macrostates have fuzzy boundaries.
The macrostate $M_{0}$ represents the initial low-entropy condition described
by WPH. $X_{0}$ is the actual initial microstate. The picture is not drawn to
scale.

There is a systematic way to think about the vagueness of the thermodynamic
macrostates in general and the vagueness of $M_{0}$ in the WPH. In the
Boltzmannian account of classical statistical mechanics, macrostates and
microstates can be understood as certain structures on phase space (Figure
10).

  * •

Phase space: in classical mechanics, phase space is a 6N-dimensional space
that encodes all the microscopic possibilities of the system.

  * •

Microstate: a point in phase space, which is a maximally specific description
of a system. In classical mechanics, the microstate specifies the positions
and the momenta of all particles.

  * •

Macrostate: a region in phase space in which the points inside are
macroscopically similar, which is a less detailed and more coarse-grained
description of a system. The largest macrostate is thermal equilibrium.

  * •

Fuzziness: the partition of phase space into macrostates is not exact; the
macrostates have fuzzy boundaries. Their boundaries become exact only given
some choices of the “C-parameters”, including the size of cells for coarse-
graining and the correspondence between distribution functions and macroscopic
variables.

  * •

Entropy: $S(x)=k_{B}\text{log}|M(x)|,$ where $|\cdot|$ denotes the standard
volume measure in phase space. Because of Fuzziness, in general, the
(Boltzmann) entropy of a system is not exact.

We can translate WPH into the language of phase space: at one temporal
boundary of space-time, the microstate of the universe $X_{0}$ lies inside a
particular macrostate $M_{0}$ that has low volume in phase space.

Fuzziness is crucial for understanding the vagueness and higher-order
vagueness of macrostates. Without specifying the exact values (or exact ranges
of values) of the C-parameters, the macrostates have fuzzy boundaries: some
microstates are borderline cases for certain macrostates, some are borderline
borderline cases, and so on. The fuzzy boundary of $M_{0}$ illustrates the
existence of borderline microstates and higher-order vagueness. There will be
a precise identification of macrostates with sets of microstates only when we
exactly specify the C-parameters (or their ranges). In other words, there is a
precise partition of microstates on phase space into regions that are
macroscopically similar (macrostates) only when we make some arbitrary choices
about what the C-parameters are. In such situations, the WPH macrostate
$M_{0}$ would correspond to an exact set $\Gamma_{0}$ on phase space, and the
initial microstate has to be contained in $\Gamma_{0}$.

However, proponents of the WPH do not specify a precise set. A precise set
$\Gamma_{0}$ would require more precision than is given in statistical
mechanics—it requires the specific values of the coarse-grained cells and the
specific correspondence with distribution functions. (In the standard quantum
case discussed in Appendix, it also requires the precise cut-off threshold for
when a superposition belongs to a macrostate.) The precise values of the
C-parameters could be added to the theory to make WPH into a precise statement
(which I call the Strong Past Hypothesis in Chen (2022b)). But they are
nowhere to be found in the proposal, and rightly so.373737For example, see
descriptions of SWPH and WPH in Goldstein 2001, Albert 2000, and Carroll 2010.

Some choices of the C-parameters are clearly unacceptable. If the coarse-
graining cells are too large, they cannot reflect the variations in the values
of macroscopic variables; if the coarse-graining cells are too small, they may
not contain enough gas molecules to be statistically significant. Hence, they
have to be macroscopically small but microscopically large (Albert (2000)
p.44(fn.5) and Goldstein et al. (2020)). However, if we were to make the
parameters (or the ranges of parameters) more and more precise, beyond a
certain point, any extra precision in the choice would seem completely
arbitrary. They correspond to how large the cells are and which function is
the correct one when defining the relation between temperature and sets of
microstates. That does not seem to correspond to any objective facts in the
world. (How large is large enough and how small is small enough?) In this
respect, the arbitrariness in precise C-parameters is quite unlike that in the
fundamental dynamical constants. (In Chen (2022b), I discuss their differences
in terms of a theoretical virtue called ‘traceability.’) Moreover, not only do
we lack precise parameters, we also lack a precise set of permissible
parameters (hence no exact ranges of values for the C-parameters). There
shouldn’t be sharp boundaries anywhere. Suppose size $m$ is borderline large
enough and size $n$ is determinately large enough. Small changes from $m$ will
eventually get us to $n$, but it is implausible that there is a sharp
transition from borderline large enough to determinately large enough.
Similarly, there shouldn’t be a sharp transition between borderline large
enough to borderline borderline large enough, and so on. That is higher-order
vagueness.

Because of higher-order vagueness, we need to take standard mathematical
representation of WPH with a grain of salt. The macroscopic
variables—adjustable parameters in WPH—need to be coarse-grained enough to
respect the vagueness. For example, we may represent the temperature of
$M_{0}$ as $10^{32}$ degrees Kelvin. But temperature does not have the
exactness of real numbers. A more careful way to represent the vague
temperature should be “$10^{32}$-ish degrees Kelvin,” where the “-ish”
qualifier signifies that temperature is vague and the number $10^{32}$ is only
an imperfect mathematical representation.383838Thanks to Alan Hájek for
discussions here. Its exactness is artificial. Hence, WPH should be
characterized as a macrostate $M_{0}$ specified by $S_{0}$-ish entropy,
$T_{0}$-ish temperature, and so on.

The vagueness here is appropriate, since macroscopic variables only make sense
when there are enough degrees of freedom (such as a large number of
particles). In practice, however, such vagueness rarely matters: there will be
enough margins such that to explain the thermodynamic phenomena, which are
themselves vague, we do not need the extra exactness. The vagueness disappears
_for all practical purposes_. Nevertheless, WPH is a genuine case of
fundamental nomic vagueness and it is a possibility to take seriously.

###  6.4 Summary

I have suggested that fundamental laws of physics can be vague, and PH
provides an example. It turns out that this may be an artifact of classical
mechanics and can be naturally avoided in quantum mechanics (Chen 2022b, sect.
4). Hence, nomic exactness may be a metaphysically contingent feature of the
universe that depends on the actual laws.

##  7 Objectivity

The final hallmark of laws I want to discuss is their objectivity. Laws are
objective features of reality that do not depend on our beliefs or desires.
They are mind-independent. However, the metaphysics of laws can make a
difference to how we understand their objectivity.

###  7.1 Ratbag Idealism

Humean Reductionism and non-Humeanism offer different understandings about the
objectivity of laws. Let us focus on BSA and MinP. On BSA, assuming that we
are relying on the right theoretical virtues and have appropriate access to
the mosaic, the best summaries will be the true laws. There is a certain sense
that, in principle, we are guaranteed to be right. On MinP, even if we rely on
the correct theoretical virtues and the correct scientific methodology, we can
still be mistaken about what the true laws are. Epistemic guides are
defeasible and fallible indicators for truth: they do not guarantee that we
find the true laws (although we may be rational to expect to find them). There
are fundamental, objective, and mind-independent facts about which laws govern
the world, and we can be wrong about them. This is not a bug but a feature of
MinP, symptomatic of the robust kind of realism that we endorse. For realists,
this is exactly where they should end up; fallibility about the fundamental
reality is a badge of honor.

On BSA, since the best systematization is constitutive of lawhood, and what
counts as best may depend on us, then lawhood can become mind-dependent. In a
passage about “ratbag idealism,” Lewis (1994) discusses this worry and tries
to offer a solution:

> The worst problem about the best-system analysis is that when we ask where
> the standards of simplicity and strength and balance come from, the answer
> may seem to be that they come from us. Now, some ratbag idealist might say
> that if we don’t like the misfortunes that the laws of nature visit upon us,
> we can change the laws—in fact, we can make them always have been
> different—just by changing the way we think! (Talk about the power of
> positive thinking.) It would be very bad if my analysis endorsed such
> lunacy….
>
> The real answer lies elsewhere: if nature is kind to us, the problem needn’t
> arise…. If nature is kind, the best system will be robustly best—so far
> ahead of its rivals that it will come out first under any standards of
> simplicity and strength and balance. We have no guarantee that nature is
> kind in this way, but no evidence that it isn’t. It’s a reasonable hope.
> Perhaps we presuppose it in our thinking about law. I can admit that if
> nature were unkind, and if disagreeing rival systems were running neck-and-
> neck, then lawhood might be a psychological matter, and that would be very
> peculiar. (p.479)

For Lewis, the solution is conditionalized on the hope that nature is kind to
us in this special way: the best summary of the world will be far better than
its rivals. That may be a generous assumption, but it seems consistent with
scientific practice.

In contrast, the worry about ratbag idealism does not arise on MinP; the
objectivity of laws can be secured without appealing to the hope that nature
is kind to us. On MinP, fundamental laws are what they are irrespective of our
psychology and judgments of simplicity and informativeness. Even though the
epistemic guides provide some guidance for discovering and evaluating them,
they do not guarantee arrival at the true fundamental laws. Moreover, changing
our psychology or judgments will not change which facts are fundamental laws.
Hence, MinP respects our conviction about the objectivity and mind-
independence of fundamental laws.

###  7.2 A Package Deal

Despite the difference regarding the metaphysical objectivity of laws, there
is a convergence in methodological principles underlying MinP and BSA, which
may be traced back to epistemic principles that are somewhat mind-dependent.
In §7.3, we discuss the epistemic version of ratbag idealism.

We have mainly focused on fundamental laws. To be sure, they are related to
the fundamental material ontology (fundamental entities and their properties).
On MinP, we regard both fundamental laws and fundamental material ontology as
metaphysical primitives and evaluate them in a package. In this respect, MinP
is similar to Loewer’s Package Deal Account (PDA), a descendent of BSA that
regards both as co-equal elements of a package deal (Loewer 2020c, 2021), but
they also have significant differences.

On PDA, we look for the best systematization in terms of a package of laws and
(material) ontology; the package is supervenient on the actual world. Thus,
fundamental laws and fundamental ontology enter the discussion in the same
way, at the same place, and on the same level. MinP shares this feature,
although fundamental ontology and fundamental laws are merely discovered by us
and not made by us or dependent on us. On PDA, given the actual world (of
which we have very limited knowledge), we evaluate different packages of laws
+ ontology, and we evaluate them based on our actual scientific practice.
Hence, there will be some degree of relativism. Relative to different
scientific practice or a different set of scientists, the judgement as to the
actual laws + ontology would have been different. Consequently on PDA,
fundamental laws and fundamental ontology are dependent on us in a significant
way.

On MinP, we may use the best package-deal systematization as a guide to
discover the laws and ontology; given the actual world (of which we have very
limited knowledge), we evaluate different packages of laws + ontology, and we
evaluate them based on our actual scientific practice. Hence, there will be
some degree of uncertainty. Relative to different scientific practice or a
different set of scientists, the judgement as to the actual laws + ontology
would have been different. Still, what they are is metaphysically independent
of our belief and practice.

Here I quote a passage from Loewer (2021). Although we disagree with him on
the metaphysics, we agree on how the enterprise of physics should be
understood:

> The best way of understanding the enterprise of physics is that it begins,
> as Quine says, “in the middle” with the investigation of the motions of
> macroscopic material objects e.g., planets, projectiles, pendula, pointers,
> and so on. Physics advances by proposing theories that include laws that
> explain the motions of macroscopic objects and their parts. These theories
> may (and often do) introduce ontology, properties/relations, and laws beyond
> macroscopic ones with which it began and go onto to posit laws that explain
> their behaviors…… The ultimate goal of this process is the discovery of a
> theory of everything (TOE) that specifies a fundamental ontology and
> fundamental laws that that cover not only the motions of macroscopic objects
> with which physics began but also whatever additional ontology and
> quantities that have been introduced along the way. (pp. 30-31)

From the perspective of MinP, this is an excellent description of how
fundamental laws and ontology are discovered—in a package.

###  7.3 Humeanism and non-Humeanism

For non-Humeans who endorse methodological principles as epistemic guides, the
problem of ratbag idealism may show up in the epistemology of laws. (Hall
2009, §5.6) suggests that, facing the problem that the simplicity criterion in
the BSA is too subjective, Lewis and other Humeans can “perform a nifty judo
move.” If non-Humeans regard simplicity as an epistemic guide to laws,
“central facts of normative epistemology are also up to us.” Hall argues that
this is more objectionable than the ratbag idealism of BSA. A defender of BSA
may reasonably embrace ratbag idealism and take laws to be pragmatic tools to
structure our investigation of the world. With that viewpoint, we can expect
that what laws are is somewhat up to us. However, there is no reason on non-
Humeanism why epistemological and normative facts should be up to us. So the
non-Humeans might face a worse problem of ratbag idealism.

My analysis in §5 suggests that both Humeans and non-Humeans need to adopt
fairly strong epistemological principles such as PNS. On Humeanism, there are
two independent appeals to simplicity (among other theoretical virtues). The
metaphysical analysis of laws with BSA requires laws be no more complicated
than the mosaic. But not all mosaics support simple laws; in fact, many
metaphysically possible mosaics may not have any regularity that deserves the
label of laws. To believe, on BSA, that simple laws exist is to believe that
the actual mosaic is very special. The epistemic guide to simple laws serves
as an epistemic and normative restriction of possible candidate mosaics
Humeans ought to consider. Why the Humean mosaic should be so nice on BSA is
basically the same question as why fundamental constraints should be so simple
on MinP. So on both BSA and MinP, epistemological and normative facts are up
to us.

This reveals a deeper difference between Humeanism and non-Humeanism. On MinP,
we should assume that certain fundamental facts of the world are simple. On
BSA, in contrast, we should assume that certain superficial facts of the world
(best-system laws), grounded in a complex fundamental reality, are simple. The
assumption, it seems to me, is much more believable on MinP than on BSA. It is
much easier to believe that nature at some deep level is very simple. It is
much more difficult to believe that nature at some deep level is very
complicated in a certain way to give rise to a simple appearance. Of course,
this would not persuade the committed Humeans, for presumably they are willing
to accept the consequence. However, for many people on the fence or coming to
this debate for the first time, the choice between Humeanism and non-Humeanism
should be clear.393939I am indebted to discussions with Tyler Hildebrand and
Boris Kment about this point.

###  7.4 Summary

Metaphysical objectivity is a feature of laws on non-Humeanism such as MinP
but not on Humean views such as BSA. However, on both theories the
epistemological principles may be somewhat up to us.

##  8 Conclusion

Laws occupy a central place in a systematic philosophy of the physical world.
They can be regarded as fundamental facts that govern the universe by
constraining its physical possibilities. With this minimal primitivist
account, one accepts that laws transcend the concrete physical reality they
govern, but need not presume a fundamental direction of time or require a
fundamental ontology of universals, dispositions, or counterfactuals. The
account allows one to contemplate a variety of candidate fundamental laws, and
to understand the marks of the nomic as arising from methodological and
epistemological principles. It is not the only viewpoint available, but it is
one I recommend to anyone looking for an account that illuminates metaphysics
but is not unduly constrained by it.

## Acknowledgements

I am indebted to Sheldon Goldstein for invaluable discussions about laws of
physics over the years. This volume is connected to several issues explored in
our joint paper on MinP. I thank Shelly Yiran Shi and Bosco Garcia for their
research assistance and helpful comments on several earlier drafts. For
discussions and encouragements, I am also grateful to Emily Adlam, David
Albert, Jeffrey Barrett, Craig Callender, Eugene Chua, Christopher Dorst, Nina
Emery, Ned Hall, James Hartle, Tyler Hildebrand, Boris Kment, Marc Lange,
Barry Loewer, Tim Maudlin, Chris Meacham, Kerry McKenzie, John Roberts, Carlo
Rovelli, Charles Sebens, Elliott Sober, Eric Watkins, and James Owen
Weatherall. This project is supported by an Academic Senate Grant from the
University of California, San Diego.

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