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Governing Without A Fundamental Direction of Time:
Minimal Primitivism about Laws of Nature
Eddy Keming Chen∗and Sheldon Goldstein†
September 19, 2021
Abstract
The Great Divide in metaphysical debates about laws of nature is between
Humeans who think that laws merely describe the distribution of matter and non-
Humeans who think that laws govern it. The metaphysics can place demands on
the proper formulations of physical theories. It is sometimes assumed that the
governing view requires a fundamental / intrinsic direction of time: to govern,
laws must be dynamical, producing later states of the world from earlier ones, in
accord with the fundamental direction of time in the universe. In this paper, we
propose a minimal primitivism about laws of nature (MinP) according to which
there is no such requirement. On our view, laws govern by constraining the phys-
ical possibilities. Our view captures the essence of the governing view without
taking on extraneous commitments about the direction of time or dynamic pro-
duction. Moreover, as a version of primitivism, our view requires no reduction /
analysis of laws in terms of universals, powers, or dispositions. Our view accom-
modates several potential candidates for fundamental laws, including the principle
of least action, the Past Hypothesis, the Einstein equation of general relativity,
and even controversial examples found in the Wheeler-Feynman theory of elec-
trodynamics and retrocausal theories of quantum mechanics. By understanding
governing as constraining, non-Humeans who accept MinP have the same freedom
to contemplate a wide variety of candidate fundamental laws as Humeans do.
Key words: laws of nature, fundamentality, explanation, non-Humeanism, Humeanism,
governing, constraint, production, direction of time, causation, primitivism, ob-
jective probability, typicality
∗Department of Philosophy, University of California San Diego, 9500 Gilman Dr, La Jolla, CA 92093,
USA. Email: eddykemingchen@ucsd.edu
†Departments of Mathematics, Physics, and Philosophy, Rutgers University, Hill Center, 110 Frel-
inghuysen Road, Piscataway, NJ 08854-8019, USA. Email: oldstein@math.rutgers.edu
1
Contents
1 The Great Divide 2
2 Some Existing Approaches 5
2.1 HumeanReductionism ............................ 5
2.2 Platonic Reductionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Aristotelian Reductionism . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Maudlinian Primitivism . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Minimal Primitivism (MinP) 18
3.1 TheView ................................... 18
3.2 MinPandExplanation............................ 22
3.3 Examples and Further Clarifications . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 DynamicalLaws ........................... 25
3.3.2 Non-Dynamical Constraint Laws . . . . . . . . . . . . . . . . . . 28
3.3.3 Probabilistic Laws . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Comparisons 32
4.1 Comparison with Humean Reductionism . . . . . . . . . . . . . . . . . . 32
4.2 Comparison with Platonic Reductionism . . . . . . . . . . . . . . . . . . 35
4.3 Comparison with Aristotelian Reductionism . . . . . . . . . . . . . . . . 36
4.4 Comparison with Maudlinian Primitivism . . . . . . . . . . . . . . . . . 37
5 Conclusion 38
1 The Great Divide
The goal of this paper is to articulate minimal primitivism about laws of nature (MinP),
a minimalist and primitivist view about laws, and to contrast it with some leading
alternatives. MinP captures our conviction that the universe is governed by laws of
nature in a way that does not presuppose a fundamental direction of time. Here we
focus on laws of physics and particularly those suitable for being fundamental laws.
To begin, let us list a few paradigm examples of candidate laws of physics:
•Newton’s laws of motion
•The Schr¨odinger equation
•The Dirac equation
They are all dynamical laws concerning how physical systems evolve in time. Here are
some other equations or principles that, for one reason or another, may be controversial
examples as candidate laws of physics:
2
•The Einstein equation (of general relativity)
•The Wheeler-DeWitt equation
•Conservation laws
•Symmetry principles
•The principle of least action
•The Past Hypothesis (of a low-entropy boundary condition of the universe)
•Equations of motion in Wheeler-Feynman electrodynamics
In physics, a significant amount of work has been devoted to the discovery of its
true fundamental laws: the basic principles that govern the world.1The collection
of all of these laws may be called the axioms of the final theory of physics or the
Theory of Everything (TOE). The fundamental laws cannot be explained in terms of
deeper principles (Weinberg, 1992, p.18); from them we can derive theorems of great
importance and explain all significant observable regularities. Some of the equations
and principles in the above lists, with suitable adaptation, may be included in such a
collection. In this paper, we assume that there are fundamental laws and they play
important roles in scientific explanations. But what kind of things are fundamental
laws? Most people believe that laws are different from material entities such as particles
and fields, because, for one thing, laws seem to govern the material entities. But what
is this governing relation? What makes material entities respect such laws? What is
the role of laws in scientific explanations? Such questions do not have straightforward
answers, and they cannot be directly tested in experiments. They fall in the domain of
metaphysics.
The great divide in metaphysical debates about laws of nature is between Humeans,
who think that laws are merely descriptions, and non-Humeans, who think that laws
govern. Humeans maintain that laws merely describe how matter is distributed in the
universe. In Lewis’s version, laws are just certain efficient summaries of the distribution
of matter in the universe, also known as the Humean mosaic. Nothing really enforces
the patterns in the mosaic. A common theme in non-Humean views is that laws govern
the distribution of matter. By appealing to the governing laws, the patterns are ex-
plained. How laws perform such a role is a matter of debate, and there are differences
of opinion between reductionist non-Humeans such as Armstrong (1983) and primitivist
non-Humeans such as Maudlin (2007).
One’s metaphysical position can shape one’s expectations about what physical laws
should look like. It is sometimes assumed that the governing view of laws requires a fun-
damental direction of time: to govern, laws must be dynamical laws that produce later
states of the world from earlier ones, in accord with the direction of time that makes a
1In this paper, we use “fundamental laws” and “laws” interchangeably unless noted otherwise.
3
fundamental distinction between past and future. Call this conception of governing dy-
namic production. It is suggested in Maudlin (2007) and discussed at length in Loewer
(2012). For Maudlin, primitivism about laws and primitivism about the direction of
time should be postulated together, with this package supporting a particular kind of
explanation associated with dynamic production. This emphasis on dynamic produc-
tion is not unique to Maudlin and is important to some other types of non-Humeans.
Although we subscribe to the governing view and the primitivist view about laws of
nature, we do not share the view that a fundamental direction of time is essential to
either.
In this paper, we propose a minimal primitivist view (MinP) about laws of nature
that disentangles the governing conception from dynamic production. On our view, fun-
damental laws govern by constraining the physical possibilities of the entire spacetime
and its contents.2They need not exclusively be dynamical laws, and their governance
does not presuppose a fundamental direction of time. For example, they can take the
form of global constraints or boundary condition constraints for spacetime as a whole;
they can govern even in an atemporal world; they may permit the existence of tem-
poral loops. Our minimal view captures the essence of the governing view without
taking on extraneous commitments about the direction of time. Moreover, as a ver-
sion of primitivism, our view requires no reduction or analysis of laws into universals,
powers, or dispositions. Because of the minimalism and the primitivism, our view ac-
commodates several candidate fundamental laws, such as the principle of least action,
the Past Hypothesis, and the Einstein equation (which in its usual presentation is non-
dynamical). It is also compatible with more controversial examples of fundamental laws
in the Wheeler-Feynman theory of electrodynamics and retrocausal theories of quantum
mechanics. The flexibility of MinP is a virtue. From our viewpoint, it is an empirical
matter what forms the fundamental laws take on; one’s metaphysical theory of laws
should be open to accommodate the diverse kinds of laws entertained by physicists. It
may turn out that nature employs laws beyond those expressible in the form of dif-
ferential equations that admit (Cauchy) initial value formulations or can be given a
dynamic productive interpretation. The metaphysics of laws should not stand in the
way of scientific investigations. Our view encourages openness.
The idea that fundamental laws produce later states of the world from earlier ones is
related to causal fundamentalism, the idea that causation, or something like causation
(such as dynamic production), is fundamental in the world.3If causation is fundamental
2Throughout this paper, for simplicity, we assume that spacetime is fundamental. This assumption
is not essential to MinP. One can consider non-spatio-temporal worlds governed by minimal primitivist
laws. For those worlds, one can understand MinP as suggesting that laws constrain the physical
possibilities of the world, whatever non-spatio-temporal structure it may have. Indeed, if one regards
time itself as emergent, one may find it natural to understand governing in an atemporal and direction-
less sense.
3Causal fundamentalism does not imply that everyday causality is metaphysically fundamental. For
example, Maudlin’s notion of dynamic production is different from everyday causality (Maudlin, 2007,
ch.5). For some recent works on causal fundamentalism, physics, and everyday causality, see Blanchard
4
and asymmetric, then either it defines a direction of time or the direction of time itself
is metaphysically fundamental (such that it has no deeper explanation). On MinP,
temporally asymmetric relations such as causation and dynamic production are not
constitutive of how laws govern. We do not think causal fundamentalism is true. Neither
do we think there must be a fundamental direction of time. However, rejecting them is
not part of our view about laws; agnosticism about them is sufficient for our purposes.
On MinP, there can be but need not be fundamental causal relations, and there can
be but need not be a fundamental direction of time. Even if they existed, they would
not be essential to how laws explain. Hence, MinP carves out conceptual space for
non-Humeans such as ourselves who believe that laws govern but do not demand causal
fundamentalism, a fundamental direction of time, or dynamic production.
This paper has been written by a philosopher of physics and a mathematical physi-
cist. It is written for mathematicians, physicists, and philosophers who are interested in
the nature of physical laws. Since the topic is deeply philosophical, the first few sections
are intended as a non-technical introduction to some philosophical background.
We start with a review of four leading approaches to laws of nature: Humean reduc-
tionism, Platonic reductionism, Aristotelian reductionism, and Maudlinian primitivism.
Readers familiar with the philosophical literature on laws may skim the review. Next,
we state the two central theses of MinP and suggest how minimal primitivist laws can
explain natural phenomena without presupposing a fundamental direction of time. We
illustrate MinP by providing interpretations of several types of candidate physical laws:
dynamical laws, non-dynamical constraint laws, and probabilistic laws. Finally, we illus-
trate the key differences between MinP and the alternatives. Many working physicists,
mathematicians, and philosophers of science may appreciate our view precisely because
of its minimalism and primitivism. We also list some open questions for future work.
2 Some Existing Approaches
In this section, we survey some existing approaches to laws of nature.4We highlight the
key motivations that underpin such approaches, the explanatory principles they employ,
and the kinds of laws that they accommodate.
2.1 Humean Reductionism
A popular approach to laws in contemporary philosophical literature is that of Humean
Reductionism. On this view, laws do not govern but merely describe by summarizing
what actually happen in the world. Inspired by writings of Hume, Mill, and Ramsey,
(2016) and Weaver (2018).
4This survey is by no means exhaustive of the rich literature on laws. For example, against the view
that there are fundamental laws that are universally true, Cartwright (1994a) advocates a patchwork
view of laws where they are, at most, true ceteris paribus. Van Fraassen (1989) advocates a view where
there are no laws of nature. See Carroll (2020) for a more detailed survey.
5
David Lewis pioneered the contemporary versions of this view. On Humean Reduction-
ism, the fundamental ontology is that of a Humean mosaic, a concrete example of which
is a 4-dimensional spacetime occupied by particles and fields. At the fundamental level,
laws of nature do not exist and do not move particles and fields around. There are no
“necessary connections” forged by governing laws. Laws of nature are derivative of and
ontologically dependent on the actual Humean mosaic. The laws are the way they are
because of what the actual trajectories of particles and histories of fields are, not the
other way around, in contrast to the governing picture of laws. Laws are reducible to
the Humean mosaic.
Lewis (1986) calls this view Humean supervenience.5Following Ramsey, Lewis pro-
poses a “best-system” analysis of laws that shows how laws can be recovered from the
Humean mosaic. The basic idea is that laws are certain regularities of the Humean
mosaic. However, not any regularity is a law, since some are accidental.6Hence, one
needs to be selective about which regularities to count as laws. Lewis suggests we pick
those regularities in the best system of true sentences about the Humean mosaic. The
strategy is to consider various systems (collections) of true sentences about the Humean
mosaic and pick the system that strikes the best balance between various theoretical
virtues, such as simplicity and informativeness.
To get an intuitive grasp of this balancing act, consider an example. Let the Humean
mosaic (the fundamental ontology) be a Minkowski spacetime occupied by massive,
charged particles and an electromagnetic field. The locations and properties of those
particles and the strengths and directions of the field at different points in spacetime
is the matter distribution, which corresponds to the local matters of particular fact.
Suppose the matter distribution is a solution to Maxwell’s equations. Consider three
systems of true statements (characterized below using the axioms of the systems) about
this mosaic:
•System 1: {Spacetime point (x1, y1, z1, t1) has field strengths E1and B1with
directions ~v1and ~v10and is occupied by a particle of charge q1; spacetime point
(x2, y2, z2, t2) has field strengths E2and B2with directions ~v2and ~v20and is not
occupied by a charged particle; ......}
•System 2: {“Things exist.”}
•System 3: {Maxwell’s equations}
System 1 lists all the facts about spacetime points one by one. It has much informational
content but it is complicated. System 2 is just one sentence that says there are things
but does not tell us what they are and how they are distributed. It is extremely simple
5Whether contemporary Humean position in the metaphysics of science represents the historical
Hume has been debated. See for example Strawson (2015).
6For example, the regularity that all uranium spheres are less than one mile in diameter may be
a law or a consequence of some law, but the regularity that all gold spheres are less than one mile in
diameter is not a law or a consequence of a law.
6
but has little informational content. System 3 lists just four equations of Maxwellian
electrodynamics. It has less information about the world than System 1 but has much
more than System 2. It is more complicated than System 2 but much less so than
System 1. System 1 and System 2 are two extremes; they have one virtue too much at the
complete expense of the other one. In contrast, System 3 strikes a good balance between
simplicity and informativeness. System 3 is the best system of the mosaic. Therefore,
according to the best-system analysis, Maxwell’s equations are the fundamental laws of
this world.
To emphasize, on Humean Reductionism, laws are descriptive of the Humean mosaic.
Laws are not among the fundamental entities that push or pull things, enforce behaviors,
or produce the patterns. Laws are just winners of the competition among systematic
summaries of the mosaic. Beebee (2000) calls it the “non-governing conception of laws
of nature.” Laws are merely those generalizations which figure in the most economical
true axiomatization of all the particular matters of fact that obtain.
Despite the simplicity and appeal of Lewis’s analysis, there is an obstacle. The
theoretical virtue of simplicity is language-dependent. For example, suppose there is
a predicate Fthat applies to all and only the things in the actual spacetime. Then
consider the following system:
•System 4: {∀xF (x)}
This is informationally equivalent to System 1 and more informative than System 3, and
yet it is simpler than System 3. If we allow competing systems to use predicate F, there
will be a system (namely System 4) that is overall better than System 3. Given the best-
system analysis, the actual laws of the mosaic would not be Maxwell’s equations but
“∀xF (x).” To rule out such systems, Lewis places a restriction on language. Suitable
systems that enter into the competition can invoke predicates that refer to only natural
properties. For example, the predicate “having negative charge” refers to a natural
property, while the disjunctive predicate “having negative charge or being the Eiffel
Tower” refers to a less natural property. Some properties are perfectly natural, such as
those invoked in fundamental physics about mass, charge, spacetime location and so on.
It is those perfectly natural properties that the axioms in the best system must refer
to. The predicate Fapplies to all and only things in the actual world, which makes
up an “unnatural” set of entities. Fis not perfectly natural. Hence, System 4 is not
suitable. The requirement that the axioms of the best system refers only to perfectly
natural properties is an important element of Lewis’s Humeanism.
Over the years, Lewis and his followers have, in various ways, extended and modified
the best-system analysis of laws on Humean Reductionism. Let us summarize some of
the developments and call the updated view Reformed Humeanism about Laws:
Reformed Humeanism about Laws The fundamental laws are the axioms of the
best system that summarizes the mosaic and optimally balances simplicity, infor-
mativeness, fit, and degree of naturalness of the properties referred to. The mosaic
7
contains only local matters of particular facts, and the mosaic is the complete col-
lection of fundamental facts.
Reformed Humeanism can accommodate various kinds of laws of nature. Without going
into too much detail, we note the following features:
1. Chance. Although chance is not an element of the Humean mosaic, it can appear
in the best system. Humeans can introduce probability distributions as axioms of the
best system (Lewis, 1980). This works nicely for stochastic theories such as the Ghirardi-
Rimini-Weber (1986) theory of spontaneous localization. Humeans can evaluate the
contribution of the probability distributions by using a new theoretical virtue called
fit. A system is more fit than another just in case it assigns a higher (comparative)
probability than the other does the history of the universe. For certain mosaics, the
inclusion of probability in the best system can greatly improve the informational content
without sacrificing too much simplicity. Hence, fit can be seen as the probabilistic
extension of informativeness. Humeans can also allow what is called “deterministic
chance” (Loewer, 2001). Take a deterministic Newtonian theory of particle motion and
add to it the Past Hypothesis and the Statistical Postulate (Albert, 2000), which can
be represented as a uniform probabilistic distribution, conditionalized on a low-entropy
macrostate of the universe at t0. The Humean account of chance (both stochastic and
deterministic) is arguably one of the simplest and clearest to date.
2. Particular facts. Lewis (1983) maintains that “only the regularities of the system
are to count as laws” (p.367). However, there is no reason to limit the Humean account
to laws about general facts. Physicists have entertained candidate physical laws about
particular facts. For example, the Past Hypothesis is a candidate physical law about one
temporal boundary of the universe (“t0”). Such laws are uncommon, but conceptually
we do not see any obstacle. If a particular place or a particular time in the universe
is sufficiently significant, then it may be appropriate to have a physical law about the
particular place or time. Other examples of such laws include Tooley’s case of Smith’s
garden (1977) and the Aristotelian law about the center of the universe. Callender (2004)
suggests that a Humean analysis can do away with Lewis’s restriction to laws of general
facts. In fact, this flexibility seems a significant advantage Humean Reductionism has
over some other accounts of laws.
3. Flexibility with respect to perfect naturalness. For Lewis, perfect naturalness is a
property of properties. Perfectly natural properties pick out the same set of things as
Armstrong’s theory of sparse universals (more on that in §2.2). However, the chief mo-
tivation of Lewis’s use of perfect naturalness is to rule out systems that use “gruesome”
predicates. If that is the issue, then perhaps, as Hicks and Schaffer (2017) suggest, we
can simply require that “degree of naturalness” of the predicates involved be a factor
in the overall ranking of competing systems, and the best system should also optimally
balance degree of naturalness of the predicates together with the rest of the theoretical
virtues, such as simplicity, informativeness, and fit. The flexibility with respect to per-
fect naturalness also allows the best system to refer to non-fundamental properties such
as entropy, as may be necessary if the Past Hypothesis is a fundamental law.
8
4. Theoretical virtues. Humeans do not provide a full account of the theoretical
virtues. There are certain theoretical virtues scientists do and should consider signifi-
cant. With that in mind, perhaps Humeans can leave them open-ended. As such, there
is also some vagueness in how systems are compared and in some cases there may be
vagueness about which system is best.7
Reformed Humeanism is perhaps the most flexible view on the market for its accom-
modation with multiple kinds of candidate laws of physics. There is no problem with
giving lawhood status to non-dynamical facts such as the principle of least action, the
Einstein equation, or even a version of the Past Hypothesis that refers to a particular
time (t0) and a non-fundamental property (entropy). Because of its accommodation
of the Past Hypothesis and deterministic chance, Reformed Humeanism also accommo-
dates reductionism about the direction of time. The ingredients for such a reduction
can all be interpreted as axioms of the best system summarizing the mosaic. Hence,
Humeans can do away with a fundamental direction of time (Loewer, 2012).
2.2 Platonic Reductionism
With Humean Reductionism, nothing ultimately explains the patterns in the Humean
mosaic. For illustration, suppose F=ma is a fundamental law of our world. Humeans
maintain that “F=ma” expresses a fundamental law in virtue of its being an axiom in
the best system of the Humean mosaic. It merely summarizes what actually happens:
the trajectories of all massive particles are solutions to F=ma. Those with a governing
conception of laws may seek to find a deeper explanation. In virtue of what is every
massive particle in the world behaving according to the same formula? What, if any-
thing, enforces the pattern and makes sure nothing deviates from it? In other words,
what provides the necessity or oomph that is usually associated with laws?
Dretske (1977), Tooley (1977), and Armstrong (1983) propose an intriguing answer
based on a metaphysics of universals. The universals that they accept are in addition to
things in the Humean mosaic. They are “over and above” the Humean mosaic. In tradi-
tional metaphysics, universals are repeatable entities that explain the genuine similarity
of objects. Let us start with some mundane examples. Two cups are genuinely similar
in virtue of their sharing a universal Being a Cup. The universal is something they
both instantiate and something that explains their genuine similarity. A cup is different
from a horse because the latter instantiates a different universal Being a Horse. Now,
those universals are not fundamental, and they may be built from more fundamental
universals about physical properties. Dretske, Tooley, and Armstrong use universals
to provide explanations in science. For them, the paradigm examples are universals
that correspond to fundamental physical properties, such as mass and charge. On their
7Another issue concerning theoretical virtues is how we should use them to compare different sys-
tems. As noted earlier, simplicity is language relative. Cohen and Callender (2009) suggest that the
comparisons should be relativized to languages. Their relativized account (called the Better Best Sys-
tem Account) perhaps can be used to support Fodor (1974)’s vision of the autonomy of the special
sciences (e.g. biology, psychology, economics) from fundamental physics.
9
view, laws of nature hold because of a certain relation obtaining among such universals.
This theory of laws has connection to Plato’s theory of forms.8We thus call it Platonic
Reductionism.9
Consider again the world where F=ma holds for every massive particle. In such
a world, any particle with mass minstantiates the universal having mass m, any par-
ticle under total force Finstantiates the universal being under total force F, and any
particle with acceleration F/m instantiates the universal having acceleration F/m. The
universals are multiply instantiated and repeated, as there are many particles that share
the same universals. Those universals give unity to the particles that instantiate them.
The theory also postulates, as a fundamental fact, that the universal having mass m
and the universal being under total force Fnecessitate the universal having acceleration
F/m. Hence, if any particle instantiates having mass mand being under total force F,
then it has to instantiate having acceleration F/m. It follows that every particle has to
obey F=ma.10 This adds the necessity and the oomph that are missing in Humean
Reductionism.
With Platonic Reductionism, the regularity is explained by the metaphysical postu-
late of universals and the necessitation relation Nthat hold among universals. Following
Hildebrand (2013), we may summarize it as follows:
Necessitation For all universals Fand G,N(F, G) necessitates the regularity that all
Fs are Gs.
A few clarificatory remarks:
1. Universals. (1) The appeal to universals is indispensable in this theory of laws.
The theory is committed to a fundamental ontology of objects (particulars) and a fun-
damental ontology of universals. Hence, Platonic Reductionism is incompatible with
nominalism about universals. (2) Defenders such as Armstrong appeal to a sparse
theory of universals, where the fundamental universals correspond to the fundamental
properties we find in fundamental physics. The sparse universals correspond to the per-
fect natural properties that Lewis invokes in his account. Consider Lewis’s example of
the predicate Fthat corresponds to the property of all and only things in the actual
world. For Armstrong,“∀xF (x)” does not express a fundamental law because objects
with property Fare not genuinely similar, and Fis a property that does not correspond
to one of the fundamental, sparse universals.
2. Necessity. (1) The necessity relation among universals is put into the theory by
hand. It is a postulate that such a relation holds among universals and does necessitate
regularities. (It is also postulated that the relation among universals is itself a universal.)
8For an overview of Plato’s theory of forms, see Kraut (2017).
9In the literature it is sometimes called the DTA account of laws or the Universalist account of
laws. Calling it Platonic reductionism may be controversial. But see the discussion in (Carroll, 1994,
appendix A1).
10We note that this example about F=ma does not exactly fit in Armstrong’s schema of “All F’s
are G.” See (Armstrong, 1983, ch.7) for a strategy to accommodate “functional laws.”
10
To some commentators, it is unclear why the postulate is justified.11 In response, a
defender of Platonic Reductionism may take the necessity relation simply as a primitive
and accept it as unanalyzable (although this response would not satisfy the critics).
(2) The Nrelation, though called a necessity relation, holds contingently among
universals. Thus, if N(F, G) holds in the actual world, then in some possible world Fis
not connected to Gvia N.Nis only nomologically necessary but metaphysically con-
tingent. This has the consequence that laws of nature on Platonic Reductionism, while
nomologically necessary, are metaphysically contingent. This respects a widespread
judgment about the metaphysical contingency of laws. (In §2.3 we see that Aristotelian
Reductionism violates it.)
(3) Armstrong (1983) makes room for probabilistic laws as follows:
Irreducibly probabilistic laws are also relations between universals. These
relations give (are constituted by) a certain objective probability that indi-
vidual instantiations of the antecedent universal will necessitate instantiation
of the consequent universal. They give a probability of a necessitation in the
particular case...Deterministic laws are limiting cases of probabilistic laws
(probability 1). (p.172)
What is “a probability of a necessitation?” Conceptually, whether Fnecessitates G
seems like a matter that does not admit of degree. What does this probability mean,
and how does it relate to actual frequencies and why should it constrain our credences?
Even if one accepts the intelligibility of the necessitation relation, one may be unwilling
to accept the intelligibility of objective probability of a necessitation and one may be
puzzled by how the probability of a necessitation can explain the regularities. This
may be an instance of the general phenomenon that, it is difficult to give a unified and
intelligible non-Humean account of probabilistic laws and non-probabilistic laws. It is
much easier (if one sets aside the worry about the lack of governing) to do so on Humean
Reductionism: just put them all in the best system.
3. Explanation. For those who are antecedently sympathetic to a theory of uni-
versals, Platonic Reductionism may offer an attractive metaphysical explanation of the
patterns in nature. Its enlarged ontology provides extra explanatory resources. If two
particles both have mass m, then there literally is something they have in common—the
universal having mass m. That the two particles move in the same way can be partly ex-
plained by their genuine similarity to each other—their shared universals. The relation
that obtains among such universals, the necessitation relation N, exists over and above
the mosaic (the trajectories of particles in spacetime). Since the state of affairs that
11In a famous passage, Lewis (1983) raises this objection: “Whatever N may be, I cannot see how it
could be absolutely impossible to have N(F,G) and Fa without Ga...The mystery is somewhat hidden
by Armstrong’s terminology. He uses ‘necessitates’ as a name for the lawmaking universal N; and who
would be surprised to hear that if F ‘necessitates’ G and a has F, then a must have G? But I say that
N deserves the name of ‘necessitation’ only if, somehow, it really can enter into the requisite necessary
connections. It can’t enter into them just by bearing a name, any more than one can have mighty
biceps just by being called ‘Armstrong’ ” (p.366).
11
Nobtains among universals of mass, force, and acceleration does not supervene on the
objects, it can be said to govern the objects. In contrast, on Humean Reductionism, at
the fundamental level there is nothing that exists except the Humean mosaic. However,
the explanation on Platonic Reductionism may not be transparent to those who are not
sympathetic to a theory of universals.
Because Platonic Reductionism analyzes laws in terms of universals and relations
among them, it places certain restrictions on the forms of physical laws. If universals
are repeatable entities with multiple locations in space or time, Platonic Reductionism
does not seem compatible with laws that are about particular places or times. In our
view, that is a problem as it limits physical laws to general facts. For example, the
account seems incompatible with taking the Past Hypothesis to be a fundamental law
even though we have good arguments for doing so. We return to this point in §4.2.
On Platonic Reductionism, it is unclear how we should think about the direction of
time. Even though there is a strong connection between the necessitation relation Nand
causation, it does not seem that the main defenders build the direction of time into the
relation N. However, Tooley (1997) seems to think that the direction of time is reducible
to the direction of causation, and causal facts are fundamental in his metaphysics. If that
is the case, then causal fundamentalism is true and the direction of time is close to being
fundamental. Perhaps that is an optional add-on to his theory of laws. Nevertheless,
if Platonic Reductionism does not have room for treating the Past Hypothesis as a
fundamental law, it may need to invoke a fundamental direction of time for worlds
like ours. Perhaps Platonic Reductionism is best paired with a primitivism about the
direction of time.
2.3 Aristotelian Reductionism
The view about laws to which we now turn is most commonly associated with contem-
porary defenders of dispositional essentialism. On this view, laws, even if they exist,
do not govern the world in any metaphysically robust sense. Laws do not push or pull
things around. Instead, the patterns we see are explained by the fundamental prop-
erties that objects instantiate. Those properties are the seats of metaphysical powers,
necessity, and oomph. Those properties make objects, in a certain sense, “active” (Ellis,
2001, p.1). Such properties are often called “dispositions,” and also sometimes called
“powers,” “capacities,” “potentialities,” and “potencies.”12 Importantly, they are differ-
ent from the universals in Platonic Reductionism or the natural properties in Humean
Reductionism, which may be viewed as “passive.” If there are any laws (and there is an
internal debate about this question among defenders of this fundamental dispositional
ontology), they derive from or originate in the fundamental dispositions of material
objects.
Roughly speaking, objects with dispositions have characteristic behaviors (also called
manifestation) in response to certain stimuli (Bird, 2007, p.3). For example, a glass has
12For an overview of the metaphysics of dispositions, see Choi and Fara (2021).
12
a disposition to shatter when struck; an ice cube has a disposition to melt when heated;
salt has a disposition to dissolve when put into water. On this view, fundamental
properties are similarly dispositional: negatively charged particles have a disposition to
attract positively charged particles; massive particles have a disposition to accelerate
in a way that is proportional to the total forces on them and inversely proportional
to their masses. Moreover, a dispositional essentialist holds that some properties have
dispositional essences, i.e. their essences can be characterized in dispositional terms.13
In contrast to Humean Reductionism and Platonic Reductionism, on this view the
fundamental ontology is no longer “passive” but is “active and reactive” (Ellis, 2001,
pp.1-2). We confess that we do not fully understand such locutions. Perhaps the idea is
that material objects move in virtue of the dispositions they possess and not in virtue
of something outside (such as a law) that governs them. Among those who endorse
a dispositionalist fundamental ontology, not everyone accepts that fundamental laws,
which are usually taken to be universally valid and always true, arise from dispositions.
For example, Cartwright (1983, 1994b) and Mumford (2004) deny the need for laws.
Nevertheless, the dispositional essentialists need not abandon laws. They can maintain
that laws supervene on or reduce to dispositions. Because of its Aristotelian roots
(Ellis, 2014), we call such a view Aristotelian Reductionism about laws.14 Bird (2007)
characterizes it as follows:
According to this view laws are not thrust upon properties, irrespective,
as it were, of what those properties are. Rather the laws spring from within
the properties themselves. The essential nature of a property is given by
its relations with other properties. It wouldn’t be that property unless it
engaged in those relations. Consequently those relations cannot fail to hold
(except by the absence of the properties altogether, if that is possible). The
laws of nature are thus metaphysically necessary. (p.2)
Aristotelian Reductionists maintain that (1) the metaphysical powers, necessity, and
oomph reside in the fundamental dispositions; (2) laws are metaphysically derivative of
the dispositions; (3) laws are metaphysically necessary.
How are laws derived from dispositions? Bird proposes that we can derive laws
from certain counterfactual conditionals associated with dispositional essences. A more
recent approach is that of Demarest (2017, 2021) and Kimpton-Nye (2017) that seek to
combine a dispositional fundamental ontology with a best-system-analysis of lawhood.
Here we focus on the approach of Demarest. She proposes that dispositions (she follows
Bird and calls them potencies) do metaphysical work. They produce their characteristic
behaviors, resulting in patterns in nature. Their characteristic behaviors, in different
possible worlds, can be summarized in simple and informative axiomatic systems, and
13Some, such as Bird (2007), go further and claim that all perfectly natural properties in Lewis
(1986)’s sense or all sparse universals in Armstrong (1983)’s sense have dispositional essences.
14Many defenders of this view suggest that even though it has roots in Aristotle, it is not committed
to many aspects of Aristotelianism.
13
the best one contains the true laws of nature. That is like Humean Reductionism except
that (1) Demerest’s fundamental ontology includes potencies and (2) the summary is not
of just the actual potencies but also merely possible ones. In this way, her proposal may
be an elaboration of Bird’s suggestion that we can derive laws from potencies, though
she does not rely on counterfactuals. In her most recent work (2021), she proposes the
following account:
Dynamic-Potency-BSA (DPBSA): The basic laws of nature at ware the axioms of
the simplest, most informative, true systematization of all w-potency-distributions,
where a w-potency-distribution is a possible distribution of potencies that is gen-
erated by a possible initial distribution of only potencies appearing in w. (p.9,
emphasis original)
In contrast to Humean Reductionism, here the patterns are ultimately explained by
the potencies. How do potencies explain? Demarest provides this answer:
I think the most promising solution is to appeal to production—dynamic,
metaphysical dependence. According to my view, the fundamental ground
includes spacetime and an initial arrangement of particles and potencies.
And the subsequent behavior of the particles (further potency instantiations
as well as trajectories through spacetime) is dynamically, metaphysically de-
pendent upon that base. Since the potency-BSA systematizes those trajecto-
ries, the laws of nature are not fundamental, and do not govern, but rather
depend upon the behavior of the particles and potencies. To summarize
what (metaphysically) explains what: on my view, the initial distribution of
particles and their potencies dynamically ground the subsequent behaviors
of particles and subsequent property instantiations. And, all of the possi-
ble initial distributions and evolutions determine the (metaphysically inert)
laws. (Demarest, 2017, pp.51-52)
The potencies at an earlier time explain how things move at a later time by dynami-
cally producing, determining, or generating the patterns. We note that Demarest’s view
seems committed to a fundamental direction of time. The account of dynamic explana-
tion presupposes a fundamental distinction between past and future, i.e. between the
initial and the subsequent states of the world. The initial arrangement of particles and
potencies metaphysically ground subsequent behaviors of particles. The commitment of
a fundamental direction of time does not seem optional on her view.
Moreover, the metaphysical framework of fundamental dispositions already seems
committed to a fundamental direction of time, independently of the issue of laws. For
example, it is natural to interpret the discussions by Ellis, Bird, Mumford as suggesting
that the manifestation of a disposition cannot be temporally prior to its stimulus, which
presupposes a fundamental direction of time.15 Therefore, although Aristotelian Reduc-
15In contrast, Vetter (2015) is open to a temporally symmetric metaphysics but assumes temporal
asymmetry in her account of dispositions (which she calls potentialities).
14
tionism does away with the governing conception of laws, the view seems committed to
a fundamental direction of time twice over.
2.4 Maudlinian Primitivism
In his book The Metaphysics Within Physics (2007), Maudlin develops and defends a
primitivist view about laws.16 As a primitivist, 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. He characterizes his primitivism as follows:
My analysis of laws is no analysis at all. Rather I suggest we accept
laws as fundamental entities in our ontology. Or, speaking at the conceptual
level, the notion of a law cannot be reduced to other more primitive notions.
(p.18)
As a motivation for adopting primitivism over reductionism (especially Humean Reduc-
tionism), he writes:
[Nothing] in scientific practice suggests that one ought to try to reduce
fundamental laws to anything else. Physicists simply postulate fundamental
laws, then try to figure out how to test their theories; they nowhere even
attempt to analyze those laws in terms of patterns of instantiation of physical
quantities. The practice of science, I suggest, takes fundamental laws of
nature as further unanalyzable primitives. As philosophers, I think we can
do no better than to follow this lead. (p.105)
Maudlin is also committed to primitivism about the direction of time: that the dis-
tinction between past and future is metaphysically fundamental and not reducible to
anything else. There is in effect a fundamental arrow or orientation at every space-
time point that points to the future. Maudlin combines the two commitments into a
metaphysical package:
Let’s call the idea that both the laws of physics (as laws of temporal
evolution) and the direction of time are ontological primitives Maudlin’s
Non-Humean Package. According to this package, the total state of the
universe is, in a certain sense, derivative: it is the product of the operation
of the laws on the initial state. (p.182)
16Carroll (1994) seems to endorse a version of primitivism about laws, though recently (Carroll, 2018)
he distances his view from that of Maudlin and suggests a non-Humean reductive analysis of laws.
15
There are several reasons that Maudlin is committed to both. They become clear as we
consider how laws explain 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. Production is closely related to causation, and just like (paradigm cases
of) causation it is time asymmetric. Future states are 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. Similar productive explanations
can be given in more complicated cases.
The operation of the primitive laws depends on the primitive direction of time.
Primitive laws act on past states to produce future states. Maudlin thinks that his
package yields an attractive picture by being closer to our initial conception of the
world:
The universe started out in some particular initial state. The laws of
temporal evolution operate, whether deterministically or stochastically, from
that initial state to generate or produce later states. (p.174)
This sort of explanation takes the term initial 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). (p.176)
The non-Humean package [described above] is, I think, much closer to
the intuitive picture of the world that we begin our investigations with.
Certainly, the fundamental asymmetry in the passage of time is inherent in
our basic initial conception of the world, and the fundamental status of the
laws of physics is, I think, implicit in physical practice. Both of the strands
of our initial picture of the world weave together in the notion of a productive
explanation, or account, of the physical universe itself. The universe, as well
as all the smaller parts of it, is made: it is an ongoing enterprise, generated
from a beginning and guided towards its future by physical law. (p.182)
This intuitive picture of the world require certain restrictions on the form of fundamental
laws. They have to be, what Maudlin calls, fundamental laws of temporal evolution
(FLOTEs). Examples include Newton’s F=ma, Schr¨odinger’s equation, and Dirac’s
equation on our first list in §1 but exclude most examples on our second list.
Let us summarize Maudlin’s metaphysical package as follows:
16
Maudlinian Primitivism Fundamental laws are certain primitive facts in the world.
Only dynamical laws (in particular, laws of temporal evolution) can be fundamen-
tal 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 in-
volve 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 im-
plying 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 propen-
sity is as intelligible. (Recall the earlier point about “probabilistic necessitation” in
Platonic Reductionism.) This may be another instance of the general phenomenon that
objective probability (or chance) is conceptually murkier on non-Humean metaphysics
than on Humean metaphysics.
At first glance, Maudlin’s view is intuitive. It is attractive to those who accept a
fundamental direction of time. According to Maudlinian Primitivism, there is a funda-
mental distinction between past and future that is not reducible to entropic arrow of
time, the distribution of matter in the universe, or special boundary conditions. This
distinction picks out an initial state of the universe, in the literal sense of “initial,” that
is earlier than any other states. Nonetheless, Maudlin is committed to a “block universe”
picture of time on which all times (past, present, and future) are equally real. Maudlin
rejects presentism, the moving spotlight view, the growing block view, and the shrinking
block view. So it is not the same as those pictures where spacetime is “dynamic” or the
present moment is metaphysically privileged.
However, the view may not be as intuitive as it first seems. First, in relativistic
spacetimes, there is no absolute simultaneity or a physically privileged notion of “now.”
The fundamental distinction between past and future needs to be understood without
referring to a preferred foliation and should not involve an objective present. For any
spacetime event, it requires an objective fact about which light cone points to the past
and which one points to the future. Second, we may wonder how dynamic production
extends to spacetimes with no “first” moment of time, such as those with an “initial”
singularity or without temporal boundaries. If there is no initial state, perhaps the
oomph of dynamic production, though having no beginning, always comes from earlier
states. Third, the very notion of dynamic production is a bit unclear. We return to this
issue in §3.2.
If one believes in Maudlinian Primitivism and its associated principle of (dynamic)
productive explanation, then one needs to place restrictions on the form of laws. They
can only take the form of FLOTEs. We return to this issue in §4.4.
17
3 Minimal Primitivism (MinP)
Having surveyed four existing approaches to laws, we propose our own view, which we
call Minimal Primitivism (MinP).
3.1 The View
According to MinP, fundamental laws are ontological primitives that are metaphysically
fundamental.17 They do not require anything else to exist. They are not analyzable
into (relations among) universals, powers, or dispositions. They are not reducible to
(or supervenient on) the Humean mosaic. Rather, if the Humean mosaic describes
spacetime and its contents, then the mosaic is governed by the laws, in a metaphysically
robust sense. For governing laws, we do not require them to dynamically produce or
generate later states of the universe from earlier states, and neither do we presuppose
a fundamental direction of time that distinguishes between past states and future ones.
On MinP, laws govern by constraining the physical possibilities (often called nomological
possibilities in the metaphysics literature). This places no in-principle demands on the
form of fundamental laws. To summarize, the first part of our view is a metaphysical
thesis:
Minimal Primitivism Fundamental laws of nature are certain primitive facts about
the world. There is no restriction on the form of the fundamental laws. They
govern the behavior of material objects by constraining the physical possibilities.
Even though there is no metaphysical restriction on the form of fundamental laws,
it is rational to expect them to have certain nice features, such as simplicity and infor-
mativeness. On Humean Reductionism, those features are metaphysically constitutive
of laws, but on our view they are merely epistemic guides for discovering and evaluating
the laws. At the end of the day, they are defeasible guides, and we can be wrong about
the fundamental laws even if we are fully rational in scientific investigations. The second
part of our view is an epistemic thesis:
Epistemic Guides Even though theoretical virtues such as simplicity, informativeness,
fit, and degree of naturalness are not metaphysically constitutive of fundamental
laws, they are good epistemic guides for discovering and evaluating them.
Let us offer some clarifications:
(1) Primitive facts. Fundamental laws of nature are certain primitive facts about
the world, in the sense that they are not metaphysically dependent on, reducible to, or
17Bhogal (2017) proposes a “minimal anti-Humeanism” on which laws are ungrounded (true) universal
generalizations. It may be a version of primitivism, but it is not as minimal as ours. For one thing, on
Bhogal’s view laws cannot be singular facts about particular times or places. However, Bhogal (p.447,
fn.1) seems open to relax the requirement that laws have to be universal generalizations. It would be
interesting to see how to extend Bhogal’s view to do so.
18
analyzable in terms of anything else. Depending on one’s metaphysical attitude towards
mathematics and logic, there might be mathematical and logical facts that are also
primitive in that sense. For example, arithmetical facts such as 2 + 3 = 5 and the
logical law of excluded middle may also be primitive facts that constrain the physical
possibilities, since every physical possibility must conform to them. However, we do not
think that fundamental laws of nature are purely mathematical or logical. Hence, we
stipulate that fundamental laws of nature are not such kinds of primitive facts.
(2) The governing relation. We suggest that laws govern by constraining the phys-
ical possibilities. More precisely, laws govern by limiting the physical possibilities and
constraining the actual world (history) to be one of them. In other words, the actual
world (history) is constrained to be compatible with the laws. To use an earlier ex-
ample, F=ma governs by constraining the physical possibilities to exactly those that
are compatible with F=ma. If F=ma is a governing law of the actual world, then
the actual world (history) is a possibility compatible with F=ma. This notion of
constraint does not require a fundamental distinction between past and future, or one
between earlier states and later states. What the laws constrain is the entire spacetime
and its contents. In some cases, the constraint can be expressed in terms of differential
equations that can be interpreted as determining future states from past ones. But not
all constraints need be like that. We discuss this point in §3.2.18
For a concrete example, consider the Hamiltonian equations of motion for Npoint
particles with Newtonian masses (m1, ..., mN) moving in a 3-dimensional Euclidean
space, whose positions and momenta are (q1, ..., qN;p1, ..., pN):
dqi(t)
dt =∂H
∂pi
,dpi(t)
dt =−∂H
∂qi
(1)
18For those metaphysically inclined, here are some formal details. Consider w, the complete history
of a possible world describable in terms of matter in spacetime. Let Ωwbe the non-empty set of worlds
that are physically possible (from the perspective of w). It is a priori that w∈Ωw. Consider fact L,
which may be Newton’s equation of motion with Newtonian gravitation. Let ΩLbe the set of models
generated by L. Now, suppose Lgoverns w. Then the following is true:
Equivalence ΩL= Ωw
Equivalence makes precise the idea that on MinP governing laws limit the physical possibilities. Since
w∈Ωw, it follows that:
Constraint w∈ΩL
Constraint makes precise the idea that, on MinP, laws constrain the actual world. For MinP, we
postulate that the above notions and derivations make sense. A few epistemological remarks: the
fact that ΩL= Ωwis knowable a posteriori ; consequently, the fact that w∈ΩLis also knowable a
posteriori. A careful reader might raise a consistency worry here: what if a single world (history) wis
compatible with two different laws Land L0with non-empty overlap in their solution spaces, such that
w∈ΩL∩ΩL0? The worry is handled by the earlier postulates. Having w∈ΩL0is not sufficient for
L0to be the governing law or for ΩL0to be the set of physical possibilities. MinP assumes that, from
the perspective each world, there is a single set of physical possibilities. Hence, for w, at most one of
ΩLand ΩL0is equivalent to Ωw. Moreover, since Ωwis non-empty, the laws that govern wmust be
consistent with each other.
19
where H=H(q1, ..., qN;p1, ..., pN) is specified in accord with Newtonian gravitation:
H=
N
X
i
pi2
2mi
+X
1≤j<k≤N
Gmjmk
|qj−qk|(2)
Suppose equations (1) and (2) are the fundamental laws that govern our world α. Let ΩH
denote the set of solutions to (1) and (2). Let Ωαdenote the set of physical possibilities
of the actual world. Saying that (1) and (2) govern our world entails that ΩH= Ωα.
Since α∈Ωα, we have α∈ΩH, which means our world is constrained by (1) and (2). In
this example, the dynamical equations are time-reversible. For every solution in ΩH, its
time reversal under t→ −tand p→ −pis also a solution in ΩH. Since the concept of
governing in MinP does not presuppose a fundamental direction of time, two solutions
that are time-reversal of each other can be identified as the same physical possibility.19
We should not think of a law as necessarily equivalent to the set of possibilities
it generates. The two can be different. For example, there are many principles and
equations that can give rise to the same set of possibilities denoted by ΩH. But we
expect laws to be simple. One way to pick out the set ΩHis by giving a complete (and
infinitely) long list of possible histories contained in ΩH. Another is by writing down
simple equations, such as (1) and (2), which express simple laws. Hence, the equivalence
of physical laws is not just the equivalence of their classes of models. For two laws to
be equivalent, it will require something more.20
Humeans might object that our notion of governing is entirely mysterious (Beebee,
2000). The notion of governing seems derived from the notion of government and the
notion of being governed. But laws of nature are obviously not imposed by human
(or divine) agents. So isn’t it mysterious that laws can govern? To that we reply
that a better analogy for governing laws is not to human government, but to laws of
mathematics and logic. Arithmetical truths such as 2 + 3 = 5 and logical truths such
as the law of excluded middle can also be said to constrain our world. That is, the
actual world cannot be a world that violates those mathematical or logical truths. In
fact, every possible world needs to respect those truths. In a similar way, laws of physics
constrain our world. The actual world cannot be a world that violates the physical laws,
and every physically possible world needs to respect those laws. Those modal claims
reflect physical laws and mathematical laws. We can also make sense of the difference
in scope between those laws. Mathematical laws are more general than physical laws,
in the sense that the former are compatible with “more models” than the latter. In any
case, mathematical laws and logical laws can also be said to govern the universe in the
sense of imposing formal constraints. They generate a class of models and constrain the
19If one prefers the representation where the set of physical possibilities contains each possibility
exactly once, one can derive a quotient set Ω∗
αfrom Ωαwith the equivalence relation given by the
time-reversal map.
20It is an interesting question, on MinP, what more is required and how to understand the equivalence
of physical laws. We do not provide such an account as it is orthogonal to our main concerns in the
paper. For a survey of the related topic of theoretical equivalence, see Weatherall (2019a,b).
20
actual world to be one among them. There is also a difference in epistemic access. In
some sense, we discover mathematical and logical laws a priori, without the need for
experiments or observations, but we discover physical laws a posteriori, empirically.
We do not claim that the analogy with mathematical and logical laws completely
eliminate the mystery of how physical laws govern. However, we think it dispels the
objection as previously stated, in terms of how something can govern the world without
being imposed by an agent. If there is more to the mystery objection, it needs to be
stated differently. On MinP, laws govern by constraining, and constraining is what they
do. This provides the oomph behind scientific explanations. (We return to this shortly.)
However, such an oomph is very minimal. It does not require dynamic production, and
it does not require an extra process supplied by a mechanism or an agent.
(3) Epistemic access. On MinP, even though the Humean criteria for the best system
are not metaphysically constitutive for lawhood, they are nonetheless excellent epistemic
guides for discovering and evaluating them. Lewis is right that in scientific practice, in
the context of discovery, we do aim to balance simplicity and informativeness (among
other things).
Regarding Epistemic Guides, one might ask in virtue of what those theoretical virtues
are good guides for finding and evaluating laws. This is a subtle issue, and we are not
prepared to give a complete answer here. Unlike Humeans, we cannot appeal to a reduc-
tive analysis of laws. We can offer an empirical justification: the scientific methodology
works. In so far as those theoretical virtues are central to scientific methodology, they
are good guides for discovering and evaluating laws, and we expect them to continue to
work. Can they fail to deliver us the true laws? That is a possibility. However, if that
turns out to be the case and if the true fundamental laws are complicated and messy,
scientists would not be inclined to call them laws. We return to this point in §3.2.
(4) We now address several other questions that arise concerning MinP.
•According to MinP, can laws change with time? In particular, can fundamental
laws be time dependent in such a way that different cosmic epochs are governed
by different laws? In principle, we are open to that possibility. If there is scientific
motivation to develop theories in which laws take on different forms at different
times (or in different epochs), then that is sufficient reason to consider a set of
laws that govern different times, or a single law that varies in form with time. As
a toy example, if we have empirical or theoretical reasons to think that the laws of
motion are different on the two sides of the Big Crunch, say Newtonian mechanics
and Bohmian mechanics, then different sides of the Crunch can be governed by
different laws, or by a single law with a temporal variation.
•According to MinP, can fundamental laws refer to non-fundamental properties,
such as entropy? Most of the fundamental laws we discover refer only to funda-
mental properties. But it is reasonable to consider candidate fundamental laws
that refer to non-fundamental properties. Our principle of Epistemic Guides al-
lows for this, as long as the non-fundamental properties are not too unnatural (all
21
things considered). In the case of the Past Hypothesis, for example, we may sacri-
fice fundamentality of the property involved but gain a lot of informativeness and
simplicity if we invoke the property of entropy. The version of the Past Hypothesis
that refers to entropy can still govern by constraining the physical possibilities.
(Another strategy is to revise our definition of fundamental property such that any
property mentioned by a fundamental law is regarded as fundamental, although it
may be analyzable in terms of other fundamental properties. However, this may
present a problem for certain views of fundamentality.)
•According to MinP, how are fundamental laws distinguished from non-fundamental
laws? We prefer a reductionist picture where non-fundamental laws, when properly
understood, are reducible to fundamental laws. We can distinguish them in terms
of derivability: non-fundamental laws can be (non-trivially) derived from funda-
mental laws. For example, the ideal gas law is less fundamental than Newton’s
laws of motion, in the sense that the ideal gas law can be derived from them in
suitable regime. However, derivability may not be sufficient for non-fundamental
lawhood, as other factors, such as counterfactual and explanatory robustness, may
also be relevant.
•How does MinP compare to the other views in §2? We discuss this in §4.
3.2 MinP and Explanation
On MinP, laws explain, but not by accounting for the dynamic production of successive
states of the universe from earlier ones. They explain by expressing a hidden simplicity,
given by compelling constraints that lie beneath complex phenomena. A fundamental
direction of time is not required for our notion of explanation. (This type of explanation,
sometimes called “constraint explanation,” has been explored in the causation literature
by Ben-Menahem (2018) and non-causal explanation literature by Lange (2016). Their
accounts, with suitable modifications, may apply here. We leave a comparative analysis
to future work.)
In a world governed by Newtonian mechanics, particles travel along often complicated
trajectories because that is implied by the simple fundamental law F=ma. Laws
explain only when they can be expressed by simple principles or differential equations.
It is often the case that the complicated patterns we see in spacetime can be derived
from simple rules that we call laws.
Fundamental laws need not be time-directed or time-dependent. They may govern
purely spatial distribution of matter. For example, Gauss’s law
∇ · E=ρ
0
(3)
in classical electrodynamics—one of Maxwell’s equations—governs the distribution of
electric charges and the electric field in space.
22
Figure 1: The Mandelbrot set with continuously colored environment. Picture
created by Wolfgang Beyer with the program Ultra Fractal 3, CC BY-SA 3.0,
https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons
Often the explanation that laws provide involves deriving striking, novel, and unex-
pected patterns from simple laws. The relative contrast between the simplicity of the
law and the complexity and richness of the patterns may indicate that the law is the
correct explanation of the patterns.
For a toy example, consider the Mandelbrot set in the complex plane, produced by
the simple rule that a complex number cis in the set just in case the function
fc(z) = z2+c(4)
does not diverge when iterated starting from z= 0. (For example, c=−1 is in
this set but c= 1 is not, since the sequence (0,−1,0,−1,0,−1, ...) is bounded but
(0,1,2,5,26,677,458330, ...) is not. For a nice description and visualization, see (Pen-
rose, 1989, ch.4).) Here, a relatively simple rule yields a surprisingly intricate and rich
pattern in the complex plane, a striking example of what is called the fractal structure.
Now regard the Mandelbrot set as corresponding to the distribution of matter over (a
two-dimensional) spacetime, the fundamental law for the world might be the rule just
described. What is relevant here is that given just the pattern we may not expect it
to be generated by any simple rule. It would be a profound discovery in that world to
learn that its complicated structure is generated by the aforementioned rule based on
the very simple function fc(z) = z2+c. On our conception, it would be permissible to
claim that the simple rule expresses the fundamental law, even though it is not a law
for dynamic production.
The previous examples illustrate some features of explanation on MinP:
1. Laws explain by constraining the physical possibilities in an illuminating manner.
23
2. Nomic explanations (explanations given by laws) need not be dynamic explana-
tions; indeed, they need not involve time at all.
3. Explanation by striking constraint can be especially illuminating when an intricate
and rich pattern can be derived from a simple rule that expresses the constraint
imposed by a law.
On our view, more generally, there are two ingredients of a successful scientific ex-
planation: a metaphysical element and an epistemic one. It must refer to the objective
structure in the world, but it also must relate to our mind, remove puzzlement, and pro-
vide an understanding of nature. We suggest that a successful scientific explanation that
fundamental laws provides should contain two aspects: (1) metaphysical fundamentality
and (2) simplicity.
The first aspect concerns the metaphysical status of fundamental laws: they should
not be mere summaries of, or supervenient on, what actually happens; moreover, their
truths should not depend on our actual practice or beliefs. This aspect is the pre-
condition for having a non-Humean account of scientific explanations. On MinP, the
precondition is fulfilled by postulating fundamental laws as primitive (metaphysically
fundamental) facts that constrain the world. The constraint provides the needed oomph
behind scientific explanations. Here lies the main difference between MinP and Humean
Reductionism. (We return to this point in §4.1.)
The second aspect concerns how fundamental laws relate to us. Constraints, in
and of themselves, do not always provide satisfying explanations. Many constraints are
complicated and thus insufficient for understanding nature. What we look for in the
final theory of physics is not just any constraint but simple, compelling ones that ground
observed complexities of an often bewildering variety. The explanation they provide
corresponds to insight or realization that leads us to say, “Aha! Now I understand.”
Often, simplicity is related to elegance or beauty. As Penrose reminds us:
Elegance and simplicity are certainly things that go very much together.
But nevertheless it cannot be quite the whole story. I think perhaps one
should say it has to do with unexpected simplicity, where one imagines that
things are going to be complicated but suddenly they turn out to be very
much simpler than expected. It is not unnatural that this should be pleasing
to the mind. (Penrose, 1974, p.268)
The sense of unexpected simplicity is illustrated in the toy example of the Mandelbrot
set as well as the physical laws discovered by Newton, Schr¨odinger, and Einstein.
Moreover, the second aspect of scientific explanation illuminates our principle of
Epistemic Guides. It is obvious that fundamental laws should be empirically adequate
and consistent with all phenomena. But why should we expect them to be simple?
On our view, it can partly be answered by thinking about the nature of scientific ex-
planations. If successful scientific explanations require simple laws, then laws should
be simple to perform the explanatory role. One might press further and ask why laws
24
should perform such roles and why scientific explanations can be successful. But they
can be raised for any account of laws. We note that it is a difficult issue, one that may
be related to Hume’s problem of induction (Henderson, 2020).
3.3 Examples and Further Clarifications
To further clarify MinP, we discuss some examples of dynamical laws and non-dynamical
constraint laws. On our view, there is no difficulty accommodating them, as they can
be understood as laws that constrain physical possibilities. We also consider laws that
involve intrinsic randomness, as they are less easy to accommodate on our view. We
offer five interpretive options for further consideration.
3.3.1 Dynamical Laws
We take a dynamical law to be any law that determines how objects move or things
change. (Here we focus on non-probabilistic laws and leave probabilistic ones to §3.2.3.)
Thus, our notion of dynamical laws is wider than Maudlin’s notion of FLOTEs.
Hamilton’s Equations. Consider classical mechanics for the Nparticles, described
by Hamilton’s equations of motion (1) with a Hamiltonian specified in (2). Hamilton’s
equations are differential equations of a particular type: they admit initial value formu-
lations. An intuitive way of thinking about dynamical laws is to understand them as
evolving the initial state of the world into later ones. However, this view is not entirely
natural for such a system. The view requires momenta to be part of the intrinsic state
of the world at a time; but it seems more natural to regard them as aspects of extended
trajectories, spanning continuous intervals of time. Regarding governing as dynamic
production leads to awkward questions about instantaneous states and whether they
include velocities and momenta.
The situation becomes even more complicated with relativistic spacetimes having no
preferred foliation of equal-time hypersurfaces. If there is no objective fact about which
events are simultaneous, there is no unique prior Cauchy surface that is responsible
for the production of any later state. This seems to detract from the intuitive idea of
dynamic production as a relation with an objective input, making it less natural in a
relativistic setting.21
Instead of demanding that laws govern by producing subsequent states from earlier
ones, we can regard laws as constraining the physical possibilities of spacetime and its
contents. There is no difficulty accommodating the above example or any other type of
dynamical laws. A dynamical law specifies a set of histories of the system and need not
be interpreted as presupposing a fundamental direction of time. The histories the laws
allow can often be understood as direction-less histories, descriptions of which events
are temporally between which other events.
21Christopher Dorst raised a similar point in personal communication.
25
A dynamical law such as (1) governs the actual world by constraining its history to
be one allowed by (1). And MinP requires no privileged splitting of spacetime into space
and time, as the physical possibilities can be stated in a completely coordinate-free way
in terms of the contents of the 4-dimensional spacetime.
Principles of Least Action. Besides dynamical laws of Hamiltonian form, other kinds
of equations and principles are often employed even for Hamiltonian systems. Consider,
for example, Hamilton’s principle of least action: this requires that for a system of N
particles with Cartesian coordinates q= (q1, q2, ..., qN):
δS = 0 (5)
where S=Rt2
t1L(q(t),˙q, t)dt, with ˙q=q(t)/dt,δthe first-order variation of Scor-
responding to small variation in q(t), and Lthe Lagrangian which is taken to be the
kinetic energy minus the potential energy of the system of Nparticles. While mathemat-
ically equivalent to Hamilton’s equations, the principle of least action feels very different
from a law expressing dynamic production. For those who take dynamic production to
be constitutive of governing, the principle of least action cannot be the fundamental
governing law. They would presumably need to insist that the universe is genuinely
governed by some law of a form such as (1), with the principle of least action arising
as a theorem. For us, we have no problem regarding the principle of least action as a
candidate fundamental law, with no need for it to be derived from anything else. For a
universe to obey the principle, its history must be one compatible with (5). That is the
sense in which it would govern our universe.
Wheeler-Feynman Electrodynamics. Physicists have also considered dynamical equa-
tions that cannot be reformulated in Hamiltonian form. On MinP, there is no prohibition
against laws expressed by such equations. For example, Wheeler and Feynman (1945,
1949) considered equations of motion for charged particles that involve both retarded
fields (Fret) and advanced ones (Fadv). On their theory, the trajectory of a charged
particle depends on charge distributions in the “past” (corresponding to Fret ) as well
as those in the “future” (corresponding to Fadv ). Since the total field acting on particle
jis Ftot =Pk6=j
1
2((k)Fret +(k)Fadv ), the equation of motion for particle jof mass mj,
charge ej, and spacetime location qjis
mj¨qµ
j=ejX
k6=j
1
2((k)Fµν
ret +(k)Fµν
adv) ˙qj,ν (6)
with the dot the time derivative with respect to proper time, (k)Fr et the retarded field
contributed by particle k, and (k)Fadv the advanced one. (For more details, see Deckert
(2010) and Lazarovici (2018).) It is unclear how to understand the above equation in
terms of dynamic production. In contrast, it is clear on MinP: the fundamental law
corresponding to such equations can be regarded as a constraint on all trajectories of
charged particles in spacetime.
Retrocausal Quantum Mechanics. There have been proposed reformulations of quan-
tum mechanics that involve two independent wave functions of the universe: Ψi(t)
26
evolving from the “past” and Ψf(t) evolving from the “future.” Some such proposals,
motivated by a desire to evade no-go theorems or preserve time-symmetry, implement
retrocausality or backward-in-time causal influences (Friederich and Evans, 2019). Con-
sider Sutherland (2008)’s causally symmetric Bohm model, which specifies an equation
of motion governing Nparticles moving in a 3-dimensional space under the influence of
both Ψi(t) and Ψf(t):
dQj(t)
dt =Re(~
2imjaΨ∗
f∇jΨi)
Re(1
aΨ∗
fΨi)(Q(t), t) (7)
with Q(t)=(Q1(t), ..., QN(t)) ∈R3Nthe configuration of the Nparticles at time
t,mjthe mass of particle j, and athe normalization factor. It is unclear whether
Sutherland’s theory is viable; it also has many strange consequences. Nevertheless,
MinP is compatible with regarding the above equation, understood as a constraint of
particle trajectories in spacetime, as expressing a fundamental law (even though we have
other reasons to not endorse the theory).
Similarly, MinP is compatible with Goldstein and Tumulka (2003)’s model of two
opposite directions of time. To reconcile relativity (Lorentz invariance) and non-locality,
the model contains a macroscopic boundary-condition law on the “future” end of time
and a microscopic dynamical equation acting on “past” data. Together, they constrain
the particle trajectories in spacetime.
The Einstein Equation. In general relativity, the fundamental equation is the Ein-
stein equation:
Rab −1
2Rgab =k0Tab + Λgab (8)
where Rab is the Ricci tensor, Ris the Ricci scalar, gab is the metric tensor, Tab is
the stress-energy tensor, Λ is the cosmological constant, k0= 8πG/c4with GNewton’s
gravitational constant and cthe speed of light. Roughly speaking, the Einstein equation
is a constraint on the relation between the geometry of spacetime and the distribution
of matter (matter-energy) in spacetime. On MinP, we have no problem taking the
equation itself as expressing a fundamental law of nature, one that constrains the actual
spacetime and its contents. If equation (8) governs our world in the minimal primitivist
sense, then that is a fundamental fact that does not supervene on or reduce to the actual
spacetime and its contents.
There are ways of converting equation (8) into FLOTEs that are suitable for a
dynamic productive interpretation. (A famous example is the ADM formalism (Arnowitt
et al., 1962).) However, they often discard certain solutions (such as spacetimes that are
not globally hyperbolic). For non-Humeans who take dynamic production as constitutive
for governing or explanations, those reformulations will be necessary. For them, the true
laws of spacetime geometry should presumably be expressed by equations that describe
the evolution of a 3-geometry in time. In contrast, on MinP there is no metaphysical
problem for taking the original Einstein equation as a fundamental law. The Einstein
27
equation is simple and elegant and is generally regarded as the fundamental law in
general relativity. We prefer not to discard or modify it on metaphysical grounds.22
The Einstein equation allows some peculiar solutions. A particularly striking class
of examples are spacetimes with closed timelike curves (CTCs). For MinP, there would
seem to be no fundamental reason why such a possibility should be precluded. But
the possibility of CTCs is precluded by Maudlinian Primitivism, since they may lead
to an event that dynamically produces itself (Maudlin, 2007, p.175). And it is hard
to see how Demarest’s version of Aristotelian Reductionism can allow them. Humean
Reductionism should be compatible with CTCs, just as MinP is. It is unclear whether
they are compatible with Platonic Reductionism.
3.3.2 Non-Dynamical Constraint Laws
The examples mentioned earlier are explicitly related to time. There are also important
equations and principles that are not. For example, some purely spatial constraints on
the universe may be thought of as physical laws. We call them non-dynamical constraint
laws. The minimal notion of governing easily applies to them. In §3.1 we considered
two examples of such laws—(3) and (4). Here we consider a few more.
The Past Hypothesis. In the foundations of statistical mechanics and thermodynam-
ics, followers of Boltzmann have proposed a candidate fundamental law of physics that
Albert (2010) calls the Past Hypothesis (PH). It is a special boundary condition that
is postulated to explain the emergent asymmetries of time in our universe, such as the
Second Law of Thermodynamics. Here is one way to state it:
PH At one temporal boundary of the universe, the universe is in a low-entropy state.
This statement of PH is vague. We may be able to make it more precise by specifying the
low-entropy state in terms of the thermoydnamic properties of the universe or in terms
of some geometrical properties (Penrose, 1979). Penrose’s version in general relativity
renders it as follows: the Weyl curvature Cabcd vanishes at any “initial” singularity. Let
us use ΩP H to denote the set of worlds compatible with PH. If it is plausible that PH
is a candidate fundamental law (Chen, 2020), then the metaphysical account of laws
should make room for a boundary condition to be a fundamental law. On MinP, such
an account is no problem. Together, PH and dynamical laws can govern the actual world
by constraining it to be one among the histories compatible with all of them. They deem
that the actual world (history) is a member of the intersection ΩPH ∩ΩDL, where the
latter denotes the set of histories compatible with the dynamical laws. However, PH is
not a governing law in the sense of dynamic production. So its fundamental lawhood is
22Making a similar point, Callender (2017, p.139) writes: “[The] ten vacuum Einstein field equations
separate into six “evolution” equations Gij = 0 and four “constraint equations,” G00 = 0 and G0i= 0,
with i= 1,2,3.The latter impose nomic conditions across a spacelike slice. To decree that four
of the ten equations that constitute Einstein’s field equations are not nomic without good reason is
unacceptable.”
28
incompatible with Maudlinian Primitivism. And it is not a natural fit for Aristotelian
Reductionism (but see Demarest (2019) for a recent proposal for how they might fit).
It is possible that the vagueness in PH cannot be eliminated. If PH (or something
like it) turns out to be a fundamental law, then there can be vagueness in the funda-
mental laws. See Chen (2022) for a discussion of this possibility, called “fundamental
nomic vagueness.” On MinP, fundamental nomic vagueness implies that there are vague
fundamental facts. Whether fundamental nomic vagueness exists in our world is a subtle
question. (It is related to the issue of whether the ontic quantum state of the universe
is pure or impure (Chen, 2022, sect.4).)
Conservation Laws and Symmetry Principles. According to a traditional perspec-
tive, symmetries such as those of rotation, spatial translation, and time translation are
properties of the specific equations of motion. By Noether’s theorem, those symmetries
yield various conservation laws as theorems rather than postulates that need to be put in
by hand. On that perspective, symmetries and conservation laws can be regarded as on-
tologically derivative of the fundamental laws, and are compatible with all metaphysical
views on laws.
According to a more recent perspective, symmetries are fundamental. See for ex-
ample: Wigner (1964, 1985) and Weinberg (1992). Lange (2009) calls them metalaws.
For example, Wigner describes symmetries as “laws which the laws of nature have to
obey” (Wigner, 1985, p.700) and suggest that “there is a great similarity between the
relation of the laws of nature to the events on one hand, and the relation of symmetry
principles to the laws of nature on the other” (Wigner, 1964, p.957). Here we do not
take a firm stance on this perspective. Nevertheless, we note that it is compatible with
MinP. If there is a symmetry principle Kthat a fundamental law of nature Lmust
obey, then both Kand Lare fundamental facts, where Kconstrains Lin the sense that
the physical possibilities generated by Lare invariant under the symmetry principle K,
and any other possible fundamental laws are also constrained by K. This introduces
further “modal” relations in the fundamental facts beyond just the constraining of the
spacetime and its contents by L.
3.3.3 Probabilistic Laws
Candidate fundamental physical theories can also employ probability measures and dis-
tributions. Such measures and distributions can be objective, and they may be called
objective probabilities. The probabilistic postulates in physical theories may well be
lawlike, even though the nature of those probabilities is a controversial matter. As
mentioned earlier in §2, Humean Reductionism can accommodate those probabilistic
postulates as axioms in the best system achieving the optimal balance of a simple and
informative summary. It is not so clear what objective probability means in the non-
Humean accounts considered earlier. On MinP, the extension from non-probabilistic
laws to probabilistic ones is also not straightforward. We propose several strategies for
consideration, but which one is most promising depends on further interpretive questions
about probabilities.
29
There are two types of probabilistic postulates in physics: (1) stochastic dynamics
and (2) probabilistic boundary conditions. We will start with the former as they are
more familiar. Consider the GRW theory in quantum mechanics, a theory in which
observers and measurements do not have a central place and in which the quantum
wave function spontaneously collapses according to precise probabilistic rules. On the
GRW theory, the wave function of the universe Ψ(t) evolves unitarily according to the
Schr¨odinger equation but is interrupted by random collapses. The probabilities of where
and when the collapses occur are fixed by the theory. (For details, see Ghirardi et al.
(1986) and Ghirardi (2018).) Another example is Nelson’s stochastic mechanics that
describes particle motion in accord with a stochastic differential equation. (See Nelson
(1966) and Bacciagaluppi (2005).)
For an example of a probabilistic boundary condition, consider Albert and Loewer’s
Mentaculus theory of statistical mechanics, where they postulate in addition to the
dynamical equations (such as (1) and (2)) and PH, a probabilistic distribution of the
initial microstate of the universe:
Statistical Postulate (SP) At the temporal boundary of the universe when PH ap-
plies, the probability distribution of the microstate of the universe is given by
the uniform one (according to the natural measure) that is supported on the
macrostate of the universe (compatible with PH).
At first glance, it is not obvious what SP is intended to convey. One may understand it
in terms of typicality: that we regard the initial probability distribution to pick out a
measure of almost all or the overwhelming majority—a measure of typicality (Goldstein,
2001, 2012). On this way of thinking, SP says the following:
SP’ At the temporal boundary of the universe when PH applies, the initial microstate
of the universe is typical inside the macrostate of the universe (according to the
natural measure of typicality).
On the basis of SP’, one can then explore what the theory says about typical histo-
ries and apply it to our universe. A similar probabilistic boundary condition appears
in Bohmian mechanics, where one can interpret the initial probability distribution of
particle configuration as representing a typicality measure:
ρt0(q) = |Ψ(q, t0)|2(9)
where t0is when PH applies and Ψ(q, t0) is the wave function of the universe at t0.
Based on this measure, almost all worlds governed by Bohmian mechanics will exhibit
the Born rule. (For more details, see D¨urr et al. (1992) and Goldstein (2017).)
In fact, it is also possible to interpret stochastic dynamics as yielding a typicality
measure: the GRW theory specifies a probability distribution over entire histories of
the quantum states, and what matters is the behavior of “almost all” of those possible
histories. Although we are sympathetic to the typicality interpretation of both stochastic
dynamics and probabilistic boundary conditions, we do not insist on it here.
30
Probability measures and typicality measures are not straightforwardly understand-
able in terms of MinP: it is not clear how they should be understood in terms of con-
straints. The difficulty is greater for stochastic dynamics. On the typicality approach,
one has the option to regard the measures picked out by the probabilistic boundary con-
ditions as referring to something methodological instead of nomological—how in practice
one decides whether a law is supported or refuted by evidence. However, the probabil-
ities in the stochastic dynamics are clearly nomological and not just a methodological
principle of theory choice. Here we offer five interpretive options for how to understand
probabilistic laws in MinP.
Option 1: Humeanism. Probabilistic laws are second-class citizens that supervene
on the distribution of matter. On this interpretation, probability and typicality mea-
sures are given a Humean best-system analysis. This is compatible with taking the
non-probabilistic laws as fundamental facts. This option is a Humean and non-Humean
hybrid: it is Humean about probabilistic laws but non-Humean about other laws. We
find this option, while viable, unsatisfactory as probabilistic laws and non-probabilistic
laws seem on a par with respect to explaining patterns. Leaving the former superve-
nient on the mosaic renders the statistical patterns ultimately unexplained. The hybrid
strategy goes against the non-Humean conviction that motivates our adoption of MinP.
Option 2: Primitivism. Probabilistic laws are primitive facts but they are not di-
rectly related to constraints. On this interpretation, they are fundamental irreducible
facts in the world, but they do not simply constrain the world. They are somehow
connected to frequencies and credences but not via constraint. But then how are they
connected? This option needs to be developed further for evaluation.
Option 3: Gradable Constraint. Probabilistic laws are primitive facts that constrain
the world, but their constraining admits of degree. This interpretation extends the
concept of constraining from allowed / forbidden to degrees of constraints (between 0
and 1, inclusive), with the categorical ones (allowed / forbidden) taking the extreme
values. This notion of degrees of constraint is not entirely clear. However, for those who
are fine with probabilification (Platonic Reductionism) or propensities, perhaps this is
also acceptable. This notion of degree of constraining is essentially propensity without
a fundamental direction of time.
Option 4: Typicality constraint. Probabilistic laws are primitive facts about typi-
cality that entail categorical constraint (which does not come in degree). If the right
approach to probabilistic laws is in terms of typicality, and if typicality relates to cat-
egorical constraint, then we can relate both to the notion of constraint in MinP. A
typicality statement is about particular kinds of behaviors that satisfy certain proper-
ties. We can regard such properties as imposing a constraint on possible histories. This
makes typical histories the only physically possible histories. There are two potential
problems. First, such a property may be complicated to specify and the complexity
makes the statement a bad candidate for a law. Second, it seems to express too strong a
requirement. We usually think that atypical histories are still physically possible. They
are just expected not to happen.
31
Option 5: Dual modalities. Probabilistic laws are primitive facts about typicality,
which is another kind of modality distinct from possibility. On this interpretation, there
is a dualism between modal notions of possibility and typicality. Non-probabilistic laws
govern by constraining the space of possibilities. Probabilistic laws govern by constrain-
ing which possibilities are typical. Neither is reducible to the other. Some worlds are
possible but atypical. However, every typical world is possible. Both typicality and
possibility should influence our expectations, and both play roles in scientific explana-
tions. In this way, we may think of probabilistic laws as imposing a narrower kind of
constraints on the world. The actual world must be a member of the physical possibil-
ities delineated by non-probabilistic laws. The actual world must also be a member of
the possibilities delineated by probabilistic laws.
Those issues above have not been much explored in the literature, and there are
many open problems here. We do not take a firm stance, merely noting the above
five strategies, with the acknowledged qualifications and subtleties, may be available to
a defender of MinP. The problem of probabilistic laws is difficult for all non-Humean
accounts of laws. Solving it may turn on questions about the relation between probability
and typicality, and their relation to physical possibility.
4 Comparisons
We have argued that MinP is a minimalist version of non-Humeanism about laws that
is flexible enough to naturally accommodate the diverse kinds of laws entertained in
physics. In this section, we highlight some differences between MinP and the alterna-
tives.
4.1 Comparison with Humean Reductionism
Although MinP is a non-Humean view, in several respects it is similar to Humean
Reductionism. First, neither requires a fundamental direction of time, and both permit
a reductionist understanding of it. MinP shows that anti-reductionism about laws does
not require anti-reductionism about the direction of time. Second, both views are flexible
enough to accommodate the distinct kinds of laws entertained in physics, although
Humean Reductionism might have an upper hand in understanding probabilistic laws.
Third, both views highlight the importance of simplicity (and other super-empirical
virtues) in laws and scientific explanations.
We turn now to the key differences between the two. As mentioned earlier, the main
differences are whether laws are reducible to the mosaic and whether laws depend on
our practice and beliefs. Humean Reductionism answers Yes while MinP answers No to
both.
Ultimate explanation. On Humean Reductionism, the patterns in the Humean mo-
saic have no ultimate explanation; after all, the mosaic grounds what the laws are.
Many see this as a problem for the view. (For example, see (Armstrong, 1983, p.40)
32
and (Maudlin, 2007, p.172) for further characterization of this worry.) Loewer (2012)
responds by distinguishing between scientific explanations and metaphysical ones. He
argues that Humean laws can scientifically explain the patterns even though they do not
metaphysically explain them. On Loewer’s view, a scientific explanation just requires
that the explanans be simple, unifying, and exemplifying other theoretical virtues. A
mere summary of the mosaic, satisfying those theoretical virtues, can be such an expla-
nation. In contrast, metaphysical explanations go deeper and require more.
On MinP, suitable explanations of the patterns must not be merely summaries of
the mosaic. On our view, fundamental laws are metaphysically fundamental facts that
exist in addition to the mosaic. They govern the mosaic and explain its patterns by
constraining it in an illuminating manner. Loewer’s Humean scientific explanation, on
our view, lacks the metaphysical element that provides the needed oomph.
Non-supervenience. Another metaphysical difference concerns non-supervenience.
This has been much discussed in the literature (see for example Carroll (1994) and
Maudlin (2007)). On MinP, since fundamental laws are primitive facts, there can be a
physically possible world corresponding to an empty Minkowsi spacetime governed by
the Einstein equation. However, on Humean Reductionism, that world is one where
the simplest summary is just Special Relativity, and it is impossible to have such a
world where the law is the Einstein equation (Maudlin, 2007, pp. 67-68). To allow two
worlds with the same mosaic (empty Minkowski spacetime) but different laws (special
relativity and general relativity) is to allow non-supervenience of the laws on the mosaic.
To endorse non-supervenience is to endorse the view that the laws cannot be reduced
to non-nomic things and properties.
Objectivity and Mind-Independence. We take it that a hallmark of metaphysical re-
alism about something is to believe in its objectivity and deny its mind-dependence. As
metaphysical realists, we think that we are fallible and can be wrong about the ultimate
reality, including the fundamental laws of nature. This brings out an epistemological
difference between MinP and Humean Reductionism. On Humean Reductionism, as-
suming 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 cannot be mistaken. On MinP, even if we rely on the correct theo-
retical virtues as our guide and rely on 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
ultimate reality is a badge of honor!
Moreover, if the best systematization is constitutive of lawhood, and if what counts
as best dependent on us, then lawhood can become mind-dependent. In a passage about
33
“ratbag idealism,”23 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. Without a precise theory of which standards of simplicity
and informativeness are permissible, and which are not, it is difficult to ascertain the
assumption and determine what can be confirming evidence for it and disconfirming
evidence against it.
In contrast, on MinP, fundamental laws are what they are irrespective of our psychol-
ogy and judgments of simplicity and informativeness. Even though the epistemic guides
provide some guidance for discovering and evaluating them, they do not guarantee ar-
rival 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.
In this paper, we 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 meta-
physical primitives and evaluate them in a package. In this respect, MinP is similar
to Loewer’s Package Deal Account (PDA), a descendent of Humean Reductionism that
regards both as co-equal elements of a package deal (Loewer, 2020, 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
23See also Gordon Belot’s paper on ratbag idealism in this volume.
34
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 evalu-
ate 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 knowl-
edge), 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.
We end this comparison with a quote 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 be-
gins, 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 be-
yond macroscopic ones with which it began and go onto to posit laws that
explain their behaviors...... The ultimate goal of this process is the discov-
ery 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 our perspective, this is an excellent description of how fundamental laws and
ontology are discovered—in a package. We leave its analysis for future work.
4.2 Comparison with Platonic Reductionism
MinP and Platonic Reductionism agree that there are governing laws that do not super-
vene on the Humean mosaic, but disagree on whether governing laws should be analyzed
in terms of or are reducible to relations among universals.
While Platonic Reductionism is ontologically committed to fundamental universals,
MinP is not. We do not think that universals offer additional explanatory benefits. The
motivating idea of Platonic Reductionism is that universals are properties that genuinely
similar objects share, and it is partly in virtue of the universals shared by those objects
that the objects behave in the same way everywhere and everywhen. However, it is really
35
the necessitation relation Nthat does the explanatory work in Armstrong’s theory. It is
crucial that the state of affairs N(F, G) is understood as a universal. The metaphysics
of Nis a complicated business, and in our opinion it seems to create more mystery
than it dispels. In contrast, on MinP we maintain that there are fundamental laws
that govern the world by constraining the physical possibilities. Explanation in terms
of simple laws seems clear enough to vindicate the non-Humean intuition that there is
something more than the mosaic that governs it. Moreover, MinP is compatible with
various metaphysical views about properties such as realism and nominalism. We do not
think that a realist attitude towards laws requires a realist attitudes towards properties.
Platonic Reductionism places certain restrictions on the form of fundamental laws.
On Platonic Reductionism, all laws need to be recast in the form of relations among
universals, and it is unclear how to do so for the majority of laws in modern physics.
Here we agree with Wilson (1987)’s criticism that Armstrong’s discussion is removed
from concrete scientific practice and focuses mainly on schema of the “All F’s are G”
type. Consider a differential equation that expresses a candidate fundamental law such
as (1). What are the universals that they actually relate? Assuming that velocity
and acceleration are derived quantities, what are the universals that correspond to the
derivatives on either side of the equations? Armstrong argues that universals must
be instantiated in some concrete particulars. As Wilson observes (p.439), differential
equations conflict with Armstrong’s principle about the instantiation of universals, as
the values of the derivatives are calculated from values possessed by non-actual states
(those in the small neighborhood around the actual one) that are not instantiated. In
contrast, MinP has no difficulty accommodating laws expressed by differential equations.
Moreover, some candidate fundamental laws involve properties that do not seem to
correspond to universals. For example, PH is a temporally restricted law that applies
to only one moment in time. As such, it is a spatiotemporally restricted law that
seems in tension with the approach involving universals (universal, repeatable, and
multiply instantiated). Tooley (1977) considers an example of Smith’s Garden, and
there he seems open to accept spatiotemporally restricted laws if they are significant
enough. But the tension needs a lot of work to remove, and we do not know if that is
compatible with the metaphysics of universals that they are committed to. In contrast,
spatiotemporally restricted laws can function perfectly as constraints on the universe
that are about specific places or times. There is no in-principle obstacle of letting them
be fundamental laws on MinP.
4.3 Comparison with Aristotelian Reductionism
There are several differences between MinP and Aristotelian Reductionism. Aristotelian
Reductionists do not think that laws govern in a metaphysically robust sense.24 In
contrast, MinP vindicates the conviction that laws do so.
24However, Bird (2007) talks about laws supervening on dispositions and allows that laws can still
govern in a weaker sense.
36
Aristotelian Reductionism is committed to a fundamental ontology of dispositions.
MinP is not. Most physicists today may be unfamiliar with the concept of fundamental
dispositions. In contrast, physicists are familiar with the concept of fundamental laws
and how they figure in various scientific explanations. Hence, MinP seems more science-
friendly.
It is natural to read dispositional essentialists such as Bird, Mumford, and Ellis
as having an implicit commitment to a fundamental direction of time.25 Demarest’s
account is more explicit in linking the dispositional essentialist ontology and the ac-
count of nomic explanations to that of dynamic metaphysical dependence, or what we
call dynamic production. As discussed in §3.3.1, we do not understand how dynamic
production works even in simple cases such as Hamilton’s equations and much less in
relativsitic spacetimes. Requiring dynamic production presumably rules out theories
that permit closed timelike curves, as well as purely spatial laws, or even worlds for
which spacetime is emergent. In contrast, MinP is not committed to a fundamental
direction of time, and MinP is entirely open to those possibilities (even though we may
have other considerations, beyond the conception of laws, to not consider them).
Finally, there are problems specific to accounts (such as Bird’s) that analyze laws
in terms of dispositions. Bird (2007) lists four problems (p.211): (1) fundamental con-
stants, (2) conservation laws and symmetry principles, (3) principles of least action, and
(4) multiple laws relating distinct properties. Problem (1) arises because slight differ-
ences in the constants do not require the properties to be different; problem (2) because
conservation laws and symmetry principles do not seem to be manifestations of dispo-
sitions; problem (3) because the principles seem to commit to the physical possibilities
of alternate histories, something not allowed on dispositional essentialism; problem (4)
because a third law relating two properties will not be the outcome of the dispositional
natures of those properties. These problems may be solvable on Aristotelian Reduction-
ism, and Demarest’s version may be especially well posed to do that. In any case, such
problems do not arise on MinP.
4.4 Comparison with Maudlinian Primitivism
MinP agrees with Maudlinian Primitivism that fundamental laws are metaphysically
fundamental and that they govern. However, we 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
25In a recent book, Vetter (2015) is open to the idea that there can be past-directed dispositions but
still suggests that there is a temporal asymmetry: past-directed dispositions are trivial.
37
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, the Past Hypothesis cannot be a Maudlinian
law. And neither can a purely spatial constraint such as Gauss’s law or the simple rule
responsible for the Mandelbrot world. On MinP, all of these examples can be understood
as fundamental laws that express simple constraints.
Our difficulty with dynamic production is not just it precludes certain candidate
fundamental laws. We also have difficulty understanding 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, we are
unclear how 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, re-
gardless 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.)
5 Conclusion
We suggest that MinP is an intelligible and attractive proposal for understanding funda-
mental laws of nature. It vindicates the non-Humean conviction that laws govern while
remaining flexible enough to accommodate the variety of kinds of laws entertained in
physics. MinP illuminates metaphysics but is not unduly constrained by it.
Acknowledgments. For helpful discussions, we thank David Albert, Craig Callender,
Christopher Dorst, Tyler Hildebrand, Barry Loewer, Kerry McKenzie, Shelly Yiran Shi,
Nino Zangh`ı, and participants in the graduate seminar “Rethinking Laws of Nature” at
the University of California San Diego in spring 2021. EKC received research assistance
from Shelly Yiran Shi and is supported by an Academic Senate Grant from the University
of California San Diego.
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