# Laws of Nature as Constraints

Emily Adlam  University of Western Ontario

##  1 Introduction

The laws of nature have come a long way since the time of Newton: quantum
mechanics and relativity have given us good reasons to take seriously the
possibility of laws which may be non-local, atemporal, ‘all-at-once,’
retrocausal, or in some other way not well-suited to the standard dynamical
time evolution paradigm. But many extant realist approaches to lawhood are
based on the paradigm of Newtonian laws, and thus they face significant
challenges when we try to apply them to laws outside the time evolution
picture. In light of these developments, we are in need of a new approach to
lawhood which is better suited for laws outside the time evolution paradigm.
In the present paper, we argue that this can be achieved within an approach
which conceptualizes lawhood in terms of modal structure, and we set out a
general framework in which the modal structure associated with a wide class of
possible laws can be expressed and studied.

This project is motivated in large part by the recognition that the ideas we
have about what laws look like inevitably shape the types of laws which
scientists formulate. Thus the limitations of our current notion of lawhood
translate to limitations on scientific thought: although laws outside the
dynamical time-evolution paradigm are gaining increasing prominence in
physics, many physicists are reluctant to accept them and there remains
confusion about their conceptual significance. Moreover, the development of
these sorts of approaches is currently hampered by attempts to assimilate them
to a time-evolution picture - for example Kastner, observing that modern-day
proponents of retrocausal models often attempt to express them in dynamical
terms, comments that ‘ _the dynamical stories presented in these
interpretations are inconsistent with the ontology they require._ ’[1] So in
order to facilitate further progress within these research programmes it is
crucial to develop a conceptually clear way of thinking about lawhood which
straightforwardly accommodates laws outside the time evolution paradigm. The
framework presented in this article is intended to help emerging research on
non-dynamical laws propagate into broader scientific thought and to give
scientists better conceptual resources for their explorations of various kinds
of non-standard laws.

Moreover, there are many philosophical topics where the notion of lawhood
plays a crucial part: for example, laws are often invoked in the analysis of
causation[2], explanation[3], determinism[4], scientific confirmation[5],
prediction[6] and free will[7]. But these analyses frequently assume, either
explicitly or implicitly, that the laws in question take the standard
Newtonian time-evolution form, and thus in the light of developments in modern
physics which indicate that laws may not always take this specific form,
there’s a clear need for us to re-evaluate the role that laws play in these
applications. Furthermore, in many of these applications it makes a difference
whether the account of lawhood is realist or non-realist - for example, one
might well draw different conclusions about determinism and its relationship
with free will in the realist setting where laws _make_ things happen versus
in the Humean best-systems setting where laws are simply read off whatever
does happen. So in order to make progress on these topics we are in need of a
realist way of thinking and talking about laws which accommodates various
sorts of laws outside the time-evolution paradigm: the framework presented in
this article is intended as a first step towards addressing these questions.

At this juncture, we should clarify that this paper is focused on the
fundamental laws of physics - Schrodinger’s equation, the Dirac equation, the
Einstein equations and their ilk, as well their as yet unknown successors. We
have no doubt that interesting issues may arise in attempting to expand the
framework suggested here to the laws of the special sciences and ceteris
paribus laws, but that is a task for another day.

##  2 Physics Beyond Time Evolution

Newton bequeathed to us a picture of physics in which the fundamental role of
laws is to give rise to time evolution: the Newtonian universe can be regarded
as something like a computer which takes in an initial state and evolves it
forward in time[8]. Newton’s vision has been enormously influential on the
philosophy of science and lawhood, and even today many philosophical
discussions about lawhood use Newton’s laws as an example, presumably on the
grounds that $F=ma$ is a simple case which is presumed to share most of the
relevant features of more complex laws.

But there have been several major conceptual revolutions in physics since the
time of Newon, and thus we should not necessarily expect that accounts of
lawhood based on a Newtonian time-evolution picture will be well-suited to the
realities of modern physics. In this section, we will discuss several
important examples of laws appearing in modern physics which look very
different to the traditional Newtonian time evolution laws. This will provide
motivation for the claim that a more general realist framework is needed, and
also provide some inspiration for the overall structure of that framework.
Note that it is not my intention to argue that all the putative laws mentioned
in this section are definitely fundamental laws of physics, or indeed that
these examples prove beyond doubt that we must abandon the time-evolution
picture. The point is merely that all of these types of laws are taken
seriously by at least some sectors of the scientific community, and thus if we
accept the naturalistic notion that our account of lawhood should be led by
scientific practice rather than pre-existing metaphysical commitments, we need
an account of lawhood which is capable of accommodating the same range of
possibilities as science itself.

###  2.1 Lagrangian approaches

We don’t even need to go beyond classical mechanics to encounter laws which
don’t work well within the standard time-evolution picture. For although it is
most common to conceptualise classical mechanics in terms of the Newtonian
schema[9] in which laws act on states to produce time evolution, there is also
an alternative Lagrangian description of classical mechanics in which systems
are required to take the path which optimizes a quantity known as the
Lagrangian[10]. Lagrangian methods are well-recognised as valuable
mathematical tools, but they have not usually been taken seriously as possible
descriptions of reality, presumably because the Lagrangian is optimized over
an entire history and thus the Lagrangian description can’t straightforwardly
be understood within the standard time-evolution picture[11, 8].

However path integrals - the analogue of the Lagrangian method within quantum
mechanics[12] \- have become so important to quantum field theory that
increasingly we are seeing calls to reconsider the idea that the Lagrangian
description is purely a mathematical trick[13, 14]. And indeed, there seems no
reason other than a prejudice in favour of the time evolution picture to
insist that the real laws must take a Newtonian form. Moreover, as argued by
Wharton[8], taking Lagrangian methods seriously leads to a novel ‘all-at-once’
approach to lawhood in which we think of laws applying externally and
atemporally to the whole of history at once[11]. Of course, in an ‘all-at-
once’ picture there is by construction no time evolution, so if we are to make
sense of all-at-once laws of this form we are in need of an approach to
lawhood which is not predicated on the time evolution picture.

###  2.2 Non-dynamical approaches

There is a practice within physics of presenting theories in terms of their
‘kinematics’ (state space) and ‘dynamics’ (laws of evolution)[15], and thus it
is common for philosophical accounts of laws of nature to conceptualise laws
in this way. But there are a number of theories in modern physics where it
doesn’t make a great deal of sense to have a rigid division between ‘state
space’ and ‘evolution laws.’ For example, the solution to the Einstein
equations of General Relativity is not a state at a time but an entire history
of a universe[16], so it doesn’t seem to require any concept of time
evolutionl. A time-evolution formulation of the Einstein equations does
exist[17, 18], but the original global formulation remains central to research
in the field and there seems no obvious reason to think that the time-
evolution formulation must be more fundamental. Similarly, a number of
approaches to quantum gravity, including quantum loop gravity[19], postulate
laws which apply ‘all-at-once’ to the whole of history, rather than taking a
state at a time and evolving it forwards.

Note that the terms ‘kinematics’ and ‘dynamics’ are still sometimes used in
the context of General Relativity and other theories without a notion of time
evolution, where ‘kinematically possible’ now refers to the set of models
which are considered to be of the right mathematical type to represent a
physically possible world, and ‘dynamically possible’ refers to the set of
models which do in fact represent a physically possible world[20]. But as
Spekkens points out, there may be many different ways of splitting a theory
into ‘kinematics’ and ‘dynamics’ and moreoever, the kinematics and dynamics of
a theory can only ever be tested together, so there is no hope of empirically
distinguishing which distinction between kinematics and dynamics is the
correct one. Spekkens argues that we should therefore refrain from making this
distinction altogether and instead think in terms of causal structure[21] (or
perhaps _modal_ structure for greater generality). This line of argument
suggests that simply decoupling the notion of dynamics from the time-evolution
picture may not be enough to save traditional accounts of lawhood: the
increasing importance of non-dynamical approaches in physics gives us good
reason to take seriously the possibility that the next generation of
fundamental laws won’t be parsed within the kinematics/dynamics framework at
all, and thus we are in need of some alternative account of lawhood which does
not lean so heavily on that distinction.

###  2.3 Retrocausal approaches

Retrocausal approaches to the interpretation of quantum mechanics have been
attracting significant attention in recent years; see for example the two-
state vector interpretation[22], the transactional interpretation[23], Kent’s
approach to Lorentzian beables[24], Wharton’s retrocausal path integral
approach[25], and Sutherland’s causally symmetric Bohmian model[26]. The
proliferation of such models provides good reason to suppose that a correct
understanding of quantum mechanics may require us to rethink some of our ideas
about time and temporal evolution. Of course, one can always choose to rewrite
retrocausal laws in a non-retrocausal way by adding appropriate hidden-
variables and/or fine-turning initial states; but it has been argued that in
certain cases the resulting non-retrocausal law will fail to obey time
symmetry[27, 28] or will contain inexplicable coincidences[29] and thus even
if the non-retrocausal description always remains technically possible, there
are compelling reasons to take the retrocausal descriptions seriously.

As noted in ref [11], there are two importantly different notions of
retrocausality currently employed in the field. Some approaches (such as the
two-state vector interpretation) explicitly postulate two distinct directions
of dynamical causality, such as a forwards-evolving state and also a
backwards-evolving state, and in such cases we might hope to accommodate
retrocausality within standard pictures of lawhood by making a small
alteration which allows for evolution backwards as well as forwards. But other
approaches (such as Wharton’s models) are more at home within the kind of
‘all-at-once,’ block universe model that we have already encountered in
section 2.1. Moreoever, there are strong indications that the latter approach
is the more coherent one: ref [11] argues that only the ‘all-at-once’ picture
can circumvent the conceptual difficulties associated with retrocausality, and
ref [1] argues that even in accounts which appear to promote two directions of
dynamical causality, ‘ _such dynamical stories are just narrative overlays on
a static ontology_.’ Thus these developments in physics suggest strongly that
we are in need of new philosophical approaches to lawhood which are better
aligned with the ‘all-at-once’ picture.

###  2.4 Quantum information and foundations

Large sectors of the growing field of quantum information science are
concerned with discovering general constraints on what information-processing
tasks can be achieved using quantum systems. For example, there is the ‘no-
signalling principle’ - that is to say, the fact that measurements on distinct
quantum systems commute entails that quantum mechanics can’t be used to send
signals faster than light[30]. A generalisation of no-signalling known as
‘information causality’ has been used to derive the exact value of the limits
on quantum non-local correlations[31]. We also have ‘monogamy,’ which places
limits on how strongly quantum systems can be entangled with two or more
distinct systems[32, 33], and ‘no-cloning,’ which places limits on the ways in
which quantum information can be copied[34]. These sorts of constraints are
regarded by many researchers in the field as being deep and fundamental
features of physical reality, and yet they are certainly not dynamical laws or
time evolution laws in the usual sense, since they are primarily concerned
with describing what is possible or impossible.

Similarly, in the adjacent field of quantum foundations, there has been a
great deal of interest in operational theories[35, 36] and a related family of
research programmes including nonlocal correlations[37, 31, 32], generalized
probabilistic theories[38, 39, 40, 41], empirical models[42], and device-
independent approaches[43, 44, 45, 46]. These frameworks are often used to
prove relations of the form ‘every theory which has property $A$ also has
property $B$,’ or to come up with operational axiomatisations of quantum
mechanics. The properties and axioms that feature in these sorts of approaches
are, once again, not usually concerned with dynamics or time evolution: they
frequently take the form of constraints delimiting the the bounds of the
possible, such as ‘the existence of a continuous reversible transformation
between any two pure states’[38] or ‘the existence of an information unit.’
[41] Again, proponents of these approaches often seem to be of the opinion
that these properties and axioms are deep, fundamental features of the
physical world, but they are clearly not dynamical or time evolution laws.
Thus it seems desirable to have some kind of framework in which we could at
least entertain the possibility that these sorts of information-theoretic and
operational constraints might be genuinely lawlike.

###  2.5 Consistency Conditions

Einstein’s equations of general relativity admit solutions which contain
closed timelike curves, i.e. points in spacetime which lie in their own causal
future[47]. The existence of closed timelike curves raises the spectre of
logical paradoxes like the grandfather paradox, where someone travels into the
past via a closed timelike curve and shoots their grandfather in order to
prevent their own conception. To rule out the possibility of such paradoxes it
is necessary to impose consistency constraints on the fields at points along
the closed timelike curve. Earman considers that the best way of
conceptualizing these consistency constraints is to suppose that they ‘ _have
law status_ ’ and moreover, he argues that it’s unlikely these constraints can
be expressed in wholly local terms, and thus _‘in general, the consistency
constraints may have to refer to the global structure of spacetime.’_[48]
Therefore if consistency constraints are indeed laws, it’s likely that they
are not local time evolution laws but rather some new sort of global, all-at-
once laws. Earman suggests understanding these sorts of laws within the best-
systems paradigm (see section 3.1, but in principle there is no reason why one
could not entertain global laws of this kind within a realist approach to
lawhood, and thus there is a clear need for a realist framework which
accommodates constraints of this kind.

###  2.6 Constraints

The notion of expressing laws as constraints has attracted interest in other
areas of physics as well. For example, Deutsch and Marletto’s constructor
theory expresses physical law in terms of what tasks are possible and
impossible[49]. The second law of thermodynamics is a law of this kind - in
the Kelvin formulation, it tells us that processes which convert heat
completely into work are _impossible_ in the constructor theory sense, i.e.
the conversion of heat into work might occasionally occur by fluke but it
can’t be reliably achieved. Note that the concepts of ‘possible’ and
‘impossible’ employed here are in a sense non-local, since they refer not to
individual instances but to behaviour over repeated instances: _possible_
tasks are those which can be executed repeatably, in a cycle, with arbitrary
accuracy, whilst _impossible_ tasks are those which can’t, though they might
still occur as a one-off. Moreover, we know from past failed attempts at
deriving the second law from the time evolution equations used in statistical
mechanics[50, 51] that it is typically very difficult to account for such
impossibility constraints in terms of purely time-evolution laws, so it seems
likely that in order to take Deutsch and Marletto’s construction seriously we
will need an account of lawhood which is not predicated on time evolution.

Similarly, Filomeno discusses various phenomena within physics where the
dynamics exhibit typicality properties such that we get emergent stable
patterns which are almost independent of the details of the underlying
dynamics, and therefore it is the _constraints_ on the kinematically possible
states (e.g. symmetries of the state space) which really determine the
behaviour. Filmeno thus argues that we can explain the existence of
regularities by committing to modal constraints on the kinematic state space
rather than being ontologically committed to specific dynamical laws[52].
Constraints of this sort are clearly not time-evolution laws, and although we
might attempt to rewrite them in that form, doing so would seem to erase the
specific modal content of the claim that some sort of process is either
possible or impossible. Constraint-based laws seem much more at home within
some sort of block universe picture, where we can imagine them applying all-
at-once to prohibit the relevant sort of process across the whole of history.
Thus, given the increasing importance of constraint-based language in physics,
it seems reasonable to require that our account of lawhood should allow the
possibility of laws which take this sort of form.

##  3 Existing Accounts

The examples of section 2 reveal a general trend toward laws of physics which
don’t fit into a traditional time-evolution picture - laws which apply ‘all-
at-once,’ which are non-dynamical, which are phrased in the language of
constraints. How do traditional accounts of lawhood fare with these sorts of
laws? In this section, we will examine several popular views and it will
become clear that none of them is sufficient to accommodate the wide variety
of non-Newtonian laws described in section 2.

###  3.1 Humeanism

Humeanism is the name given to a family of metaphysical attitudes which insist
that nothing exists other than the ‘Humean mosaic,’ i.e. the set of local
matters of particular fact instantiated within our world, and that therefore
all modal features of reality simply supervene on the physical. The most
common Humean approach to lawhood is the best-systems view, which holds that
laws of nature are simply the axioms of the best systematisation of the Humean
mosaic, where the ‘best systematisation’ is supposed to be the one with the
best combination of simplicity, robustness, and strength[53]. The best-systems
account thus tells us that laws are merely _descriptive_ and have no modal
force of their own.

The best-systems approach does accommodate a fairly large range of possible
laws, as it allows that anything which can be written as the axiom of a best-
system is a potential law. But there is one sort of case where the Humean may
struggle - laws dealing with possibility or impossibility, as discussed in
sections 2.4, 2.6. It’s difficult for the Humean to do justice to the modal
force of such laws, given that a central tenet of the Humean approach is the
denial of modal concepts: the best option within the best-systems picture
appears to be analysing ‘X is possible’ as ‘X occurs’ and ‘X is impossible’ as
‘X does not occur.’ But in some cases we might want to say that ‘X is
possible’ is a law even though X does not actually occur. For example, in
several of the axiomatisations discussed in section 2.4 it is shown that the
fundamental structure of quantum mechanics can be derived from a set of axioms
including _continuous reversibility:_ ‘there exists a continuous reversible
transformation between any two pure states,’ (i.e. every transformation
between pure states _is possible_ in both directions )[38, 41]. Yet it may
well be the case that not every transformation between pure states is actually
instantiated in our world - indeed quantum mechanics defines an infinite
number of such transformations, so the actual Humean mosaic can instantiate
them all only if it contains an infinite number of events, and we are not yet
sure whether or not the actual Humean mosaic is infinite. So it’s likely that
in this sort of case the analysis of ‘X is possible’ as ‘X actually occurs’ is
not available, which makes it hard for the Humean to maintain that a principle
like continuous reversibility could possibly be lawlike. Yet the
axiomatisations suggested in ref [38, 41] give us at least heuristic grounds
to argue that the lawlike nature of this possibility statement is an important
part of the reason why our world obeys quantum mechanics; it isn’t my
intention to claim here that this argument is necessarily correct, but
arguments of this sort are taken seriously by at least some sectors of the
physics community and therefore it’s important that our account of laws of
nature should be capable of making sense of them. Humeanism therefore seems a
poor fit for at least some directions that the physics community is currently
pursuing.

In any case the Humean approach is not a realist approach to lawhood - that
is, it does not allow us to say that laws _make things happen_ and have modal
force of their own. Yet the governance paradigm for lawhood still has a high
degree of intuitive plausibility and moreover presents an importantly
different picture of the relationship between laws and reality which may offer
alternative insight into some of the philosophical topics which make use of
the notion of lawhood. So even if it ultimately turns out that the Humeans can
solve the problem with possibility-based laws and accommodate the full
spectrum of laws postulated in modern physics, there would still be good
reasons to carry on looking for a _realist_ account of lawhood which can do
the same, where by ‘realist’ we mean an account which holds that modality does
not simply supervene on the physical and laws form part of the objective modal
structure of reality. Let us therefore proceed to realist approaches,

###  3.2 Universals

One of the most prominent realist approaches to the analysis of laws,
developed by Armstrong[54], Dretske[55], Tooley[56] and others, tells us that
laws are relations between universals. In particular, Armstrong’s account
depends on a concept of universals as _perfectly natural properties_ , so a
law is to be understood as a special necessitation relation that holds only
between perfectly natural properties. For example, Newton’s second law $F=ma$
is to be analysed as the claim that a relation of necessitation holds between
the universals ‘force,’ ’mass’ and ‘acceleration,’ which all look
superficially like good candidates to be considered perfectly natural
properties.

Various objections to the universals account have been raised, but from the
perspective of the present article, the most important problem is that it has
an unavoidably reductionist flavour, because ‘perfectly natural properties,’
are those which ‘slice nature at the joints,’ and this slicing metaphor is
often taken very literally: the perfectly natural properties are considered to
be the most basic properties of the smallest things in nature, whatever they
might be. Moreoever, this necessitation approach is most naturally situated in
a context where we have a clear distinction between kinematics and dynamics,
because the universals themselves typically describe possible features of
kinematical states and the relations between them define the dynamical laws
determining how these features evolve over time. So the universals account
seems to work well if we limit ourselves to dynamical laws governing the local
evolution of the most fundamental entities, but starts to look fairly weak if
we consider any other types of law. For example, it has been argued that the
universals approach struggles to account for conservation laws[57, 58],
functional laws[59], laws concerning irreversibly defined properties[60] and
laws about the retention of dispositions[61]. And these are just laws which
exist already within mainstream modern physics - if we try applying the
universals picture to some of the emerging ideas discussed in section 2 we
will undoubtedly have even greater difficulties.

One could, of course, attempt to modify the universals account to be less
reductionist in character. This would involve making a change in what we
consider to be candidates for ‘perfectly natural properties’ - rather than
including only basic properties of very small things, we should allow this
category to include properties of spatially distributed entities and
processes, such as ‘the Lagrangian.’ It seems, however, that such a move would
carry us too far from what we would naturally understand as a ‘universal,’ or
‘perfectly natural property.’ In particular, one obvious desideratum for a
perfectly natural property is that it should not exhibit multiple
realizability: it should be exactly the same kind of thing wherever it occurs.
But many of the laws described in section 2 concern properties which _do_
appear to exhibit multiple realizability - for example, the no-signalling
principle. We could perhaps try to write no-signalling as a law of the form
‘(signalling) necessitates (timelike or lightlike connectedness),’ but are
‘signalling’ and ‘timelike or lightlike connectedness,’ really universals?
‘Signalling’ is a general property that applies to any type of process which
allows the transmission of information, and there are many different ways to
achieve signalling - electromagnetic waves, the spoken word, carrier pigeon,
and so on. It seems implausible to suppose that there’s any one ‘perfectly
natural property’ that all these very different processes have in common -
certainly, they do not seem to satisfy the desideratum of ‘exact similarity.’
In a similar vein, we also see multiple realizability in the phenomenon of
‘universality’ in physics - the fact that strikingly similar behaviours can be
exhibited by physical systems of very different types[62]. One might
reasonably hope to subsume all these different processes under a single
governing law, but regarding these very different systems as instantiations of
the same universal would seem to be straining at the boundaries of what we
could reasonably regard as a ‘perfectly natural property.’ Thus it seems
unlikely that we can relax our definition of ‘universal’ enough to accommodate
the variety of laws which appear in modern physics without losing hold of what
makes the account in terms of universals compelling in the first place.

###  3.3 Powers

An alternative realist account of laws of nature, due in particular to
Cartwright, regards them as simply statements about the
powers/capacities/dispositions of entities[63, 64, 65, 66, 57]. These
powers/capacities/dispositions are regarded as being real, irreducible, and
causally responsible for the way in which things turn out: when entities
interact, their individual powers jointly determine what the outcome of the
interaction will be.

The power account does seem a little more flexible than the universals
approach: for example, if we are willing to to accept claims about what powers
entities do _not_ have as part of the powers account (a controversial point in
itself), we could imagine expressing the no-signalling principle in the form
‘no entity has the power to produce superluminal signals,’ However, this
approach to lawhood is highly local in character, for the entities which are
the bearers of these powers are usually supposed to be the most fundamental
entities, whatever they might be - for example, one might refer to the power
of one fundamental particle to attract another. So powers are to be attached
to specific objects in specific local regions, which doesn’t work well for
laws applying in a non-local way across large regions of spacetime, such as
constraint-based laws or the optimization of a Lagrangian. Moreoever, the
plausibility of regarding laws as powers is dependent on a clear distinction
between kinematics and dynamics, because our idea of ‘powers’ is an
essentially anthropomorphic one: we think of powers as being exercised at
particular moments and we take it that the exercise of powers produces effects
in a temporally ordered sequence. Thus it has been noted[67] that the powers
account has difficulty dealing with conservation laws and symmetry laws, and
the same seems true for other sorts of non-dynamical laws. It’s not obvious to
see how one could describe the optimization of a Lagrangian as a power: to
which entity do we ascribe the power, and at which moment is this power
exercised?

Again, one could attempt to modify this account to make it suitable for a non-
dynamical context. This would involve making a change in what we consider to
be the candidates for bearers of nomic powers - rather than including only
very small things, we should allow this category to include spatially
distributed entities and processes. For example, refs [68, 57] suggest that
the powers approach should understand conservation laws and symmetries as
resulting from the ‘dispositional essence of the property of being a world.’
However, as argued in [67], once one starts ascribing laws to powers of the
‘whole world’ it’s unclear what need there could be to invoke powers of
individual entities at all, so the central thesis of the powers approach
starts to look superfluous. Moreover, once we start talking about powers of
the world as a whole, the idea of powers being associated with _individuals_
which act _at times_ on _other individuals_ loses all meaning, and it becomes
unclear in what sense these ‘powers’ can be described as such at all.
Similarly, it’s unlikely that any generalisation of the powers view which
entirely put paid to the distinction between kinematics and dynamics would
involve anything that would look sufficiently like a ‘power’ to merit the
term; so continuing to describe this picture in terms of powers only serves as
obfuscation.

It’s also worth nothing that powers on their own are not adequate to
reconstruct most scientific laws. For after all, idiosyncratic individual
causal powers are not much use to us as epistemic agents; it is necessary that
each instance of a particular type of entity should have broadly the same
powers, so that we can, for example, draw conclusions about the behaviour of
electrons in the future based on our observations of electron behaviour in the
past. Different accounts achieve this in different ways - for example, ref
[67] offers an explicitly dualist account postulating both fundamental powers
and fundamental laws governing the ascription of the powers to appropriate
entities. The dualist approach does somewhat increase the flexibility of the
powers account, making it possible, for example, to accommodate conservation
laws, and therefore one might think that such a dualist account would give us
the resources we need to model non-dynamical laws. But although this approach
does allow some level of global coordination, ultimately it still requires us
suppose that these global laws take effect only via individual local
interactions: ‘properties are ‘told’ what to do by the nomic relation,’ [67]
where the properties and the things that they do are understood to be local
and specific to individual entities. So this dualist approach ultimately looks
like an attempt to ‘localize’ the global laws by siting their effects in
properties of specific entities rather than allowing that laws may govern
holistically and all-at-once; if we want an account of lawhood which
accommodates genuinely global, atemporal laws, the halfway house offered by
the dualist account should not satisfy us.

###  3.4 Primitivism

So far we have focused on reductionist accounts of lawhood, but an alternative
exists in the form of primitivism, i.e. the idea that lawhood cannot be
analysed at all and must be considered as a primitive fact. This approach is
advocated by Maudlin[69], Carroll[70], and Lange[71] among others. Now, one of
the difficulties associated with primitivism about lawhood is that it is by
nature stipulative, and this means that if we are not careful, we are in
danger of simply building into the definition of lawhood whatever assumptions
about the laws of nature our ancestors happen to have made about laws. For
example, although Maudlin believes that laws are primitive and unanalysable,
he asserts that the fundamental laws are all ‘fundamental laws of time
evolution’[69] and thus the time evolution assumption is built into his
concept of a primitive law, so Maudlin’s particular take on anti-reductionism
with respect to laws looks to be incompatible with many of the types of laws
we considered in section 2. Of course other anti-reductionist approaches would
not necessarily have this feature, but they might build in other assumptions:
the general point is that if we are not careful, the assertion that laws are
unanalysable will just amount to an excuse for being dogmatic about what laws
look like.

That said, there is no reason why a sufficiently general primitivist account
could not provide a good way of thinking about non-dynamical laws. Indeed, the
approach we advocate here might be regarded as a version of primitivism. We do
not intend to be fully committed to primitivism because we don’t necessarily
want to claim that no correct reductionist account of lawhood could possibly
exist; we are simply agnostic on this point, and moreover even if such an
account did exist we are sceptical that we could ever find out what it is.
However, the distinction between this position and primitivism is fairly
small, and thus the framework that we set up here is likely to be of interest
to primitivists too. A primitivist could certainly maintain that ultimately
laws are primitive entities, but hold that the framework introduced in this
paper serves to ‘axiomatize the primitive’ in Schaffer’s sense[72] by defining
a general space of possible laws from which we may draw specific individual
primitives.

###  3.5 Modal Structure

Finally, let us consider the position that laws of nature should be regarded
as part of the objective modal structure of reality. This approach to lawhood
is most commonly seen in the context of structural realism, although it might
also be regarded as a form of primitivism as discussed in section 3.4. From
the structural realist point of view there are two different ways to motivate
this position, roughly corresponding to the two distinct strands of structural
realism. First, one could argue that since we have access to laws only via
their effects on the physical world, we have no epistemic access to them in
and of themselves, and therefore we should be epistemically committed only to
beliefs about the modal structures which laws induce and which mediate their
effects on physical reality, rather than metaphysical claims about the
underlying nature of the laws. This approach is similar to traditional
arguments for epistemic structural realism (ESR)[73]. Alternatively, one could
go a step further and argue that laws _are_ nothing above and beyond modal
structure and therefore by learning about those structures we have learned all
there is to know about the laws. This approach is akin to ontic structural
realism (OSR)[74]. In this article we will make no attempt to adjudicate
between these two positions: for present purposes the important point is that
both approaches have the consequence that understanding laws comes down to
understanding modal structures.

In order to see if this view can accommodate laws beyond the time evolution
paradigm, it will be necessary to say something about what we mean by ‘modal
structure.’ The most common approach in the literature conceptualizes laws in
terms of _causal_ structure, and in accordance with the asymmetry of the
causal relation, lawhood is then understood in a temporally directed way, such
that laws ‘ _generate or produce_ ’ later states from an initial state[69].
For example, ref [75] describes modal structure in terms of the existence of
‘genuine causal powers,’ and ref [76] holds that these powers are typically
supposed to flow ‘along asymmetrical gradients.’ Similarly, ref [77] gives an
account of modal structural realism in terms of causal structure, drawing on
the well-established contrast between causal and categorical properties. But
conceptualizing modal structure in this causal, dynamical, temporally directed
way seems suited only for laws within the time-evolution picture, since many
of the more general sorts of laws mentioned in section 2 have no built-in
temporal or causal direction. Indeed, in light of developments in modern
physics it would seem foolhardy for proponents of the modal structure approach
to tie their position too strongly to _causal_ structure: if the ‘all-at-once’
view of lawhood is accepted then there probably can’t be anything like causal
structure at the most fundamental level, since events determined by all-at-
once laws will typically depend on one another in a reciprocal fashion, so
there will be no asymmetrical causal relations to be found. But the absence of
specifically _causal_ structure need not defeat the modal structure approach
to lawhood, because of course there could still be more general modal
structure in the all-at-once setting (e.g. relations like metaphysical
necessitation or ontological dependence), and therefore it should be a
priority for proponents of modal structure to develop an account of these more
general possibilities.

Another popular option is to refrain from saying anything at all about the
specific modal structures associated with the laws of nature. This approach is
linked to ‘selective realism,’ a form of realism which aims to overcome the
pessimistic meta-induction by making epistemic commitments only to carefully
selected features of our theories which we believe are very likely to survive
all future theory change[78]. Structural realism is sometimes regarded as a
form of selective realism, where the feature that is supposed to survive
theory change is simply ‘structure,’ broadly construed[79]. And clearly in
order to make selective realism work for a modal approach to laws it will be
necessary to refrain from saying anything too definite about the modal
structures associated with laws, because greater specificity leads to a
greater risk that the structures we have identified will not in fact survive
theory change. Indeed, in order to rule out underdetermination, the selective
realist about modal structure would have to insist that more or less all the
modal structures that one could imagine associating with a given theory are in
fact equivalent, which prevents them from actually specifying any concrete
structures which might be involved in this form of structural realism. This
leads to the kind of objection that Esfeld makes in ref [80], where he argues
that OSR is only a ‘partial realism’ because it provides ‘only a general
scheme for an ontology of the physical domain,’ and hence it is too vague to
say anything interesting about, for example, how the EPR correlations come
about.

But as argued in ref [81], structural realism does not _have_ to be a form of
selective realism and nor does structural realism require the selective
approach in order to overcome the pessimistic meta-induction: it is possible
to maintain that the success of past theories is explained by continuity of
structure through theory change without necessarily being committed to the
view that we can always identify _which_ specific structures will survive
_future_ theory change. And in fact, on top of the unsatisfying vagueness of
the selective approach, there are a few specific reasons why selective realism
would be the wrong route for the current project. First, moving away from time
evolution is a significant conceptual shift, and an important part of our
motivation for this project was to arrive at an approach to lawhood which
allows us to be explicit about that shift: the selective approach to modality
is too vague for this purpose. Second, another part of the motivation for this
project was to make it possible to give generalised definitions of notions
like determinism outside the time evolution paradigm; and if we are to do that
by appeal to modal structure, we will require some fairly definite framework
in which to characterise modal structure in order to describe features of
different possible modal structures. Thus for our purposes the selective
approach is not adequate: the ongoing research programme into laws outside the
temporal evolution paradigm needs a more robust approach to modal structure.

##  4 Modal Structure

None of the realist approaches we examined in section 3 are at present a good
fit for the kind of ‘all-at-once,’ non-dynamical laws that we explored in
section 2. Thus our aim in this section is to develop a realist account which
naturally accommodates these sorts of laws. In particular, we will argue that
such an account can be achieved by employing the modal structure approach as
described in section 3.5 together with a generalisation of the notion of modal
structure to go beyond merely causal structure.

We emphasize that this article is focused on articulating modal structural
realism regarding specifically _laws_ \- this approach could certainly be
coupled with a structural realism regarding objects in general, but it could
also be paired with a more conventional object-oriented realism. For example,
one could perfectly well hold that laws are nothing but modal structure whilst
also maintaining that the things governed by these modal structures are
ordinary physical objects, not structures. So although the position presented
here may seem particularly at home within a structural realist setting, other
sorts of realists are also free to adopt it.

###  4.1 Constraints

Taking inspiration from the examples in section 2, we propose characterising
the modal structure associated with laws in terms of the _constraints_ that
the laws induce. Constraints are a suitable tool because the idea of a
constraint (at least in the popular imagination) is a modal one: we typically
think about constraints not merely as _descriptions_ of the way things
happens, but as dictating the way things _must_ happen. Indeed, Ross, Ladyman
and Spurrett use language of this sort in ref [82]: ‘ _If there are structural
facts about the whole universe, and these facts constrain all the facts about
particular regions of the universe … then the only necessity in nature is
furnished by these constraints. The constraints - that is, the structures
themselves - are real patterns._ ’

A constraint places limits on the way in which things are allowed to happen -
or equivalently, it delimits those things which are _not_ allowed to happen.
Constraints are therefore in some ways similar to powers - the powers approach
proceeds by enumerating all of those things which are possible for certain
sorts of systems, while imposing constraints similarly amounts to dividing a
space of configurations or histories up into those which are possible and
those which are not possible. However, thinking in terms of powers emphasizes
the _possibility_ side of the division, whereas thinking in terms of
constraints emphasizes the _impossibility_ side of the division, thus allowing
us to take advantage of the simple logical point that if something is not
possible it definitely will not occur, whereas if something is possible it may
occur but it may also not occur. Laws defined in terms of what is _not_
possible thus have the advantage of being completely transparent in their
empirical content: it is very straightforward to identify empirically whether
or not a law of the form ‘X is impossible’ is obeyed in any given situation,
whereas there is no easy way to say whether or not a law ‘X is possible’ has
been obeyed, since the nonoccurrence of X does not entail the impossibility of
X. This feature will allow us to define the modal structure induced by
constraints in a clear and empirically meaningful way. Moreover constraints,
unlike powers, are not typically associated with individual physical systems:
Lagrangian constraints, no-signalling constraints, thermodynamical constraints
and their ilk are applied to whole histories and to composite systems rather
attached to objects and exercised at times, and therefore constraints are a
suitable way of implementing the vision of all-at-once, non-dynamical laws
presented in previous section.

One might perhaps have the worry that laws which do nothing other than specify
that certain things are not allowed to happen would not be sufficiently
powerful to explain the wide range of lawlike behaviour exhibited in our world
- surely we need a specification of what _does_ happen in addition to a
specification of what cannot happen? But in fact, specifying what doesn’t
happen can also be used as a way of determining what will in fact happen. A
standard deterministic time evolution law which says that a certain sort of
system always behaves in a certain way can be rewritten as a constraint which
says that it is not possible for systems of that sort to behave in any other
way - for example, the law $F=ma$ can be written as a constraint ruling out
all accelerations other than $F/m$. Similarly, a law which says that a certain
sort of system behaves in a certain way with high probability can be written
as a constraint ruling out all histories in which the relevant behaviour
occurs too infrequently. We will look at more examples in section 5, but for
now it is enough to observe that specifying what is impossible will suffice to
determine what does in fact happen provided that a large enough provided that
a large enough range of possibilities are ruled out.

In accordance with the structural approach discussed in section 3.5, we will
refrain from speculating on the metaphysical nature of whatever it is that
grounds constraints (if indeed anything does), and instead characterise them
entirely in terms of the role they play in defining the objective modal
structure of reality. First, clearly constraints are to be understood as
distinct from the Humean mosaic (since they are ex hypothesi modal) but as
playing some role in determining the contents of the Humean mosaic (since this
is the function for which we have postulated them). That is to say,
constraints are _ontologically prior_ to the Humean mosaic. 111To define
ontological priority, we first define ontological dependence by saying that
$A$ ontologically depends on $B$ if $A$ depends for its existence on $B$: for
example, sets depend for their existence on their members. In many cases
(indeed, some say _all_ cases) ontological dependence seems to be asymmetric -
$A$ ontologically depends on $B$ but not vice versa - and in such cases we say
that $B$ is _ontologically prior_ to $A$[83, 84]. To enliven this rather
abstract language, ontological priority may be described metaphorically as
specifying the order in which God would have to create things to make a world
like ours[85]. In order to make this analysis more concrete, we will
henceforth define constraints _extensionally_ , appealing to techniques
employed in modal logic: a constraint will simply be defined as a set of
Humean mosaics, i.e. the set of all mosaics in which that constraint is
satisfied.

The structural role of a constraint is simply to require that the actual
Humean mosaic must belong to the corresponding set, and in that sense a
constraint is indeed ontologically prior to the actual Humean mosaic, since
the constraint dictates the possible contents of the Humean mosaic. Indeed,
the extensional definition of constraints allows us to say something more
concrete here. A constraint guarantees that the actual Humean mosaic must
belong to the corresponding set; that is to say, every possible world in which
a given constraint holds is a possible world in which the Humean mosaic of
that world belongs to the associated set. So for example, if every mosaic in
the associated set exhibits a certain sort of regularity, every possible world
in which that constraint holds is a possible world in which that regularity is
exhibited, which is the definition of metaphysical necessitation (in the
possible world semantics, we say that ‘$X$ metaphysically necessitates $Y$’ if
and only if ‘every possible world in which $X$ is true is also a world in
which $Y$ is true.’). So constraints determine the contents of the Humean
mosaic in virtue of _metaphysically necessitating_ that the actual Humean
mosaic should have certain features. Metaphysical necessitation is clearly a
modal relation, but there is no need for it be attached to individual local
systems or to have any particular temporal direction, and therefore it is a
good candidate for the kind of general modal structure that we need to express
a large variety of laws outside the time evolution paradigm.

This way of defining constraints is analogous to the idea that propositions
can be defined as sets of possible worlds. Indeed, for the proponent of Humean
supervenience who believes that a ‘possible world’ just is a Humean mosaic,
the definitions are equivalent. However, the case of propositions is
complicated by the fact that propositions have intensions, and it has been
suggested that the definition in terms of possible worlds is too coarse-
grained to respect this intensionality[86]. But since we intend to understand
‘constraints’ as a matter of objective fact, we need not worry about
intensions, so the view of propositions as sets of possible worlds fits our
purposes very nicely: it is precise, clear, and allows us to set out a space
of candidate laws which assumes very little about what those laws might look
like. In some cases, a constraint will be expressible in simple English as a
requirement like ‘no process can send information faster than light,’ such
that the constraint corresponds to exactly the set of mosaics in which the
requirement is satisfied. But there are many sets of mosaics which will not
have any straightforward English characterisation, so we will not be able to
state the corresponding constraint in simple terms. Nonetheless, each set
still defines a unique constraint.

###  4.2 Laws

What then is the relationship between laws and constraints? Well, in many
cases we can simply think of laws as inducing constraints - for example, if
the no-signalling principle is considered to be a law of nature, it induces a
constraint consisting exactly of all the non-signalling mosaics. Such a law
can be interpreted as saying that the actual Humean mosaic must belong to the
corresponding constraint; and in the case where the constraint can be given a
simple English characterisation, the law can be interpreted as saying that the
actual Humean mosaic must have the corresponding property. If all of the laws
of nature induce constraints like this, then clearly the actual mosaic must
lie in the intersection of all the constraints associated with the laws.

However, simply equating laws with constraints would rule out the possibility
of laws which are genuinely probabilistic even from the external, all-at-once
point of view, such as a law assigning a probability distribution over
possible initial states of the universe. So for full generality, we will
instead suggest that laws should be thought of as inducing _probability
distributions_ over constraints. In simple cases like the no-signalling
principle, where the law is non-probabilistic, the corresponding distribution
will be trivial - i.e. it will assign probability one to a single constraint
and zero to all other constraints. In other cases, the distribution associated
with a law may be non-trivial, so we must imagine that first a constraint is
drawn according to this distribution, and then the actual mosaic is required
to belong to the corresponding set of mosaics. In this picture, we can think
of a constraint being drawn for each law according to the probability
distribution that the law induces, and then the actual mosaic must lie in the
intersection of all of the selected constraints.

The idea of a constraint being selected according to some distribution may
seem conceptually problematic, given that this process presumably only happens
once and therefore no relative frequency interpretation is possible. However,
sceptics about objective chances or single-case probabilities can always
choose to insist that all the actual laws induce trivial probability
distributions - for as we will see in section 6.2, it is perfectly possible to
observe apparently non-trivial probabilities from our local point of view
within a world governed by laws of nature which induce trivial probability
distributions from the external point of view, so this option is not
inconsistent with any current empirical evidence. Therefore scepticism about
objective chances need not derail this approach: it merely places certain
limits on the space of possible laws which we can contemplate.

Note that we could have chosen to work directly in terms of probability
distributions over Humean mosaics, rather than constraints. However, there are
several reasons for preferring the constraint-based formulation. First, it is
inspired by the increasing importance of constraint-based language in physics,
as seen in section 2. Second, it allows us to acknowledge that a law may
require (deterministically or probabilistically) that the actual Humean mosaic
belongs to a given set, but then say nothing further about which particular
mosaic within the set will be selected - that is, the law prescribes no
distribution within the set, not even the uniform distribution. For example,
we might consider analysing the no-signalling principle as a law which
requires that the actual Humean mosaic must belong to the set of non-
signalling mosaics, but it does not necessarily seem right to say that the law
assigns a uniform distribution over all non-signalling mosaics: in fact the
law simply singles out the allowed set, it does not tell us how to assign
probabilities within that set. Of course we may still end up assigning some
distributions inside the set of allowed mosaics: for example, we may start
with some priors which lead us to assign credences over Humean mosaics, and
then when we learn the existence of some law which requires that the actual
Humean mosaic belongs to some set, we eliminate all mosaics outside that set
and renormalize our credences, leading to a distribution which is non-zero
only inside the specified set. But the initial credences are nothing to do
with the law - it is only the updating process which is dictated by the law,
and hence the resulting distribution within the set of mosaics is a subjective
probability distribution based on our credences rather than an objective
chance distribution specified by a law. As we will see in section 6.2, this
distinction is important for the analysis of determinism and objective chance.

Since constraints need not be expressible in simple English, the probability
distributions induced by various possible laws need have no simple English
expression. Thus just as the best-systems picture allows anything that can be
written as an axiom to be a possible law, defining constraints as sets of
mosaics similarly allows us to define a general space of law-candidates with
very few restrictions on what sort of laws are allowed. It also easily
accommodates laws with non-local effects, because a generic constraint created
by selecting a set of Humean mosaics at random will induce massively non-local
relationships between events: for example, if we know that the world is
subject to a constraint consisting of exactly two mosaics which differ on
matters of fact at exactly two spacetime points $(x,t)$ and
$(x+\Delta,t+\Delta^{\prime})$, then when we observe what actually happens at
spacetime point $(x,t)$ we immediately know what happens at
$(x+\Delta,t+\Delta^{\prime})$ even if it is very far removed in space and
time.

###  4.3 Governance

As noted above, in this setting the relation of ‘governance’ is simply
equivalent to metaphysical necessitation, and this provides us with an
interesting new approach to the question of what it is for a law to govern.
Within traditional time evolution pictures of lawhood we must envision a set
of laws external to physical reality which somehow ‘reach in’ and at each
moment of time push particles along certain trajectories, or induce states to
evolve in certain ways, or similar; whereas in the constraint-based picture
the laws of nature select a set of constraints, which single out a set of
‘allowed’ mosaics, and then the actual Humean mosaic is selected from this
set. So rather than reaching into the universe to meddle in the course of
physical events, the modal structure associated with laws can be regarded as
simply selecting Humean mosaics all at once in an external fashion, making
lawhood consistent with a ‘block universe’ picture since the whole mosaic is
now the object of the law. Thus as hoped, this approach thus represents a
concrete way of realising the ‘all-at-once’ vision postulated in refs [25, 8,
11, 87].

Note that we have not attempted to provide any account of the mechanism by
which the laws single out some set of mosaics, nor how the actual mosaic is
drawn from this set. One could of course come up with a literal reading of
this framework akin to Lewis’s literal view of possible worlds[88], where we
take it that the other possible Humean mosaics actually exist and so, in the
deterministic case where laws are just constraints, laws are literally sets of
mosaics which actually exist. However this would seem to undermine the idea
that laws are ontologically prior to the actual Humean mosaic, as the actual
Humean mosaic is a part of all the sets which contain it, and thus we could
only say that these sets are ontologically prior to the actual Humean mosaic
if we were willing to say that sets are ontologically prior to their members,
which would certainly be contentious[85]. And in any case, a major motivation
for moving to the structuralist picture is to avoid making specific
commitments to the metaphysical nature of whatever it is that induces the
modal structure in question (if indeed anything does).

Nonetheless, even if the selection of the actual mosaic from the candidate set
is regarded only as a formal device, it still points to a different way of
thinking about how laws govern, and therefore it offers an interesting
opportunity to revisit some discussions around the notion of governance (e.g.
see refs [67, 90, 91]). For example, traditionally proponents of OSR have been
inclined to resist the notion that laws ‘govern,’ because the idea of
postulating a governing force independent from nature seems incompatible with
the naturalism which motivates OSR. The constraint approach presented here
addresses this concern by providing us with a naturalised notion of
governance, where laws ‘govern’ in exactly the same way that ordinary modal
properties ‘govern’ - a cause can be said to ‘govern’ its effect, for example.
Since proponents of OSR are typically willing to countenance modal notions and
modal structure, this is a notion of governance which should be acceptable to
them; although the constraints are ex hypothesi external to the Humean mosaic,
they are not outside of or independent from nature as a whole, because
according to the OSR worldview the objective modal structure of reality is an
intrinsic part of nature. Thus this way of thinking makes it possible to
effect a reconciliation between the traditional ‘governance’ picture and the
naturalised modal structure picture promoted within OSR.

Second, this approach might take the edge off some worries about how it is
that laws operate in a consistent way across time. For if we take it that the
laws of nature apply all at once to the whole of history, there’s some
principled reason why we should expect to see consistency across the whole of
history; conversely, if laws operate locally and instantaneously as in the
standard kinematical/dynamical picture, it’s less clear why we should expect
the laws to work in exactly the same way at every separate instance, and thus
the worry that they might suddenly stop working somewhere in the future has
more bite. Of course this is not intended to be a deductively valid argument:
the constraint framework can’t _guarantee_ that the future will be the same as
the past, because laws which change discontinuously at a certain time or which
fluctuate across history can certainly be expressed as constraints, and thus
we still have to make some sort of consistency assumption. But at least in the
global setting the laws are applied externally once and for all when the
actual Humean mosaic is selected, so we don’t have to worry about the
possibility that the laws will suddenly stop working or wonder about what
coordinates different instances of nomic powers across spacetime.

###  4.4 Humean Supervenience

Because this framework is explicitly modal in character, it is of course not
compatible with Humean supervenience: even though laws are characterised in
terms of their effects on the Humean mosaic, it is nonetheless the case that
two worlds with different laws of nature could give rise to the same Humean
mosaic. This is true even if we limit ourselves to non-probabilistic cases
where the mosaic is singled out uniquely by the laws of nature. For example,
if $A$, $B$ and $C$ are Humean mosaics, one world could have a law of nature
assigning probability $1$ to the single constraint $\\{A\\}$ and a second
world could have two laws of nature with the first assigning probability $1$
to the constraint $\\{A,B\\}$ and the second assigning probability $1$ to the
constraint $\\{A,C\\}$, and the different laws of nature of both of these
worlds will deterministically produce the same Human mosaic, $A$. So according
to this realist approach, two worlds with the same Humean mosaic may differ in
virtue of being governed by different laws. Indeed, there are an infinite
number of constraints which would uniquely single out the mosaic $A$, since
there are an infinite number of Humean mosaics from which we could select
mosaics $B$, $C$ in order to form the constraint2 $\\{A,B\\}$, $\\{A,C\\}$.
But according to the approach we have set out here, there is exactly one set
of constraints which does _in fact_ result from the actual laws of nature, and
it is this unique set of constraints which represents the objective modal
structure of reality, even though it is not the only set of constraints which
is compatible with the actual Humean mosaic. This is of course to be expected
from any non-Humean approach to modality: if we reject Humean supervenience
then it follows that modal structure can’t simply be read off the Humean
mosaic, and therefore it must be the case that not all of the possible modal
structures which are compatible with the actual Humean mosaic are correct
representations of the objective modal structure of reality.

###  4.5 Humean Mosaics

So far our discussion has left the notion of a ‘Humean mosaic’ largely
unanalysed. But it is important to be clear that our use of Humean mosaics is
not supposed to be a statement about our ontological commitments: although the
Humean mosaic consists entirely of instantiations of properties across
spacetime, we don’t mean to claim that this minimal Humean ontology
necessarily includes everything that exists. The real fundamental ontology
could be objects, fields, processes, relations, structures, or indeed there
could be no fundamental ontology at all: this approach is intended to be
agnostic as to that point. But any fundamental ontology must ultimately give
rise to some distribution of categorical properties across spacetime (even if
‘properties’ and/or ‘spacetime’ are regarded as emergent features), and this
distribution of properties across spacetime is the level at which we as
observers make contact with objective reality, so it is the appropriate level
at which to describe the modal features of laws. Any two laws which gave rise
to different modal structure for the fundamental ontology but identical modal
structure at the level of the Humean mosaic would not be interestingly
different from the point of view of human observers and our epistemic access
to reality.

For example, proponents of OSR presumably would not wish to say that nature
contains categorical, non-relational properties at a fundamental level, but
they will nonetheless need to provide an account of how structures give rise
to (at least the appearance of) distributions of properties across spacetime,
and thus they can still adopt the constraint framework at the level of these
emergent distributions of properties. Similarly, there are indications from
quantum gravity that ‘spacetime’ may be emergent from some other sort of
substratum[19], and therefore we might not want to be committed to the claim
that the spacetime on which the Humean mosaic is defined is genuinely
fundamental. But laws governing this substratum will have consequences for the
geometry of spacetime and the distribution of properties across it, and
therefore these laws will still give rise to meaningful constraints within the
constraint framework. (Of course, given that it has so far proven difficult to
use theories of quantum gravity to make many concrete empirical predictions,
it might not be straightforward to determine what these constraints look like
- but at least in principle they must exist!).

Even with this proviso, the sorts of laws which one can imagine coming up with
will depend sensitively on what sorts of categorical properties one allows
that the Humean mosaic will contain. For example, some physicists would
probably insist that the Humean mosaic should contain properties like ‘spin’
whereas others might feel that spin is an abstraction which can be completely
accounted for by keeping track of spatial position (as is done in the de
Broglie-Bohm interpretation[92]). This is in fact an intentional feature,
because the project we are engaged on is motivated in large part by the
naturalistic idea that a philosophical account of laws should be led by
science and not vice versa, and as part of this vision it’s important to leave
open the question of which categorical properties appear in the Humean mosaic;
this allows us to postulate a space of Humean mosaics which contain all of the
basic properties featured in the theories we currently take seriously, whilst
also allowing the possibility of mosaics which contain properties that we have
not yet imagined. Indeed, specifying what sorts of categorical properties may
appear in a Humean mosaic is loosely analogous to specifying an ontology, or
to setting out the kinematical space of the theory, with further constraints
that single out a smaller subset of mosaics then playing the role of the
dynamics. Moreover, this specification can itself be regarded as a constraint:
limiting ourselves to consideration of Humean mosaics which contain only
pointlike masses is functionally equivalent to imposing a constraint
consisting of all and only those Humean mosaics which contain only pointlike
masses. So in principle we can imagine that the infinite space of Humean
mosaics contains every possible mosaic featuring every possible categorical
property, and the ‘laws of nature’ then specify what sorts of properties exist
as well as how those properties behave.

One type of realist view which might seem a little more difficult to
accommodate in the constraint framework is a branching-world approach like the
Everett interpretation of quantum mechanics[93], since a single Humean mosaic
doesn’t seem adequate to fully capture what the world is like in such a
setting. However, the role of the Humean mosaic in the constraint framework is
simply to capture the ‘actual, non-modal’ contents of reality, in order that
we can identify sets consisting of different ways the non-modal contents of
reality could be. And presumably there is at least in principle some way of
specifying the actual, non-modal contents of reality within an Everettian
universe - for example, one might imagine replacing a ‘Humean mosaic’ with a
‘universal wavefunction’ and then consider a constraint to be a set of
different ‘universal wavefunctions’ with differing local or global structure.
So the constraint framework could in principle be expanded to accommodate
branching-world pictures. However, we have refrained from doing so here
because the way in which human observers actually make contact with and learn
about the objective modal structure of reality is via observations of what
appears from our point of view to be a single Humean mosaic, i.e. various
instantiations of properties across spacetime. Even if we are in an Everettian
universe we will never be able to directly observe the structure of the
universal wavefunction; at best we can form hypotheses about it based on what
prima facie appears to be a single Humean mosaic. So the attempt to make
inferences about the objective modal structure of reality encounters
additional epistemic difficulties in the context of branching-world pictures,
and we have elected here not to engage in detail with these issues - though we
would invite proponents of branching-world or multiple-world pictures to take
up these questions.

##  5 Examples

In this section, we give some examples of how various different sorts of laws
would look within the constraint framework.

###  5.1 Time Evolution Laws

Our stated aim for this analysis was to provide a realist approach which is
_more_ general than theories based only on the time evolution picture, so it
is important that our framework should also accommodate time evolution laws as
a special case. It might seem a little odd to think about a time evolution law
in terms of constraints on an entire Humean mosaic, since we typically think
of time evolution laws as acting moment-by-moment on instantaneous states -
but in fact, many sectors of the philosophy of physics community have already
adopted a constraint-based approach to time-evolution laws. For example, in
ref [20], Wallace writes ’A _history_ of the system is then a smooth function
$q:\mathbb{R}\rightarrow Q$ assigning to each time $t$ the configuration
$q(t)$ of the system at that time. The dynamical equations of Newtonian
mechanics distinguish dynamically possible from dynamically impossible
histories.’ Similar language also appears in refs [94, 95]. In this sense, the
proposal we have made here simply formalises a way of thinking about lawhood
that has been common for some time but which has not yet passed over into
mainstream philosophical analyses of lawhood (or at least, not into non-Humean
analyses of lawhood).

This approach allows us to rewrite any time-evolution theory in the form of a
constraint theory where the role of the laws is not to act moment-by-moment on
states but rather to pick out the set of possible histories. It is then
straightforward to write such laws in the constraint framework: a
deterministic evolution law which applies to systems of type $s$ is analysed
as a probability distribution which assigns probability $1$ to the constraint
consisting of the set of all Humean mosaics in which all systems of type $s$
have histories which are dynamically possible according to the law (where the
‘histories’ in question may extend over the whole of time, or the entire time
during which the system in question exists).

We can also write indeterministic time evolution laws in a similar language.
In fact, there are two possible routes for probabilistic laws. The first
option involves constructing a probability distribution over each possible
history of a given system by multiplying the probabilities assigned by the law
at each time-step, and if necessary taking a limit as the time-steps approach
size zero. Then a law which assigns probability $p_{s}(h)$ to history $h$ for
a particular system $s$ can be analysed as a law which assigns probability
$p_{s}(h)$ to the set of all mosaics in which system $s$ has history $h$, and
so on for every system of the same type across the whole Humean mosaic.
Alternatively, we can implement a form of nomic frequentism[96] where a law of
the form ‘process $P$ produces outcome $A$ with probability 0.8 and outcome
$B$ with probability 0.2,’ is rewritten as the constraint ‘across all
occurrences of process $P$ over all of spacetime, 80 percent have outcome $A$
and 20 percent have outcome $B$.’ Provided that the number of instances of
process $P$ across spacetime is sufficiently large, the effect of this sort of
constraint will be locally indistinguishable from the effects of constraints
assigning probability distributions over histories. We will discuss these
options further in section 6.2 but for now it is enough to observe that as
previewed earlier, both deterministic and probabilistic time evolution laws
can always be rewritten in terms of constraints which rule out behaviour
incompatible with the law.

###  5.2 Lagrangians

A deterministic Lagrangian theory can be written in the constraint framework
as a law which assigns probability $1$ to the set of all mosaics in which
systems always take the path which optimizes their Lagrangian; and a
probabilistic Lagrangian theory which assigns probability $p_{s}(h)$ to
history $h$ for a particular system $s$ can be analysed as a law which assigns
probability $p_{s}(h)$ to the set of all mosaics in which system $s$ has
history $h$. This description allows us to take seriously the atemporal
character of these descriptions: rather than saying that earlier parts in the
history cause later parts and not vice versa, as in the traditional time-
evolution picture, we are now able to allow that the laws select the whole
history all at once, so the relationship between earlier parts and later parts
of the history is reciprocal and holistic.

###  5.3 Retrocausal Laws

We noted earlier that it is important to distinguish between the type of
retrocausality which involves both forwards and backwards evolution, and the
type of retrocausality which arises naturally in an ‘all-at-once’ picture
without either forwards or backwards evolution. The former can be accommodated
within the constraint picture in the form of two distinct laws. In the
deterministic case, for a retrocausal description of a system of type $s$, we
have one law which induces a distribution assigning probability $1$ to the
constraint consisting of the set of all Humean mosaics in which all systems of
type $s$ have histories which are dynamically possible according to the
forwards-dynamics, and another which induces a distribution assigning
probability $1$ to the constraint consisting of the set of all Humean mosaics
in which all systems of type $s$ have histories which are dynamically possible
according to the backwards-dynamics. The probabilistic case is similar, except
the laws would now induce nontrivial probability distributions assigning
probabilities to histories according to the probabilities ascribed by the
forward and backwards dynamics respectively.

For the latter, we need only a single law which singles out a set of Humean
mosaics exhibiting appropriate correlations between events at different times.
As a simple example, consider a law inducing a distribution assigning
probability $1$ to the set of all Humean mosaics in which the result of the
measurement at position $(x,t)$ is the same as the result of the measurement
at position $(y,t+\delta)$ with $\delta>0$. If we happen to know about this
law, then when we observe the result at $(x,t)$ we know immediately the value
of the result at $(y,t+\delta)$, but it is not the case that the earlier
result causes the latter and not vice versa: the law simply requires that they
should be the same, and thus from the external, all-at-once point of view
their effect on one another is perfectly reciprocal. Obviously in this very
simple case there are no compelling theoretical reasons to prefer the all-at-
once approach to the approach where the earlier result causes the latter one
to have some value, but as noted in section 2, in more complex cases we there
are indeed several arguments in favour of the all-at-once approach, and the
constraint framework allows us to postulate such all-at-once laws in a
straighforward manner.

###  5.4 Information-Theoretic Constraints

Clearly, laws which are already written explicitly in terms of constraints,
such as the information-theoretic constraints discussed in section 2.4, will
be a natural fit for this framework. Moreover, if the law is already written
in explicitly operational terms it is straightforward to identify sets of
Humean mosaics obeying the law, since operational language can
straightforwardly be translated to assertions about distributions of
categorical properties over spacetime. For example, provided we can give a
suitably operational definition of terms like ‘signal’ and ‘faster than
light,’ then the ‘no-signalling principle’ can straightforwardly be analysed
as a law which assigns probability $1$ to the set of Humean mosaics in which
no signal travels faster than light.

###  5.5 Consistency Conditions

The case of consistency conditions for spacetimes without closed causal loops
is a particualrly interesting one for the constraint framework, because
consistency conditions don’t actually have to be imposed as laws at all within
the constraint framework: they emerge as consequences of the framework itself.
This is because the laws of nature are understood to take the form of sets of
Humean mosaics, or probability distributions over each set; and of course,
each Humean mosaic represents a complete and consistent way the arrangements
of physical fact within a world could be. Thus it is never possible to obtain
a consistency problem from a law of nature which takes the form ‘the actual
Humean mosaic must belong to a given set,’ since inevitably every mosaic in
the set is consistent. Consistency conditions are therefore a natural feature
of this framework rather than a distinct set of laws which must be imposed
externally.

Moreover, it should be clear that this is the way things must be for any
account of lawhood which does not uphold Humean supervenience. This can be
seen from the following argument, where we postulate that laws and consistency
conditions are independent and then derive a contradiction:

  1. 1.

There exists a world $W$ governed by a set of laws $\\{L\\}$ plus a law $C$
enforcing a consistency condition

  2. 2.

The law $C$ is logically independent from the laws $\\{L\\}$

  3. 3.

If law $C$ did not hold in world $W$, the laws $\\{L\\}$ would give rise to
inconsistencies

  4. 4.

The laws of a given world do not supervene on the Humean mosaic of that world

  5. 5.

1), 2) and 4) together entail: there is another possible world $W^{\prime}$
which is the governed only by the set of laws $\\{L\\}$ without the
consistency condition $C$

  6. 6.

3) and 5) together entail: there exists a possible world $W^{\prime}$ which
contains inconsistencies

But an ‘inconsistency’ here refers to a scenario in which one and the same
quantity has two different values at the same spacetime point, which is
logically impossible (unless we are willing to countenance branching world
scenarios). So, if we are not willing to countenance branching world
scenarios, there cannot be a possible world which contains inconsistencies;
thus we have derived a contradiction. It follows that any realist about laws
(i.e. anyone who does not believe in Humean supervenience with respect to
lawhood) must maintain that if a consistency condition is really
indispensable, that consistency condition can’t be independent from the laws
themselves. The constraint framework delivers this guarantee, since no set of
laws expressed within the constraint framework can ever lead to
inconsistencies.

###  5.6 Constructors

Constructor laws are a little more complex than some other sorts of
constraints, since they allow that ‘impossible’ processes may occur as a one-
off, so we can’t simply analyse a constructor law as a law assigning
probability $0$ to the set of all mosaics where the ‘impossible’ process
occurs. Rather, we must analyse such a law as a probability distribution
assigning probability $0$ to the set of all mosaics in which process $x$
occurs _in a cycle_ \- i.e. where the same setup is used to execute process
$x$ some nontrivial number of consecutive times (and each execution is
successful, or highly accurate). Fortunately this is possible within the
constraint framework because of the non-local character of constraints, which
allows us to consider laws as applying ‘all-at-once’ to some extended series
of events, rather than simply producing the events one at a time in a
temporally ordered sequence. We would simply have to specify how often the
process is allowed to occur at a given level of accuracy before we would
consider the law to be violated, and then define a constraint consisting of
all mosaics in which the number of occurrences of the process is smaller than
this cutoff.

##  6 Applications

As noted earlier, one of the main purposes of this framework is to facilitate
re-assessements of the role played by laws in various areas of science and
philosophy. In this section we briefly preview a few possible directions for
future research. In particular, we reinforce the ways in which the realist
approach set out in this paper leads to analyses that differ from comparable
Humean accounts. It is not my intention to argue here that the realist
analyses are necessarily superior to the Humean ones, but they are certainly
_different_ analyses, and it is therefore useful to have access to this
realist framework at least for the purpose of comparing and contrasting the
two approaches.

###  6.1 Realism vs Humeanism

For many people it seems very natural to suppose that laws _govern_ \- that is
to say, laws make things happen and are thus part of the objective modal
structure of reality. However, this realist picture of lawhood has to a large
extent fallen out of favour in philosophy of science, and a significant
proportion of modern philosophers favour the best-systems view. Why do
philosophers find the best-systems account so compelling? There are of course
a number of epistemic and metaphysical reasons which might lead one to
question the existence of objective modal structure, but on top of these
considerations the best-systems view also has one very important advantage: it
tells us that anything which could be written as an axiom of some
‘systematisation’ could in principle be a law, and therefore it does
reasonably well at accommodating a variety of nonstandard laws that appear in
modern physics (though as we noted in section 3.1, there may be difficulties
with laws pertaining to possibility and impossibility). Conversely we saw in
section 3 that existing realist accounts aren’t typically very friendly to
laws outside the time evolution paradigm, and thus any philosopher who accepts
that an account of laws of nature should be capable of accommodating the full
range of laws entertained by modern physics seems to have good reason to
prefer the Humean approach.

But there is really no reason that the sort of permissiveness offered by the
best-systems account should not be extended to the realist setting, and so it
appears there is a lacuna in the space of extant positions; it ought to be
possible to provide a realist account of lawhood which can accommodate at
least the same range of possible laws as the best systems account. The modal
approach that we have suggested here is just such an account. Indeed, it is
straightforward to see that any putative law which can be written as the axiom
of a ‘best-system’ can also be written in the constraint framework. To see
this, recall that Lewis’ later accounts of the best-system approach
distinguish two types of axioms[97]. Deterministic axioms simply state facts
about the actual Humean mosaic, e.g. ‘everywhere in the mosaic a body’s
acceleration is equal to the net force on the body divided by its mass.’ We
can always rewrite such an axiom as a constraint according to the recipe
‘axiom $\rightarrow$ the set of all Humean mosaics in which the axiom is
true.’ Probabilistic axioms, meanwhile, _‘pertain not only to what happens in
history, but also to what the chances are of various outcomes in various
situations’_ e.g. ‘atoms of type $A$ always have probability $x$ of decaying
within the next $h$ seconds.’ We can always rewrite such an axiom as a
probability distribution over constraints according to the recipe ‘axiom
assigning probability $p$ to some event $e$ $\rightarrow$ distribution
assigning probability $p$ to the set of all mosaics in which event $e$
occurs,’ or alternatively by using a nomic frequentist approach where ‘axiom
assigning probability $p$ to some event $e$ ’ is rewritten as the constraint
‘event $e$ must occur in a proportion $p$ of all cases of this kind across all
of spacetime.’

Thus the constraint framework can accommodate any type of law which the best-
systems framework can accommodate, which means that it is of value to the
ongoing debate over realism vs Humeanism - for if we are to make a fair
comparison between realist and Humean approaches to lawhood, we should be
comparing the versions of the two approaches which are the best fit to the
reality of modern physics as we currently understand it. Thinking about laws
in terms of constraints brings realist analyses of lawhood up to date with the
sorts of developments discussed in section 2 and thus makes it clear that at
least one of Humeanism’s apparent advantages over realism - its ability to
accommodate the large variety of laws postulated in modern science - is not a
real advantage, as an appropriate realist analysis can achieve similar
flexibility. Clearing away these sorts of considerations should allow the
debate to be more clearly focused on the central epistemic and metaphysical
distinctions which separate the realist and Humean pictures.

Is the converse true - can every probability distribution over constraints be
written as an axiom of a best-system? Well, to start with, we saw in section
3.1 that the best systems approach has trouble with laws which deal with
_possibility_ and _impossibility_ \- in particular, it’s hard for the Humean
to make sense of laws like ‘X is possible’ within worlds in which X does not
actually occur. On the other hand the constraint framework certainly has the
resources to accommodate these sorts of laws because it is structured around
laws which delimit what is and is not possible. For example, a law of the form
‘X is impossible’ will straightforwardly induce the constraint consisting of
the set of mosaics in which X does not occur. Note that this does not entail
that whenever X does not occur in the actual mosaic it is a law that X is
impossible: in such a case the actual mosaic does indeed belong to the set of
mosaics in which X does not occur, but the constraint framework tells us that
not every set of mosaics to which the actual mosaic belongs to is associated
with a law, because lawhood is a matter of objective fact which is not simply
read off the mosaic. Laws of the form ‘X is possible,’ on the other hand, will
_not_ typically be understood as inducing the constraint consisting of the set
of mosaics in which X does occur, because we have seen that sometimes it is
important to be able to insist on the lawlike nature of ‘X is possible’ even
if X does not occur in some possible world. Thus it is preferable to analyse a
law of the form ‘X is possible’ as the _absence_ of a law of the form ‘X is
impossible,’ so in the constraint framework, ‘X is possible’ is analysed as
‘there is no law inducing the constraint consisting of the set of all mosaics
in which X does not occur,’ which of course may be the case even if it so
happens that the actual mosaic is a mosaic in which X does not occur. Thus the
constraint approach is able to provide meaningful analyses of laws like ‘X is
possible’ even in cases where the best-systems approach will struggle.

Moreover, it is built into the best-systems approach that the actual laws of
nature must be ‘simple,’ whereas there are many possible constraints which
would not lead to axioms which look ‘simple’ according to any reasonable
description. We have made no requirement in our framework that the actual laws
of nature must induce constraints which are simple; so the constraint
framework allows us to consider laws which would not be simple enough to be
taken seriously within the best-systems framework. This highlights an
important distinction between realist and non-realist approaches to lawhood.
For the realist, the laws of nature are an objective fact about reality and
whether or not they are simple is something which we might hope to discover by
observation. Of course, in practice we tend to assume that the laws of nature
are reasonably simple, since otherwise the problem of discovering them becomes
intractable, but in the realist picture this simplicity principle is a
_hypothesis_ rather than a matter of definition. Whereas for the Humean, the
laws of nature are by definition whatever axioms lead to the simplest possible
description, so simplicity is non-optional. And this has practical
consequences: for the realist, simplicity is just one of many hypotheses which
we may adopt to narrow down the search space for possible laws, so it could be
partially or completely discarded if the empirical evidence gave us reason to
think the laws are in fact not simple in some particular case, whereas for the
Humean simplicity is part of what it is to be a law and so the simplicity
constraint can never be abandoned.

This feature may be of particular relevance as parts of physics begin to move
away from the time-evolution paradigm, because once we start dealing with
other sorts of laws it’s possible that we will encounter laws that look more
or less simple from different points of view. For example, we can certainly
imagine laws applying to the whole of history all-at-once in an atemporal
fashion which look simple from the external point of view but which look very
complicated if we try to write down their consequences for human observers (in
much the same way as the motion of the planets looks mathematically simple if
we consider the sun to be at rest but very complicated if we take the earth to
be at rest). It’s unclear whose point of view the axioms of the best system
are supposed to be simple from - the original formulation in terms of a ‘gods-
eye’ description of the whole of history[6] does appear to suggest that
simplicity from the external point of view is the key feature, but later
strands of Humean thought such as the ‘better best systems approach’[98] have
emphasized the importance of coming up with laws which are simple and usable
from the point of view of human observers. This ambiguity can mostly be swept
under the carpet when we are dealing with time-evolution laws, which don’t
look notably different from the external and local points of view, but as
physicists move beyond the time evolution paradigm it seems likely that this
will become an increasingly pressing question. Thus a particularly interesting
feature of the constraint framework is that it provides a new locus for these
sorts of hypotheses. For example, rather than supposing that laws will look
simple when expressed as dynamical equations, what if we instead hypothesize
that they are simple in the sense that it should be possible to demarcate the
relevant constraints in some way that is more efficient than simply specifying
all the mosaics which are in the set? That is to say, perhaps we should be
aiming to come up with laws which offer a simple description of the universe
as a whole, rather than focusing on the simplicity of their description of
moment-by-moment dynamical evolution. We know from experience that assumptions
like simplicity often work reasonably well in science, but there is no reason
to assume that the specific way in which we have applied these assumptions in
the past is optimal: applying old assumptions in a global way rather than a
local way may well open up new routes for scientific advancement.

###  6.2  Probability and Determinism

The constraint framework allows us to disambiguate several different ways in
which laws may be deterministic. The traditional notion of determinism
pertains to what is often referred to as _Laplacean determinism_ , where the
result of a measurement at a given time is wholly determined by the state of
the world at that time[99]. This definition makes sense in the time evolution
picture; but in the context of non-dynamical laws there’s no particular reason
to think that ‘the state at a time’ has any special significance, and so we
are in need of a new generalized notion of determinism, which we may arrive at
using our new generalized notion of lawhood.

For example, we might choose to say we have determinism at a ‘global’ level if
all of the laws of nature induce trivial distributions i.e. they only assign
probabilities of $1$ and $0$. In that case we have no need for any objective
chances, and each law is simply associated with a single constraint. Note that
this sort of global determinism does not entail Laplacean determinism - for
example, a law of the form ‘a certain process $P$ produces outcome $A$ with
probability 0.8 and outcome $B$ with probability 0.2,’ could be analysed as a
constraint to the effect ‘across all occurrences of process $P$ over all of
spacetime, 80 percent have outcome $A$ and 20 percent have outcome $B$,’ so
process $P$ would therefore look indeterministic from a local point of view,
but from the global point of view there would be no objective chances
involved. So the constraint-based approach makes it clear that even though our
best current theories don’t seem to obey Laplacean determinism, it could
certainly still be the case that we have something like determinism at a
global level.

In the case where we have global determinism and the intersection of the
corresponding constraints contains only a single mosaic, then the actual
Humean mosaic is singled out uniquely by the laws of nature, so we have what
one might describe as _strong global determinism_. If the probability
distributions induced by the laws are all trivial but the intersection of the
corresponding constraints contains more than one mosaic, then the actual
Humean mosaic must be selected from the intersection, so we have what one
might describe as _weak global determinism_. The latter can still be regarded
as a form of determinism because the selection of a mosaic from the
intersection is not a ‘chancy’ event: the laws do not define any particular
probability distribution over the mosaics in the intersection, not even the
uniform distribution, since any one of them will do just as well as any other.
The selection of the actual Humean mosaic from this set is therefore analogous
to the selection of the initial conditions in in Laplacean determinism: it is
arbitrary and it is not determined by anything, but it is also not chancy.
Weak global determinism is thus simply a generalisation of the idea that in
Laplacean determinism, both the initial conditions and the laws are needed to
determine a unique course of history.

Observe that the constraint framework provides us with a straightforward way
to articulate the differences between Laplacean determinism, weak global
determinism and strong global determinism - these distinctions could certainly
not be formulated in the language of dynamical time evolution laws. Note
further that it is important for this analysis that our approach is a
_realist_ one, since if laws do not have modal force there can be no fact of
the matter about whether they single the actual mosaic out uniquely, so a
Humean analysis would necessarily look quite different. So the constraint
framework does indeed seem to be fulfilling its intended purpose by providing
a means of re-assessing topics like determinism in the context of realist but
non-dynamical laws.

###  6.3 Unification

One important consequence of the generality of the constraint approach is that
it allows us to express very different sorts of laws within the same
framework. This is important for several reasons. First, if we want to perform
analyses of philosophical ideas drawing on the notion of lawhood without
leaning too strongly on specific assumptions about the form of the laws in
question, it’s necessary to do so using a framework which can accommodate a
wide variety of laws. For example, traditional definitions of determinism
intended for the time evolution picture don’t work well for non time evolution
laws, but below we will suggest a way to define determinism in a way that
makes sense for any set of laws which can be expressed in the constraint
framework. Thus the constraint approach allows us to make meaningful
comparisons between the properties of very different sorts of laws.

Second, we may sometimes want to consider a case where we have very different
sorts of laws acting within one and the same world. Prima facie it may seem
difficult to understand how a set of very heterogenous laws could be applied
together: for example, how do we combine a constructor-type law with a time
evolution law? Which law do we apply first in any given situation? Which is
the more fundamental? But the constraint framework provides a straightforward
way to combine different sorts of laws - and as long as there is at least one
mosaic in the intersection of all of their constraints we can be assured that
we have no contradictions between them. By contrast, the Humean approach can
in principle accommodate all these sorts of laws, but it may be less clear
whether we can make sense of a a wide variety of types of law all occurring
within a single world, because one would typically expect that the axioms of
the ‘best deductive system’ would all be of a roughly similar type. For
example, in any reasonable deductive system it should always be clear how to
combine any set of two or more different axioms, and one might worry that a
deductive system which combines both constructor-type laws and time evolution
laws would not necessarily have that feature. Thus in fact the constraint
framework suggested here might actually be useful to Humeans too, since a
Humean wanting to postulate a set of highly heterogenous laws could always use
the constraint language as a way of formulating their axioms, and simply omit
the supposition that constraints represent objective modal structure.

And third, we noted in section 2 that physicists and philosophers are
increasingly questioning the meaningfulness of the traditional distinction
between kinematics and dynamics. Moreoever, we saw in section 4.5 that the
requirements we would traditionally refer to as ‘kinematics’ can be written in
the form of a constraint consisting of all and only those Humean mosaics which
contain only entities of the sort featured in the relevant kinematical space.
We can then impose further constraints singling out smaller subsets of that
set of Humean mosaics, and those further constraints play the role of the
dynamics. So the constraint framework allows us to express both kinematics and
dynamics in the same formalism, and thus it represents an important step
towards delivering on the vision discussed in section 2

###  6.4 Higher Modal Structure

The framework we have suggested here has the further advantage that once we
have made some proposal about the laws of nature using the constraint
language, we can easily draw conclusions about the ‘higher-up’ modal structure
induced by these laws of nature between various less fundamental features of
reality.

For example, consider two constraints, $A$ and $B$, such that $B$ is a proper
subset of $A$ \- that is to say, every Humean mosaic which satisfies the
condition associated with $A$ must also satisfy the condition associated with
$B$. Equivalently, we can say that every possible world containing a Humean
mosaic satisfying the condition associated with $A$ must also be a possible
world containing a Humean mosaic satisfying the condition associated with $B$.
This of course is just another instance of metaphysical necessitation[100].
Thus the fact that $B$ is a proper subset of $A$ can be understood as
expressing the fact that $A$ metaphysically necessitates $B$, so the
constraint framework provides a convenient way of studying the modal structure
associated with relations between different sorts of regularities. For
example, $A$ could be the constraint associated with the no-signalling
principle and $B$ could be the constraint associated with the no-signalling
monogamy bound: it follows from the proof given in ref [32] that $B$ is a
proper subset of $A$, and thus we are licensed to say that the no-signalling
principle metaphysically necessitates the no-signalling monogamy bound.

Moreover, the same argument can be made in the case where $A$ is associated
with a law whilst $B$ is a constraint which merely expresses a contingent fact
- for example, $A$ could be the set of all mosaics in which it is everywhere
true that $F=ma$ while $B$ could be the set of all mosaics in which it is true
that on some particular occasion $F=ma$. Clearly here it will be the case that
$B$ is a proper subset of $A$ and hence in this setting the relation of
metaphysical necessitation also underlies the relations between laws and
individual instances of laws, allowing us to unify relations between
regularities, and relations between laws and their instances, in terms of a
single overarching form of modal structure.

###  6.5 The Past Hypothesis

One argument that has been put forward in favour of the best systems approach
is that it allows us to regard the past hypothesis as a law of nature rather
than a mere coincidence[101]. Now, realists about laws might well feel that
the supposed gain here is a spurious one, for the best systems account has
made the past hypothesis nomological only in virtue of making the nomological
into a brute fact: in the Humean picture laws are just descriptions of
whatever actually happens, so the fact that the past hypothesis is a law
doesn’t make it any less mysterious, and thus insofar as one finds the past
hypothesis suspiciously coincidental from the realist point of view, one
should also find it suspiciously coincidental within the Humean picture.

However, there is nonetheless an important insight here. The reason the past
hypothesis is not normally treated as nomological is that physicists typically
expect that physical systems can be decomposed into kinematics and dynamics;
the initial state of a system is an element of the kinematics and is therefore
by definition not a law, since laws are supposed to be dynamical. But as Hicks
points out, Humeanism in its standard form does not accommodate any such
distinction[102]: if including the initial state in the best-system leads to a
significant simplification, it follows on the Humean view that the initial
state is a law. Hicks regards this as a problem to be rectified, but from the
present perspective it is a significant advantage: the Humean view gives us
licence to ignore arbitrary distinctions such as the kinematics/dynamics split
and simply write down whichever laws provide the best systematization,
regardless of whether they accord with common assumptions about what laws
ought to look like.

So really the novel contribution of Humeanism to the problem of the past
hypothesis is located not in its antirealism but rather in the fact that it
licences us to take nonstandard, temporally non-local laws seriously, and thus
allows us to put the initial state into the laws of nature. But we do not have
to become Humeans in order to achieve this end - we simply have to reject the
standard kinematics/dynamics distinction. The constraint framework
straightforwardly allows us to include the past hypothesis in the laws of
nature, since it can be understood as a constraint which assigns probability
$1$ (or at least, high probability) to the set of Humean mosaics where the
initial state is a ‘low entropy’ state (or some more sophisticated
description[103]).

So the constraint framework allows us to make the past hypothesis nomological
in a more robust sense than the Humean approach: it is not merely a fact about
the universe which happens to appear in the best-system, but it is determined
by some law that is part of the objective modal structure of reality and thus
it can be given a lawlike explanation in just the same way as anything else in
the universe.

##  7 Objections

In this section, we discuss some possible objections that might be raised to
the analysis of lawhood we have presented

###  7.1 Epistemic Issues

If constraints are ontologically prior to the Humean mosaic, then it seems
that laws of nature are necessarily external to the Humean mosaic, and this
feature forms the basis of a common objection to realist accounts of lawhood:
if laws are not a part of the Humean mosaic, and we are only able to observe
directly those things which feature in the Humean mosaic, how can we ever come
to know about or have grounds for believing in laws of nature? To address this
concern, it will be important to distinguish between two different epistemic
difficulties. First, how could we ever know or have grounds to believe that
laws which are ontologically prior to the Humean mosaic exist? And second, how
could we find out about their true metaphysical nature?

With regard to the first difficulty, the answer seems straightforward: we
infer that laws exist based on our observations of their effects on the world.
That is to say, we observe regularities and then apply inference to the best
explanation to conclude that these regularities are the effect of some
objective structure, which is either the consequence of or to be directly
identified with the laws of nature. For as Hildebrand argues, if an entity is
external to the actual universe but is essential to the analysis of lawhood,
then in fact we do have a way to find out about it - to wit, by studying the
laws of nature[104]. Here, we are specifically identifying laws of nature as
those things which are external to the physical universe but which _have an
effect on it_ : that is the content of the claim of ontological priority. So
if there are any laws which satisfy this criterion, they are certainly not
disconnected from physical reality - they act on physical reality, even if
physical reality cannot act back on them, so we do have grounds for forming
beliefs about them.

With regard to the second difficulty, as we have adopted a structural realist
approach we should simply hold that we can be epistemically committed to the
real modal structure associated with laws without knowing anything about the
underlying metaphysical nature of the laws (whether because we have no
epistemic access to laws in and of themselves, in the tradition of ESR, or
because laws _are_ nothing over and above modal structure, in the tradition of
OSR). Of course, this approach might rule out some uses to which one might
want to put one’s realism about lawhood: if for example one were attempting to
use an analysis of lawhood to solve the problem of induction, then indeed one
would potentially need to answer questions about how laws actually operate on
physical reality, in order to make the argument that the existence of laws
guarantees that the future will be relevantly similar to the past. But if one
is a realist simply because the existence of governing laws seems like the
best explanation for observed regularities, or because postulating laws leads
to a more coherent picture of the content of reality, then there is no obvious
need to explain _how_ laws bring about the effects that they do.

A more specific variant of this objection to realism involves the concern that
it is hard to see how universals or capacities or any such metaphysical notion
could play the role in rational belief-formation that laws clearly do. This is
sometimes known as the inference problem - that is, the problem of giving an
account of how it is that laws permit us to make inferences[105, 106, 107]. It
seems likely that a similar complaint would be levelled at our more general
claim that laws induce constraints which are ontologically prior to the Humean
mosaic - why, the Humeans will ask, should we base our expectations about
future events on something that is not even a part of the physical universe?
But again, t the claim that the inference problem can only be answered by
giving an account of the metaphysical nature of the laws should be resisted.
According to the view we have put forward here, laws are by definition those
things which induce certain modal structures, or perhaps laws simply _are_
modal structures, so if we believe that certain laws hold then we have every
right to form expectations about the effects that they will bring about, since
that is exactly what the relevant modal terminology is designed to capture. In
much the same way, we can make inferences about the future based on our
knowledge of electrons because even if we don’t know what electrons actually
are we do have a clear understanding of them in structural and functional
terms: electrons are those things which behave in certain ways, so if we
believe that electrons do exist and are present in a given instance, we can
make inferences about the effects they will produce.

Of course, one might attempt to make a distinction on the grounds that in the
case of electrons there is at least the possibility in principle of finding
out more about the nature of the underlying constituents, whereas in the case
of laws it is most likely impossible even in principle for us to find out the
true metaphysical nature of the laws. Is this an epistemically relevant
distinction? Those who consider themselves scientific realists but who also
espouse Humean (or other non-realist) approaches to lawhood certainly seem to
think so - the claim seems to be that there is something particularly
problematic about making inferences to things which are unobservable because
they are external to the Humean mosaic, whereas it is less problematic to make
inferences to things which are unobservable for some more prosaic reason. But
this particular way of setting the boundary of the knowable seems fairly
arbitrary. Indeed, we are never going to observe the metaphysical basis for
laws ‘directly,’ if there is any such basis at all, but then we are almost
certainly never going to observe electrons ‘directly’ either, since even in a
cloud chamber what we actually observe are the effects of the electron rather
than the electron itself. And if we have learned anything from the long
history of debate over the distinction between observable and unobservable
objects, it is that if we countenance only beliefs in directly observable
features of physical reality then very little is safe. Realism about laws is
in this in this regard not importantly different from other forms of
scientific or metaphysical realism: in either case we have solid grounds for
realism provided we have a solid understanding of the modal properties of the
entity in question.

A slightly different objection regards the role of laws in explanation. It has
been argued that explaining regularities by appeal to laws does little to
remove the mystery surrounding them, because we have just swapped the question
‘Why are there regularities in the universe?’ with the equally puzzling ‘Why
should laws give rise to regularities?’ And the same point can be made about
the account in terms of constraints - there are many possible constraints
which would not give rise to any noticeable regularities, and so appealing to
constraints does nothing to explain the existence of regularities without some
further supposition about why the laws of nature happen to induce the special
sorts of constraints which produce noticeable regularities. We have not
attempted to make such a supposition because we see no way to do so without
invoking some kind of hypothesis about the underlying metaphysical nature of
the laws, and the point of focusing on structures was precisely to avoid those
sorts of commitments. Nor is such a supposition necessary, because the account
we have given is not intended to solve the explanatory problem: it is only
supposed to offer a picture of the world which allows regularities to be the
consequences of objective modal structure. The question of _why_ there should
be any objective modal structure, or structure of this particular kind, is
another matter altogether.

###  7.2 Law Selection

Going against the thrust of our approach in this paper, one might argue that
it is actually a good thing that traditional realist approaches based on a
time evolution picture severely constrain the space of possible laws, because
otherwise there will simply be too many possible laws and we will never be
able to get the underdetermination down to a reasonable level. Indeed, it’s
true that scientists often need to constrain the space of possible laws in
some way in order to make the search space tractable. That is the purpose
served by criteria such as ‘simplicity,’ ‘unification,’ ‘beauty,’ ‘locality,’
‘unitarity,’ and so on. And it’s also true that insisting that laws should
take the time evolution form (or at least, that it should be possible to
rewrite them in a time evolution form) is one way in which scientists have
historically constrained the search space.

But scientists also recognise that sometimes constraints on the search space
need to be abandoned when the empirical data demonstrates their unsuitability.
For example, in the aftermath of Bell’s theorem[108] many physicists came to
the conclusion that it was necessary to abandon the principle of locality222I
reinforce that ‘locality’ as referenced in Bell’s theorem is not the same as
the mathematical principle of ‘locality’ employed in quantum field theory;
Bell’s theorem says nothing about the latter.[109] which was previously
regarded as a natural constraint on the set of possible laws. And as we saw in
section 2 a variety of physicists have now begun exploring laws which move
away from the time evolution picture, driven in many cases by the conviction
that the empirical data have shown the time evolution form to be unsuitable.
Of course this is not yet a consensus position, and it remains possible that
science will find a way to accommodate all the existing data within a strict
time evolution model, but given the increasing prominence of these approaches
it seems the right time to begin exploring some of their consequences for the
philosophy of lawhood.

###  7.3 Cartwright’s Challenge

The constraint methodology works straightforwardly for certain very simple
laws - for example, ‘no signal can propagate faster than light.’ Subject to a
satisfactory definition of what counts as a signal, if this is a law it is
obeyed exactly, and therefore it can very straightforwardly be translated to a
constraint consisting of the set of Humean mosaics which do not contain any
signalling processes.

But what about a law such as the Newtonian gravity relation,
$F=\frac{GMm}{r^{2}}$? We cannot simply identify this law with the set of all
mosaics in which it is everywhere the case that $F=\frac{GMm}{r^{2}}$, because
a ‘force’ is not the sort of thing which is typically supposed to feature in a
Humean mosaic. We would have to combine the Newtonian gravity law with the
force law $F=ma$; but we also cannot identify this law with the set of all
mosaics in which it is everywhere the case that $ma=\frac{GMm}{r^{2}}$,
because as Cartwright famously pointed out,[110] this relationship is never
exactly satisfied, since bodies are typically subject to more than one force
and their final acceleration is given by the vector sum of all the forces
acting on them. Moreover, we also cannot make $ma=\frac{GMm}{r^{2}}$ into a
probabilistic law, because although bodies do not usually obey
$ma=\frac{GMm}{r^{2}}$ exactly, it is also not the case that their motion is
probabilistic - it is a deterministic function of all the forces acting on
them.

One possible resolution of this problem would be to identify a constraint $C$
with the set of worlds for which $C$ belongs to the axioms of the best-system,
defined in the usual Humean way as the system which optimizes simplicity,
strength and fit. But the problem with this approach is that it is subject to
the same objection often levelled against Humeanism - i.e. that ‘simplicity’
is not objective and that therefore what counts as a law will depend on our
preferred standards of simplicity[111]. This is a consequence that the Humean
might be willing to accept, but it is significantly more problematic in our
case, since we have required that laws should be ontologically prior to the
Humean mosaic and so it seems we cannot allow that they should depend on the
preferences of the inhabitants of the mosaic.

An alternative would be to reconsider the idea that forces should not feature
in the Humean mosaic; then we might be able to identify a set of mosaics in
which it is exactly true everywhere that there exists a force
$F=\frac{GMm}{r^{2}}$, and also exactly true that the resulting motion is the
vector sum of the forces. One’s willingness to accept this solution will
naturally be dependent on one’s intuitions about whether a ‘force’ counts as a
local matter of particular fact. We think it is unlikely that Lewis would have
liked this, but we do not necessarily have to follow Lewis on this point.
Moreover within the constraint picture we have a little more leeway, because
we have already abandoned strict Humean supervenience and therefore we need
not be held back by the fear of betraying our founding metaphysical
principles.

A final possibility would involve abandoning the Newtonian gravitational law
in favour of conservation laws or a Lagrangian description. For while it is
not exactly true that $ma=\frac{GMm}{r^{2}}$ anywhere, it _is_ exactly true
that momentum, energy and angular momentum are conserved for a closed system,
and from these conservation laws it is possible to deduce the individual
accelerations of the constituent systems. Of course in the real world very few
systems are truly closed, but this is not a problem for the constraint view,
because constraints are in general non-dynamical, so we can make the system as
large as necessary to ensure that the conservation is exact - even going so
far as to include the whole universe if need be. Indeed, one might argue that
the reason Newtonian force laws are very seldom exactly true is that they are
not the ‘real’ laws - it is the conservation laws or Lagrangian descriptions
which feature in the true laws of nature, and force laws are simply the result
of our attempts to parse the messy local consequences of these global laws.
From this point of view, Cartwright’s challenge results from supposing that
laws must be spatially and temporally local, whereas in fact laws are
generically non-local and therefore hold exactly only at a global level.

Of course, it may be objected that although this works for the case of
Newtonian force laws because we always have the alternative Lagrangian
description available to us, there are conceivable laws where this sort of
description is not possible - indeed, it is known that a set of dynamical laws
has a corresponding Lagrangian description if and only if, after
transformation into a certain canonical form, the right-hand sides of all the
equations are derivable by differentiation from a single function $H$[112,
113, 114]. And even if it is in fact the case that all laws in the actual
world do have a Lagrangian equivalent, the stated intention for the constraint
framework was to come up with a highly general language which avoids ruling
out large classes of laws by fiat, so it would be a problem if this framework
can’t deal with laws which don’t have a Lagrangian description.

However, for any set of deterministic laws which are both individually and
jointly deterministic, there must be some way of writing these laws (perhaps
in the form of a single highly complicated disjunctive law, and/or by
augmentation with a suitable composition rule) such that they hold exactly. If
this were not the case, then there would be some scenarios in which the set of
laws did not prescribe a single outcome, meaning that the laws would not after
all be jointly deterministic. For example, in the force law case, even if the
Lagrangian description were not available it would be possible to find a law
which is exactly true by writing $ma=\int F(x)$ where the integral (or sum in
the finite case) is taken over all sources of force. The resulting law might
be highly non-local, extremely complicated and aesthetically displeasing, but
nonetheless we would be able to pick out a set of Humean mosaics in which this
law always holds and thus it could be written as a constraint. And a similar
argument shows that any set of laws which individually and jointly prescribe
exact probabilities can be written as an exact disjunctive law assigning
probabilities over some disjunctive sample space of events, meaning that
together they can be written as a probability distribution over constraints.

###  7.4 The God Problem

It is common to criticize realist accounts of laws of nature on the grounds
that they get their plausibility from implicit (or sometimes explicit!)
invocation of a deity who personally steps in to ensure that the universe
obeys the laws of nature; thus it is argued that such accounts are a remnant
of theism and should be rejected by all right-thinking atheists[115, 116]. And
one might make a similar criticism of the approach we have suggested here, for
although we have steered clear of making any assertions about the metaphysical
underpinnings of the laws of nature, nonetheless there seem to be some vaguely
theistic ideas in play. In particular, in the discussion of constraints we
invoked the notion of ‘ontological priority,’ which is often understood in
terms of the order in which God creates things, and thus one way to motivate
the idea of the Humean mosaic being subject to constraints would be to picture
the universe as something like a sudoku grid subject to global rules, with God
stepping in to solve it.

But in fact we should take care with this metaphor, because it seems to be
based on an intuition that the parts of the universe must come into being in
some temporal order, with God’s process of solving the puzzle constituting a
process of temporal becoming. Yet the universe, considered as an atemporal
whole, already contains within it the temporal relations which are
constitutive of our experience of time; and therefore one might argue, a la
McTaggart, that we do not need an external ‘A-Series’ process of temporal
becoming in addition to the existing ‘B-Series’ determined by the temporal
relations internal to the universe[117]. Thus it seems that the metaphor which
conjures up God solving a puzzle adds an unnecessary layer of temporal
becoming to an account of lawhood which is perfectly adequate without it.

And indeed, it is easy to remove God from the picture. For after all, a sudoku
grid does not actually require a player: it has a unique correct solution
regardless of whether anyone ever comes along to solve the puzzle. We need
only suppose that the universe is the same: the constraints prescribe (either
deterministically or probabilistically) a set of possible solutions, and the
actual Humean mosaic exists and satisfies these constraints in an atemporal
block universe sense, with no need for a personal being to come along and
solve the puzzle.

Of course, in the context of _epistemic_ structural realism about laws it is
part of the deal that we can’t know what laws of nature are or how they get
enforced, so although ontic structural realists might be able to be dogmatic
about this, epistemic structural realists can’t - for all they know, perhaps
there is a God out there solving the puzzle of reality, and/or the universe
really is generated in a process of temporal becoming. But the point to be
made here is that the plausibility of the approach does not depend in any way
on the theistic metaphor, and indeed the theistic metaphor may well be more
harmful then helpful.

##  8 Conclusion

Developments in modern physics give us good reason to take seriously the
possibility of laws which are non-local, global, atemporal, retrocausal,
and/or not easily put in the standard kinematical-dynamical form. We have seen
that existing realist accounts of laws as relations between universals or
capacities/dispositions/powers are not well suited to accommodate these sorts
of non-standard laws, and thus there is a clear need for a realist account
which brings realism about laws up to date with developments in modern
physics. Ideally, this account will bring some conceptual clarity to the new
approaches currently emerging within physics, and will also give us the
resources to re-assess some common philosophical applications of the concept
of lawhood in the light of these scientific development.

We. argued that this account can be given by appealing to the idea that laws
are part of the objective modal structure of reality. However, existing
analyses of modal structure in terms of individual systems and/or causal
relations are not sufficiently general for this purpose, and thus we set out
to provide a more general account of modal structure. Drawing on a number of
physics examples, We proposed characterising laws in terms of probability
distributions over constraints, and demonstrated that this approach is capable
of accommodating a wide variety of possible laws. Finally, we gave some brief