# Determinism Beyond Time Evolution

Emily Adlam  The University of Western Ontario

###### Abstract

Physicists are increasingly beginning to take seriously the possibility of
laws outside the traditional time-evolution paradigm; yet many popular
definitions of determinism are still predicated on a time-evolution picture,
making them manifestly unsuited to the diverse range of research programmes in
modern physics. In this article, we use a constraint-based framework to set
out a generalization of determinism which does not presuppose temporal
evolution, distinguishing between strong, weak and delocalised holistic
determinism. We discuss some interesting consequences of these generalized
notions of determinism, and we show that this approach sheds new light on the
long-standing debate surrounding the nature of objective chance.

Physicists are increasingly beginning to take seriously the possibility of
laws which may be non-local, global, atemporal, retrocausal, or in some other
way outside the traditional time-evolution paradigm. Yet many definitions of
determinism in usage today are still predicated on a forwards time-evolution
picture, making them manifestly unsuited to the diverse range of research
programmes in modern physics. As physics begins to move beyond the time
evolution paradigm, is there still a meaningful notion of determinism as a
metaphysical property of the world to be recovered? In this article we argue
that there is: we propose a way of thinking about determinism which does not
rely on a time-evolution picture, and we explore some of the consequences of
this generalization for the the philosophy of determinism and chance.

We begin in section 1 by discussing some existing definitions of determinism.
After reviewing some of the flaws of the Laplacean definition, we assess
several existing generalisations of the notion. In particular, we note that
although the ‘region-based’ formulation used in the study of spacetime
theories does avoid building a temporal direction into the definition, this
model-theoretic formulation requires us to think of determinism purely as a
technical feature of a given theory, rather than a putative metaphysical
feature of the world. We therefore consider that there remains a need to
understand what sort of metaphysical picture could replace the temporally
directed Laplacean notion of determinism in a post time-evolution paradigm.

In section 3 we use a constraint-based framework to offer several new
definitions of determinism in modal terms, distinguishing between strong, weak
and delocalised holistic determinism, and we show that these definitions
successfully accommodate a range of cases outside the time evolution paradigm.
Then in section 4 we discuss some interesting consequences of these
generalized notions of determinism. In section 5, we show that this approach
sheds new light on the long-standing debate surrounding the nature of
objective chance, because it transpires that in a world satisfying holistic
determinism it is possible to have events which appear probabilistic from the
local point of view but which nonetheless don’t require us to invoke
‘objective chance’ from the external point of view. Finally, in section 6 we
discuss how holistic determinism relates to several other relevant research
programmes.

##  1 Definitions of Determinism

The notion of determinism was most famously articulated by Laplace, who
suggested that ‘ _we ought to regard the present state of the universe as the
effect of its antecedent state and as the cause of the state that is to
follow_.’(Laplace (1820)) That is, a theory is deterministic if according to
that theory, the present state of the universe, together with the laws of
nature, is sufficient to fix everything that happens in the future. We will
refer to this approach as ‘Laplacean determinism.’

Although the terminology has evolved somewhat, many modern definitions of
determinism are still predicated on the basic idea of the past ‘determining’
the future. For example, the model-theoretic definition of determinism, due
originally to Montague (1974), tells us that _‘a theory is deterministic if
for any two of its models describing the possible temporal evolution of a
system, coincidence of the state at one time brings with it coincidence at all
times from then on, where “coincidence” is to be spelled out in terms of a
suitable mapping.’_. Similarly, Butterfield (2005) suggests that ‘ _[A] theory
is deterministic if, and only if: for any two of its models, if they have
instantaneous slices that are isomorphic, then the corresponding final
segments are also isomorphic._ ’ And Müller and Placek (2018) provide a
careful classification of three different approaches to defining determinism:
we may require that the solution to the differential equations should always
be unique, we may require that if two linear temporal realizations can be
mapped at one time they can always be mapped at all future times, or we may
require that the theory’s models are not branching. As the authors note, all
three of their approaches are based on the core idea _‘that given the way
things are at present, there is only one possible way for the future to turn
out’_ and therefore all three approaches are still expressions of the
temporally directed picture originally associated with Laplacean determinism.

There is an obvious metaphysics underlying this temporally directed picture,
which enjoins us to envision the universe as something like a computer, taking
some initial state and evolving it forward in time to generate the content of
reality(Wharton (2015)). We can imagine that in Laplace’s time this
description may have been understood more or less literally. Indeed, as argued
by Chen and Goldstein (2021), many physicists and philosophers still appear to
take this kind of ‘dynamic production’ picture seriously - for instance,
Maudlin (2002) advocates a ‘ _metaphysical picture of the past generating the
future_.’ And of course, if one has a dynamic production picture in mind then
questions about whether or not our best current theories obey Laplacean
determinism have profound consequences for our understanding of the kind of
world we live in. In particular, proponents of dynamic production will
presumably consider that a failure of Laplacean determinism indicates that the
universe probably contains some processes which are intrinsically chancy - for
there must be some way of deciding what happens when the process of dynamic
production arrives at a point where events are not fixed by the laws and the
initial state.

###  1.1 The end of dynamic production?

But there are well-known problems for the Laplacean way of thinking about
determinism: in particular, it fails to properly disentangle the concepts of
determinism and predictability. Laplace’s original comments on the matter
conflated the two, describing determinism metaphorically in terms of the
predictive abilities of _‘an intellect which at a certain moment would know
all forces that set nature in motion, and all positions of all items of which
nature is composed,’_(Laplace (1820)) and this way of thinking is still
evident in many modern analyses of determinism (Popper (1992)). Yet at least
in theory, predictability and determinism are supposed to be quite distinct -
predictability is an epistemic matter, whereas determinism is metaphysical,
which is to say, it is supposed to capture something about the way the world
really is, independent of our ability to find out things about it. Thus for
example Clark and Butterfield (1987), considering the possibility that the
functional dependence of the future on the present state might fail to be
effectively computable, assert that _‘there is no reason at all to tie the
claim of determinism to a thesis of global predictability.’_ The Laplacean
definition doesn’t seem to do justice to this intuition.

Later work has sought to disentangle the notions of determinism and
predictability - for example, Montague (1974) argues that a purely ontological
characterisation of determinism would not necessarily imply deducibility,
because deducing the state at a time $t$ from the state at some other time
$t_{0}$ requires us to express those states in sentences, but there are only
denumerably many sentences whereas presumably there are more than denumerably
many instants, assuming that instants are in a one to one correspondence with
real numbers. And Earman (1986) writes, _‘determinism and prediction need not
work in tandem; for the evolution of the system may be such that some future
states are not predictable […] although any future complement than the one
fixed from eternity is impossible.’_ But as this quote makes clear, much of
the relevant work still seeks to understand determinism in terms of forwards
temporal evolution (albeit perhaps evolution which we ourselves can’t
predict), and thus it does not fully succeed in separating determinism and
predictability. For the notion of time evolution is closely linked to
prediction: we typically write down laws describing temporal evolution from a
present state to future ones, and non-coincidentally, we ourselves have a
strong practical interest in predicting the future from facts about the
present. But if deteminism is genuinely to be regarded as a property of the
world rather than as a function of our practical interests, there’s no reason
it should be defined in terms of (fowards) temporal evolution: from the point
of view of the universe as a whole, it need not be the case that things are
always determined by a kind of process closely modelled on the way in which in
which human observers usually want to predict them.

Of course, defining determinism in terms of temporal evolution is harmless so
long as it is taken for granted that all laws of nature take a dynamical,
time-evolution form as suggested by the dynamic production picture. But as
argued in refs Adlam (2021a); Chen and Goldstein (2021), the dynamic
production picture is increasingly under threat in contemporary physics. As a
case study, consider the procedure of Dirac quantisation (Dirac (2019))
employed in the canonical approach to quantum gravity (Kuchař (1993)). In this
procedure, we start with a Hilbert space consisting of all kinematically
possible states, and then apply the Hamiltonian and diffeomorphism constraints
to arrive at the physical Hilbert space, consisting of all those states which
satisfy the constraints. For example, the Hamiltonian constraint is encoded in
the Wheeler-deWitt equation, $H||\psi\rangle=0$, where $H$ is the universal
Hamiltonian and $||\psi\rangle$ is the universal quantum state. This equation
has the consequence that the state $||\psi\rangle$ does not evolve - that is
to say, it can’t be regarded as a state at a time, but must be understood as a
description of the whole of history at once (this is one manifestation of
quantum gravity’s famous ‘problem of time’). Suppose then that we take all of
this at face value and we want to consider the constraints as potential laws
of nature. But if we accept that there is no evolution in this picture,
clearly they cannot be _evolution_ laws. In fact what the constraints do is
narrow down the set of physical possibilities to a subset of the space we
might originally have envisioned - that is, the constraints act directly on
the space of possibilities. Moreover, clearly if we had enough constraints of
this kind, we might eventually find the set of physical possibilities narrowed
down to just a single physically possible universe. That, surely, ought to
count as a form of determinism: indeed, it would be an example of what Penrose
(2016) refers to as ‘strong determinism,’ where the laws of nature single out
the course of history uniquely. But the way in which the constraints work is
to rule out whole possibilities at once, so there is no guarantee that we will
find that given the content of some region of spacetimes we can
straightforwardly use the constraints to fill in the rest of the content of
spacetime; nor is that a natural thing to try to do in this context, since the
constraints do not treat the universe piecewise in this way. These laws, if
they are indeed laws, seem to apply to the whole of history ‘all-at-once,’ and
so we need a different way of thinking about what it would mean for such laws
to be deterministic.

Indeed, ‘all-at-once’ laws occur even in classical physics. Although it is
most common to conceptualise classical mechanics in terms of the Newtonian
schema (Smolin (2009)) 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 (Brizard (2008)): the Lagrangian approach assigns
probabilities to entire histories, rather than taking a state at a time and
evolving it forward. And path integrals - the analogue of the Lagrangian
method within quantum mechanics (Feynman et al. (2010)) - have become so
important to quantum field theory that increasingly we are seeing calls to
take the Lagrangian description more seriously (Hartle (1993); Sorkin (2007)).
As argued by Wharton (2015), taking Lagrangian methods seriously requires us
to adopt an ‘all-at-once’ approach to lawhood in which we think of these laws
applying externally and atemporally to the whole of history at once (Adlam
(2018)), so again, we are in need of a different way of thinking about
determinism in the context of these sorts of laws.

A multitude of other examples are to be found in contemporary physics.
Retrocausal approaches to the interpretation of quantum mechanics have been
attracting significant attention in recent years; see for example the two-
state vector interpretation (Aharonov and Vaidman (2002)), the transactional
interpretation (Cramer (1986)), Kent’s approach to Lorentzian beables (Kent
(2014)), Wharton’s retrocausal path integral approach (Wharton (2018)), and
Sutherland’s causally symmetric Bohmian model (Sutherland (2006)). It is clear
that these kinds of models do not fit comfortably into a metaphysical picture
of time evolution occurring in only one direction. Likewise, large sectors of
the growing field of quantum information science are concerned with
discovering, not evolution laws, but rather general constraints on what
information-processing tasks can be achieved using quantum systems. For
example, there is the ‘no-signalling principle’(Masanes et al. (2014)),
‘information causality’(Pawlowski et al. (2009)), and ‘no-cloning’(Scarani et
al. (2005)). And the constructor theory of Deutsch and Marletto (2015)
expresses physical law not in terms of time evolution, but by appeal to an
atemporal characterisation of what tasks are possible and impossible (where
_impossible_ is understood to mean, not that a process can never occur, but
that it can’t executed repeatably in a cycle with arbitrary accuracy). It is
not our intention in this article to argue that all of these examples will
definitely turn out to be amongst the fundamental laws of the universe, but we
do contend that approaches outside the standard time evolution paradigm have
become sufficiently prominent in contemporary physics that the philosophy of
physics would do well to start addressing them. And since the Laplacean
approach to determinism clearly isn’t a useful way of thinking about
constructor laws, retrocausal laws or constraints on information-processing
tasks, there is an obvious need for a generalisation of determinism which
makes sense once we start taking laws outside the time evolution paradigm
seriously.

At this point we should reinforce that it is not our intention to have an
argument over terminology - one could certainly make a case that determinism
should be defined in terms of evolution forwards in time because that is the
way the term has always been used, and proponents of this way of thinking
might then be inclined to suggest that worlds governed by laws outside the
time evolution paradigm should never qualify as deterministic. We have no
particular quarrel with this position: our intention here is simply to argue
that there is a meaningful distinction between worlds in which everything is
determined by laws which may fail to take a time evolution form, and worlds in
which some events are not determined by anything, so there is a need for
terminology which recognises that distinction. For convenience and familiarity
we will describe the former as ‘deterministic’ and the latter as
‘indeterministic,’ but readers who prefer to use the term determinism to refer
only to determination by forwards-evolving states are welcome to adopt
whatever alternative terminology they prefer.

###  1.2 Generalisations of Laplacean Determinism

One fairly obvious approach to generalising determinism involves allowing
evolution backwards as well as forwards - for example, Maar (2019)
distinguishes between systems that are ‘futuristically deterministic’ (their
evolution into the future is unique) and ‘historically deterministic’ (their
evolution into the past is unique). This is indeed an improvement, in that it
no longer insists arbitrarily that determination must occur in one particular
temporal direction. But clearly this approach is still predicated on an
_evolution_ picture, and thus it won’t work well for many of the types of laws
that we have just discussed. For example, consider a retrocausal world in
which the initial and final states of the universe together with the laws of
nature suffice to fix everything which happens over the whole course of
history: neither the initial or final state alone will fix the course of
history, so if we write down a theory of this world in a time-evolution form
it won’t show unique evolution into either the future or the past. And yet
there still seems to be a sense in which this world is not really
indeterministic - after all nothing that happens in it is random and we have
no need to invoke anything that looks like an objective chance 111Note that
here and throughout this article we will use ‘objective chance’ to refer
exclusively to chances which arise from the fundamental laws of nature, such
as the probabilities arising from the Born rule within indeterministic
interpretations of quantum mechanics - i.e. in this article ‘objective chance’
does not include higher-level emergent chances, chances derived via the method
of arbitrary functions, deterministic probabilities or anything else that
might in another context be called an objective chance.. This makes it clear
that the generalisation we’re seeking must involve a more radical conceptual
shift than simply allowing for some backwards evolution.

Another possible generalisation has been proposed by Dowe (2019) to deal with
the possibility of worlds in which the relationships between events are
temporally non-local. For example, consider a non-Markovian world which is
governed by laws such that the evolution at a given time depends not only on
the present state, but also on some facts about the past which are not
recorded in the present state. The non-Markovian world fails to satisfy any
definition of Laplacean determinism based on states or instantaneous time-
slices, since the state at a given time does not suffice to determine all
future states. But it seems odd to refer to such a world as ‘indeterministic’
- after all, the evolution at a given time is entirely determined by the past.
Similarly, consider the example of a world whose ontology contains only
pointlike events, so there are no ‘states’ in this world. Clearly in such a
world the present state of the universe can’t determine the future, since
there _is_ no present state, but nonetheless it might be the case that the
distribution of future events is fully determined by the distribution of past
events and under those circumstances one might be inclined to describe this
world as ‘deterministic’ even though it doesn’t satisfy any definition of
Laplacean determinism based on states or instantaneous time-slices,

In response to this problem, Dowe (2019) proposes several alternative
definitions of determinism - for example, his modal-nomic definition proposes
that a world $W$ is deterministic iff ‘ _for any time t and any other
physically possible world W’, if W and W’ agree up until t then they agree for
all times_.’ Both the non-Markovian world and the stateless world can
potentially be deterministic according to this definition. But Dowe’s approach
retains the temporally-directed features associated with dynamic production,
and thus it still excludes many types of non-time-evolution laws from
consideration. For example, the retrocausal world where initial and final
states fix the course of history will be judged indeterministic by Dowe’s
criterion. Similarly, a world governed by ‘all-at-once’ laws in the style of
the Wheeler DeWitt equation will typically be judged as indeterministic by
Dowe’s criterion, because in the all-at-once picture, past and future events
depend on one another mutually and reciprocally and therefore we will not
typically be able to write these dependence relations wholly in terms of
determination of the future by the past. Yet ‘all-at-once’ laws could in
principle fix the whole course of history uniquely, so it seems very
reasonable to think that we ought to have a definition of determinism which
allows retrocausal and all-at-once worlds to be deterministic.

###  1.3 Region-based definitions

The notion of determinism comes under particular pressure in the context of
General Relativity and other spacetime theories, for within such theories it
is often challenging to identify what counts as an ‘instantaneous slice’ or
even an ‘initial segment.’ Even in special relativity there is no unique
notion of a ‘state at a time,’ and when we move to more general cases we have
to deal with spacetimes which are not globally hyperbolic and thus have no
Cauchy surfaces, i.e. they contain not a single spacetime region which could
be regarded as an ‘initial slice’ from which deterministic evolution could
take place. Thus a generalisation of determinism which seems less closely
aligned with the time-evolution picture has become common in the study of
spacetime theories. Roughly speaking, this ‘region-based’ definition says a
theory is deterministic if ‘ _agreement on regions of a certain kind
(typically sandwiches or slices) forces agreement elsewhere’_ (Butterfield
(1989)). There has been some disagreement on the precise details of this
definition, but here is one example, from Butterfield (1989):

######  Definition 1.1.

A theory with models $\langle M,O_{i}\rangle$ is S-deterministic, where S is a
kind of region that occurs in manifolds of the kind occurring in the models,
iff: given any two models $\langle M,O\rangle$ and $\langle
M^{\prime},O^{\prime}\rangle$ containing regions S, S’ of kind S respectively,
and any diffeomorphism $\alpha$ from S onto S’, if
$\alpha^{*}(O_{i})=O_{i}^{\prime}$ on $\alpha(S)=S^{\prime}$, then there is an
isomorphism $\beta$ from M onto M’ that sends S onto S’, such that
$\beta^{*}(O_{i})=O_{i}^{\prime}$ throughout M and $\beta(S)=S^{\prime}$.

Note in particular that this definition of determinism is relativized to a
type of region S, so we are free to choose whatever kind of region seems most
interesting to us - perhaps Cauchy surfaces, if such things occur in the
relevant kinds of manifolds, or perhaps some other kind of structure
altogether. Since there is no longer an insistence that the determining region
should be in the past of the determined region, nor even that the determining
region should be of some particular type, this approach does indeed seem to be
moving away from the temporally directed dynamic production picture. But what
alternative picture does it have to offer? Are we supposed to imagine the
universe being provided with the content of the chosen region as an input and
then filling in the content of the rest of spacetime? This is not a very
compelling picture, not least because if the initial state is not privileged,
there is no longer any natural way to identify which particular region we
should start from: even in an extremely simple Newtonian spacetime we could
start from any time-slice and generate the rest of history by deterministic
evolution forwards and backwards, so if this is to be regarded as a literal
description we would seem to have a serious case of underdetermination. But if
we’re not supposed to think of the universe as being ‘produced’ from the
chosen region, it’s unclear what, if anything, we can conclude about reality
from the failure of this sort of determinism. Certainly it does not seem to
follow that if this sort of determinism fails for some choice of region then
the universe must contain processes which are arbitrary or chancy, because one
could always make the rejoinder that the content of reality is produced
starting from some other kind of region, or indeed reject a ‘production’
metaphysics altogether.

In fact, the literature seems to suggest that we are as far as possible to
avoid associating any metaphysics at all with this notion of determinism. For
example, Doboszewski (2019) contends that ‘ _the issue of determinism cannot
be totally decoupled from metaphysical questions … However one may
nevertheless hope to avoid making substantial use of metaphysics in an
analysis of determinism._ ’ And Müller (2013) observes that in the model-
theoretic approach to determinism, ‘ _the discussion (of determinism) shifted
from the metaphysical question to one about properties of theories._ ’ The
region-based formulation of determinism, being one of these model-theoretic
approaches, can only be understood as a property of a given theory; within the
model-theoretic paradigm we are actively discouraged from thinking about
determinism as a potential property of the world. But if this notion of
determinism is not to be regarded as having metaphysical content, what exactly
is its purpose? Clearly it is not to be regarded as a practical
characterisation of the predictions we can actually make with the given
theory: for as we have seen, determinism does not entail predictability, and
moreover even in cases where there _would_ be predictability in principle, the
definition offers no reason to think that the region S will be one from which
we could reasonably collect all the relevant data. Indeed, Doboszewski (2019)
observes that if the region S is supposed to be a Cauchy surface, we will not
even be able to verify the existence of the right type of region in our actual
spacetime. Thus it is somewhat unclear why this particular technical notion of
determinism should be regarded as an interesting property of a theory;
technical features of theories are typically interesting insofar as they are
presumed to reflect some fact about the reality to which the theory
corresponds, and/or some practical feature of how the theory works in
application, neither of which obviously holds here.

One might hope to look to the actual practice of physics to understand the
utility of the region-based definition. But the main application of the term
‘determinism’ in the General Relativity community appears to be in the context
of the initial-value problem, i.e. the problem of determining the course of
history given the data on some time-slice $\Sigma$. If the data on $\Sigma$
satisfies certain constraints, then it can be embedded in a unique spacetime
which is a solution of Einstein’s equations; moreover the spacetime will be
globally hyperbolic and $\Sigma$ will be a Cauchy surface of it (Choquet-
Bruhat and Geroch (1969)). This is important to physicists because under these
conditions it’s possible to come up with a ‘time-evolution’ version of General
Relativity, and also to formulate canonical quantum gravity (Kuchař (1993)),
which is intended to furnish a description of the time evolution of the
quantum state of the $3$-metric (although as we noted earlier, due to the
famous ‘problem of time’ it’s not entirely clear that the evolution picture
succeeds). That is to say, the notion of ‘determinism’ employed by physicists
working on General Relativity remains quite closely tied to Laplacean
determinism and the time evolution paradigm - and thus, perhaps, involves an
unarticulated commitment to a metaphysics of dynamic production. Therefore
it’s unclear that the justification for this region-based generalisation of
determinism can come from within mainstream physics practice.

Moreover, even if there _are_ useful applications for a definition of
determinism which is shorn of metaphysical content, we contend that there
would also be many useful applications for a definition of determinism which
_does_ aim to characterise a metaphysically interesting feature of reality.
For as Müller (2013) puts, it, ‘ _The question of whether the world is
deterministic or not, which is perceived by many to have enormous consequences
for our understanding of ourselves as free agents, is also first and foremost
a metaphysical one … determinism and indeterminism are metaphysical notions._
’ Determinism is frequently invoked in explicitly metaphysical discussions of
topics like explanation, free will and probability (we will see some examples
in section 4) and it seems unlikely that a definition of determinism which has
been deliberately uncoupled from metaphysical commitments will be particularly
useful in these applications. Furthermore, moving away from the time evolution
paradigm has interesting consequences for a range of important debates within
both physics and philosophy, and in order to fully appreciate these
consequences it will be necessary to have some idea of what we might be moving
_towards_ \- that is, we must ask what alternative sort of metaphysical
picture might replace the time evolution picture. Understanding the role of
determinism in a post-time-evolution setting is surely an important part of
that project, and for that we will need something more than the region-based
formulation.

In addition to these concerns, the region-based definition of determinism,
even considered as a purely technical criterion, becomes difficult to apply in
many physically relevant contexts. For example, Doboszewski (2019) points out
that when applying this definition to General Relativistic models, the verdict
we reach depends sensitively on our choice of the type of region S:
Doboszewski identifies ten types of cases which intuitively seem like they
should perhaps be regarded as violations of determinism, and observes that
there is no particular causal structure that they have in common, so there is
no obvious way of identifying a specific type of region which will get the
intuitively ‘right’ verdict in all ten cases. Doboszewski concludes that we
should simply adopt a pluralistic approach to determinism, but alternatively
one might regard the fact that the region-based definition is so sensitive to
a choice of region as evidence that it is perhaps not getting at any
particularly interesting or relevant feature of these theories. Even more
serious problems are likely to arise in the context of quantum gravity: for
many approaches to quantum gravity treat spacetime itself as emergent from
some underlying reality(Rovelli and Vidotto (2015); Blau and Theisen (2009)),
and therefore we will not even be able to identify ‘regions’ in the first
place, so the definition of determinism will be applicable at best at an
emergent level. This seems troubling - do we really believe that it is
conceptually impossible for a theory of quantum gravity to be deterministic in
a fundamental sense?

It’s also clear that the region-based formulation will not work well in the
context of many approaches outside the time evolution paradigm. For example,
consider the retrocausal model where the content of reality is completely
fixed by an initial and final boundary condition together, but not by either
of them individually. The region-based definition of determinism will allow us
to regard this theory as deterministic only if we are willing to accept a
‘region’ made up of two disconnected parts, which seems to be stretching the
definition of the term ‘region’ quite far: and in any case, even if we are
willing to accept the union of two disconnected regions as a region, one might
well feel that a useful definition of determinism should not make questions
about whether a possible world is deterministic depend on the essentially
linguistic question of what we’re willing to count as a ‘region.’ One possible
response to these troubles, as advocated by Earman (2007), is to adopt a form
of quietism, i.e. say that the notion of determinism may simply be
inapplicable in many cases that interest us in modern physics. But the
alternative is to say that the ‘region-based’ approach is perhaps not the
right generalisation of the original idea of Laplacean determinism, or at
least that the generalisation we propose in this article may be more
appropriate in some cases - particularly cases where we are concerned not only
with technical criteria but with metaphysical content.

##  2 The Constraint Framework

Because we are concerned here to analyse determinism as a metaphysical
property of worlds, rather than a formal property of theories, we will follow
Müller (2013) in eschewing model-theoretic language and instead characterising
the notion in modal terms, since determinism and objective chance are patently
modal notions: _‘They are not about what does or doesn’t happen, but about
what can or what has to happen and are thus built upon the modal notions of
possibility and necessity.’_( Müller (2013) ) Our approach to determinism will
therefore be predicated on a fairly robust approach to modality - we will
assume that for every possible world there exists some well-defined modal
structure which we can invoke to determine whether or not that world is
deterministic. We will not comment further on the nature of that modal
structure - it might be defined by the axioms of the systematization of that
world’s Humean mosaic which is robustly better than all other systematizations
(Lewis (1994)), or it could result from governing laws within any one of the
realist approaches to lawhood (Armstrong (1983); Bird (2005); Ellis (2001)),
or it could be understood in terms of Lewis’ modal realism (Lewis (1986)), or
it could be something else altogether.

Objective modal structure is most commonly discussed in the context of ontic
structural realism (Ladyman and Ross (2007)). In some cases this modal
structure is characterised in terms of _causal_ structure (Berenstain and
Ladyman (2012); Esfeld (2009)), but as argued in Adlam (2021a), due to the
asymmetrical nature of the standard notion of causation this approach doesn’t
work well when we’re dealing with possibilities outside the time evolution
paradigm, so in this paper we will instead adopt the more general approach to
modal structure advocated by Ladyman and Ross (2007). In order to formalise
this notion of modal structure, we will employ the framework developed in
Adlam (2021a), which characterises laws of nature in modal terms as
_constraints_. This approach is inspired by the increasing prominence of
constraint-based laws in physics, as briefly surveyed in section 1.1; a very
similar account of lawhood in terms of constraints was recently given by Chen
and Goldstein (2021).

To formalise the notion of a constraint we make use of the notion of a Humean
mosaic, which is understood to consist of all and only the matters of local,
non-modal particular fact in a given world. Obviously what counts as a matter
of non-modal particular fact will vary according to one’s other metaphysical
commitments, and indeed this is our purpose in employing the general term
‘Humean mosaic’ - our aim is to offer a definition of determinism which is
compatible with a variety of accounts of the non-modal content of reality. We
emphasize that the use of this terminology is not intended to reflect a
commitment to the standard Humean ontology consisting only of the Humean
mosaic - we use the phrase ‘Humean mosaic’ to refer to all of the actual, non-
modal content of reality, but we do not intend by that usage to rule out the
possibility that reality also has genuine modal content: for example, a
realist about modality might wish to maintain that constraints are
ontologically prior to the actual Humean mosaic and thus play a role in
determining its content.

Because we don’t wish to make our approach to determinism contingent on any
particular account of modality, 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. 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.

Using these definitions, we postulate that every world has some set of laws of
nature which are an objective fact about the modal structure of that world,
and we characterise these laws in terms of _probability distributions over
constraints_. Within this picture, we can imagine that the laws of nature
operate as follows: first, for each law a constraint is drawn according to the
associated probability distribution, and then the constraints _govern_ by
singling out a set of mosaics from which the Humean mosaic of the actual world
must be drawn - i.e. the actual mosaic must be in the intersection of all the
chosen constraints. That is to say, the laws of nature associated with a given
world are understood to operate by narrowing down the set of physical
possibilities for that world, thus dictating what properties the Humean mosaic
for that world is required to have. We note that we could have given a similar
definition in terms of probability distributions over mosaics rather than
constraints, but we have chosen to use constraints here in order to
acknowledge that a law may require (deterministically or probabilistically)
that the actual Humean mosaic belongs to a given set of mosaics, 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. This point will be a crucial feature of our approach to
determinism.

As shown in Adlam (2021a), a very large class of possible laws can be written
in this framework. The Wheeler deWitt equation can be analysed as a law
assigning probability $1$ to the constraint consisting of all Humean mosaics
instantiating a universal quantum state $||\psi\rangle$ satisfying
$H||\psi\rangle=0$. A prohibition on superluminal signalling can be analysed
as a law assigning probability $1$ to the constraint consisting of all the
Humean mosaics in which no superluminal signalling occurs, and $0$ to all
other constraints. A constructor law can be analysed as a law assigning
probability $\epsilon$ to the set of all mosaics in which some ‘impossible’
process occurs in a repeated cycle, where $\epsilon$ is very close to zero.
And given a probabilistic evolution equation, we can use the equation to
construct a probability distribution $p_{s}(h)$ over each possible history $h$
of a system $s$, and then analyse the equation as a law which assigns
probability $p_{s}(h)$ to the constraint consisting of the set of all mosaics
in which system $s$ has history $h$. Thus we can use these probability
distributions over constraints to characterise the laws of nature without
needing to be too specific about what sorts of laws we mean, and this makes it
possible for us to give a general definition of determinism which is
applicable across many different conceptions of the content of physical law.

We acknowledge that the idea of a constraint being selected according to some
probability distribution may give rise to some scepticism, given that this
process presumably only happens once and therefore no relative frequency
interpretation is possible. Indeed, we share that scepticism, and would be
inclined to adduce the difficulty of making sense of these sorts of
probability distributions as a reason to believe that the world is likely to
be deterministic in at least one of the senses described below. But we would
argue that any conception of the content of physical reality which holds that
there exist genuine objective chances is in effect committed to a probability
distribution over Humean mosaics, and therefore if we are to articulate the
notion of determinism it is necessary to make reference to the possibility of
probability distributions of this kind in order to characterise the relevant
contrast class - whether or not that possibility is actually a coherent one is
a further question, and one which we will not have space to address here.

##  3 Holistic Determinism

Now our task is to find a way of decoupling the notion of determinism from
time evolution. One obvious way to achieve this would be to appeal to the
intuitive idea that determinism and objective chance are mutually exclusive:
that is, either it is the case that the fundamental laws of nature are
deterministic, or it is the case that they involve some sort of ‘intrinsic
randomness.’ This line of thought leads to the idea that we should simply
define determinism as the absence of objective chance. But the problem with
this route is that we don’t really have a strong grasp on what objective
chances are either - the only generally agreed-upon characterisation is that
they should obey the Principal Principle (Lewis (1980)).

The constraint framework provides a precise way of formulating this idea.
Indeed, it allows us to disambiguate several different ways in which a world
may be deterministic. First, we will say a world (which, recall, can be
associated with some set of laws of nature) obeys _holistic determinism_ iff
the probability distributions induced by the laws are trivial:

######  Definition 3.1.

A world satisfies _holistic determinism_ iff every one of its fundamental laws
induces a probability distribution which assigns probability $1$ to a single
constraint and zero to all disjoint constraints.

When this condition is satisfied, we can get rid of the probability
distributions altogether and simply assign to each law of nature a constraint,
so the actual mosaic must lie in the intersection of all of these constraints.
This is the analogue in the constraint framework of the idea that determinism
is associated with the absence of objective chance: ‘no objective chance’ is
translated as ‘there are no objective probability distributions over
constraints.’ Of course the actual mosaic must still be selected from the
intersection, but this selection does not need to involve any objective
chances, since as noted in section 2 the laws do not define any particular
probability distribution over the mosaics in the intersection, not even the
uniform distribution.

However, it might be argued that defining determinism as the absence of
objective chance is in some cases too broad. Consider for example a version of
classical electromagnetism (Rousseaux (2005)) which includes the stipulation
that the electromagnetic potentials are real fields which are a part of nature
(and thus they feature in the Humean mosaic) - that is, different
configurations of the potentials represent physically inequivalent situations.
Intuitively, it seems clear that this world is not deterministic, since a
large variety of physically inequivalent configurations of the electromagnetic
potentials will be compatible with the laws in any given situation. But the
theory defines no probability distribution over the different configurations
of the electromagnetic potentials, so as far as we know there aren’t really
any ‘chances’ here, not even the uniform distribution - some configuration
must occur, and so some configuration does, but there is nothing that can be
said about how likely different configurations are to occur. We will
henceforth refer to cases of this kind, involving events which are not
determined by anything but which also are not chancy events, as ‘arbitrary.’
Thus it seems that a generalized definition of determinism should allow that
worlds containing arbitrarinesss of this kind may be indeterministic even in
the absence of objective chance.

In light of this possibility, we distinguish further between strong and weak
holistic determinism:

######  Definition 3.2.

A world satisfies _strong holistic determinism_ iff it satisfies holistic
determinism and there is only one mosaic in the intersection of the set of
constraints associated with its laws of nature

######  Definition 3.3.

A world satisfies _weak holistic determinism_ iff it satisfies holistic
determinism and there is more than one mosaic in the intersection of the set
of constraints associated with the laws of nature

In the case of strong holistic determinism the actual mosaic is singled out
uniquely by the laws and we have neither chance nor arbitrariness; whereas
weak holistic determinism allows arbitrariness, but not chance. Thus classical
electromagnetism with ontological electromagnetic potentials will satisfy weak
holistic determinism but not strong holistic determinism. One might perhaps
question whether ‘weak holistic determinism,’ really ought to count as a form
of determinism, given that it allows events to occur which are not determined
by anything. However, even worlds satisfying Laplacean determinism exhibit
some arbitrariness - typically in such a world the initial state of the
universe is ‘arbitrary’ in just the same way as the configuration of the
electromagnetic potential, since we don’t assign objective chance
distributions over initial states but we don’t usually take them to be
determined by anything either. So when we set out to generalize determinism in
a way that does not give any special status to the initial state of the
universe, we have two options - either we allow arbitrariness elsewhere as
well, or we allow no arbitrariness whatsoever. Weak holistic determinism takes
the former route and strong holistic determinism the latter, so both have at
least some claim to be the spiritual heir of Laplacean determinism.

That said, there does seem to be a sense in which a world satisfying Laplacean
determinism is more strongly deterministic than electromagnetism with
ontological electromagnetic potentials, since the arbitrariness involved in
the former is limited to the start of time, whereas in the latter, even if we
know the full content of spacetime except for some small region, we still
won’t be able to determine the electromagnetic potentials in that region
(since it is always possible to perform a gauge transformation on the
potentials which goes to zero outside the region). So ideally we would like to
offer some further definition which distinguishes between worlds satisfying
Laplacean determinism and worlds like the one governed by electromagnetism
with ontological electromagnetic potentials. Of course, we could add a special
exception for the initial state into our definition of determinism, but this
would seem like a relic of the older temporally directed approach - if we’re
no longer insisting that laws must take the forward time-evolution form
there’s no good reason to treat the initial state as special. So instead we
suggest the following definition:

######  Definition 3.4.

A world associated with a set of laws of nature satisfies _delocalised
holistic determinism_ iff it satisfies holistic determinism, and there is no
pair of mosaics in the intersection of the set of constraints associated with
its laws of nature which are identical everywhere except on some small
subregion of spacetime

Note that this definition assumes the existence of a natural map between the
two spacetimes associated with two different mosaics. Of course in the case
where two mosaics are associated with spacetimes having very different
structure it may not be the case that there exists a unique natural map
between their spacetimes (although perhaps such a thing could be arrived at
using the best-matching techniques of Barbour (2001)). However, in that case
clearly we would not want to say that the two mosaics are identical everywhere
except on some small subregion of spacetime, so there is no need to invoke a
map; this definition becomes relevant only when we are dealing mosaics with
spacetimes similar enough to support a natural map between the spacetimes,
except possibly on some small subregion.

The point of this definition is that it ensures the arbitrariness involved in
selecting the actual mosaic can’t be localised in any particular region, which
means we can’t have any indeterministic ‘holes.’ To see how the definition
works, consider a world satisfying Laplacean determinism in which the laws of
nature are time-reversal invariant. In the constraint framework, this world
has a set of laws of nature which are each associated with a single
constraint, and the intersection of these constraints is a set of mosaics such
that each mosaic in the set has a different initial condition. Moreover,
because the laws of nature of this world are deterministic and time-reversal
invariant, it must be the case that no pair of mosaics in the set are the same
on any time-slice after the beginning of time, since otherwise reversing the
direction of time would take a single state into two different states, in
violation of the assumption that the laws of this world satisfy Laplacean
determinism. Thus when the actual mosaic is drawn from this set of mosaics,
this has the effect of fixing the state of the world on the initial state, but
since the mosaics differ at all subsequent times as well, one could equally
well regard this process as fixing the state of the world on any other time
slice. So there is no fact of the matter about which _particular_ time-slice
is responsible for the rest of history: when we select one of the allowed
mosaics we determine all the time-slices at once. Therefore the arbitrariness
involved in a universe satisfying Laplacean determinism can’t be located in
any specific spacetime region, so the Laplacean universe does indeed satisfy
weak delocalised holistic determinism.

On the other hand, a world governed by electromagnetism with ontological
electromagnetic potentials will not satisfy delocalised holistic determinism,
because given some mosaic in the intersection of the constraints, we can
generate another mosaic which is identical except on some small region of
spacetime by simply performing a gauge transformation which goes to zero
everywhere outside the region in question, and since the laws don’t assign any
probabilities over gauge-related configurations, this other mosaic must also
lie in the intersection of the constraints. Thus this definition captures what
is distinctive about worlds satisfying Laplacean determinism without needing
to give special significance to the initial condition - the important thing
about such worlds is not that the arbitrariness occurs only at the start of
time, but that the arbitrariness can never be localised anywhere, and thus in
general such worlds have a high degree of predictability and consistency even
though the laws of nature do not fix everything in the world uniquely.

Evidently all of these definitions are suitable for studying worlds with laws
outside time evolution paradigm: worlds with non-Markovian laws, worlds with
an ontology that does not include states, worlds with all-at-once laws and
worlds with retrocausal laws are all capable of satisfying weak, delocalised
and/or strong holistic determinism, depending on the specific details of the
laws in question. The definitions encapsulate quite a different way of
thinking about what it is for a world to be deterministic: rather than
treating the initial conditions (or the conditions of some spacetime region)
as an input and asking if the input is sufficient to determine the rest of the
course of history, we have the laws induce constraints which select entire
mosaics at once, and then we ask if these laws involve any objective chances
and if they suffice to pick out the course of history uniquely. Thus these
definitions also realise our desideratum of making a clean distinction between
determinism and predictability, as they characterise determinism entirely in
terms of objective modal structure, without explicitly or implicitly leaning
on facts about the practical interests and epistemic limitations of human
observers. A world which is deterministic in the holistic sense may indeed
exhibit a high degree of predictability, but it also may not - the way in
which the laws determine the course of events may be highly non-local in space
and time, meaning that it may be impossible for limited local agents to gather
enough data to see the whole picture. Thus this framework fulfils the mandate
of providing a precise way of thinking about determinism as a metaphysical
property of a world rather than a formal property of a theory, thus opening
the door for that notion to be applied to a variety of ongoing metaphysical
discussions.

###  3.1 Other Comments

  1. 1.

The distinction we have made here is similar to one employed in Penrose
(2016), which distinguishes between determinism, i.e. what we have referred to
as Laplacean determinism (_‘if the state of the system is known at any one
time, then it is completely fixed at all later (or indeed earlier) times by
the equations of the theory’_) and strong determinism, i.e. what we have
referred to as strong holistic determinism (_‘ it is not just a matter of the
future being determined by the past; the entire history of the universe is
fixed, according to some precise mathematical scheme, for all time.’_).
However, Penrose’s scheme does not seem to accommodate the intermediate
possibility of worlds which satisfy weak holistic determinism but not
Laplacean determinism, such as worlds which have some degree of arbitrariness
at some point other than the initial state of the universe but nonetheless
have no objectively chancy events. Moreover, Penrose is rather vague about
what it would take for the entire history of the universe to be fixed, but the
constraint framework allows us to be precise: the history of the universe is
fixed if and only the laws of nature induce trivial probability distributions
over constraints, and the intersection of the constraints associated with
these laws contains exactly one mosaic.

  2. 2.

In one sense, weak holistic determinism, strong holistic determinism and
delocalised holistic determinism are all weaker than Laplacean determinism,
since worlds with laws outside the time evolution paradigm like those
discussed in section 1.1 could satisfy any of these forms of holistic
determinism without satisfying Laplacean determinism. However, in another
sense strong holistic determinism is much stronger than Laplacean determinism,
because Laplacean determinism allows for the existence of arbitrariness (in
the selection of initial conditions and the selection of parameter values)
while strong holistic determinism insists that everything is fixed once and
for all by the laws of nature.

  3. 3.

We reinforce that all of these definitions for determinism depends crucially
on the claim that reality has a unique, observer-independent modal structure.
Given any Humean mosaic we could come up with an infinite number of sets of
constraints whose intersection contains only this mosaic - trivially, for a
mosaic $A$ we can pick two random mosaics $B$ and $C$ and choose the
constraints $\\{A,B\\}$ and $\\{A,C\\}$. So simply finding such a set which is
consistent with our observations doesn’t tell us that the world must be
deterministic: we must make the further claim that these constraints are _in
fact_ the constraints which feature in the objective modal structure of
reality, and therefore we must believe that there is some fact of the matter
about what this objective modal structure really is. So for example, assessing
a claim that the world satisfies strong holistic determinism has several
steps: first we must verify that the proposed constraints do indeed single out
a unique Humean mosaic, second, we must verify that the proposed constraints
are consistent with our observations, and third, we must decide whether the
proposed constraints seem like plausible candidates for laws. This last part
of the process will naturally be shaped by whatever expectations we have about
laws - for example, if we think the laws of nature are simple then we will
tend to expect that the constraints induced by the laws can be expressed in a
closed form in simple terms.

  4. 4.

Our definitions also depend crucially on the stipulation that the laws of
nature assign no probability distribution over the mosaics in the constraints
that they induce, nor over the mosaics in the intersection of all of those
constraints. However, this certainly does not entail that we ourselves may not
assign any distribution over mosaics in the intersection. For we may have
Bayesian priors which lead us to assign some distribution over mosaics before
taking any of the laws of nature into account - for example we might simply
assign the uniform distribution, or we might choose a distribution which
favours simpler mosaics. Thus when we learn about a constraint induced by a
law, we will update our beliefs by eliminating all mosaics not consistent with
the corresponding constraint and renormalizing our credences, so after
conditioning on all of the laws we will be left with a normalized probability
distribution which is non-zero only on mosaics in the intersection of the
constraints. In the case of strong holistic determinism, no matter what priors
we start out with, after conditioning on the laws we will be left with a
distribution that assigns probability $1$ to a single mosaic; but in the case
of weak holistic determinism we will in general end up with a non-trivial
distribution, which will be the uniform distribution if the original prior
distribution was the uniform distribution, but which may be non-uniform if our
original priors were non-uniform. But this distribution is a _subjective_
probability distribution derived from our original subjective priors, not an
objective chance distribution arising from the laws themselves: so in this
case we still have determinism in the sense of ‘the absence of objective
chance,’ even though observers may still assign non-trivial probabilities over
mosaics compatible with the laws. This is precisely analogous to the way in
which we might assign subjective probability distributions over initial
conditions in the context of Laplacean determinism, for example in analyses of
statistical mechanical systems based on the principle of indifference (Meacham
(2010)).

###  3.2 Humean Mosaics

None of our definitions are intended as a direct replacement of model-
theoretic definitions of determinism, because our approach does not simply
take a theory and decide algorithmically whether that theory is deterministic.
For a start, the definition applies to worlds, not theories, so to apply it to
a theory (expressed either as a set of models or a set of sentences) we would
have to postulate a world governed by that theory, which would require us to
translate the theory into putative ‘laws of nature’ in the constraint
formulation, and typically there will be more than one way of doing that. We
would also have to decide whether we need to add additional laws of nature
from outside the theory, since our definitions require us to consider the
complete set of laws of nature associated with a given world. So in
circumstances where one wants nothing more than a simple judgement as to
whether some particular _theory_ is deterministic, the model-theoretic
approach is obviously more suitable. This is not a weakness of our approach,
because it was never our intention to define determinism as a formal property
of theories. It would be interesting in future work to develop formal
expressions of this approach to determinism which can be applied within a
model-theoretic framework, but that is beyond the scope of the present
project.

Applying our definitions to a specific theory would also require a judgement
about what sorts of categorical properties feature in the set of Humean
mosaics associated with the theory. For example, consider quantum mechanics:
if we suppose that quantum mechanics is a complete theory of reality and we
take it that wavefunction collapse is a part of the Humean mosaic, then the
theory does not satisfy weak or strong holistic determinism, whereas if we say
that only the unitarily evolving quantum state is part of the Humean mosaic,
then we get the Everett interpretation (Wallace (2003)), which satisfies weak
holistic determinism. Note that the Everett interpretation in its usual form
does not satisfy strong holistic determinism, since the initial state of the
universe is arbitrary; however, the Everett interpretation coupled with
something like Chen’s ‘quantum Wentaculus,’ which features a lawlike initial
state (Chen (2018)), could potentially satisfy strong holistic determinism.

One might worry that this ambiguity is a problem for our approach. However, it
is in fact a deliberate choice, because our intention is to define in a
general, metaphysical sense what it means for a world to be deterministic, and
therefore it is important to avoid predicating the definition on specific
ontological assumptions. Moreover, this definitional quirk is not unique to
our approach - as emphasized by Earman (2004), judgements about whether a
theory satisfies Laplacean determinism likewise depend on our presuppositions
about the possibility space that we’re dealing with. Thus, for example,
Laplacean determinism is similarly noncommittal about quantum mechanics.
Moreover, this ambiguity has not prevented Laplacean determinism from yielding
generally meaningful results in many cases, although of course there are a few
interesting grey areas and ongoing debates (e.g. see Earman (2004); Santo
(2021)). We are happy for our definition to inherit this particular form of
ambiguity - it would not make sense for our definition of determinism to be
stricter than Laplacean determinism in this regard, since our intention here
was to _generalize_ Laplacean determinism and therefore it is important for us
to be able to recognise worlds satisfying Laplacean determinism as a subclass
of worlds that are deterministic in our sense.

###  3.3 Examples: Spacetime theories

In order to show that the proposed approach to determinism captures some of
the intuitive content that is not well addressed by the region-based
formulation, we discuss how some of the problem cases cited by Doboszewski
(2019) can be accommodated by the constraint approach.

#### Spacetimes violating weak causality conditions

Spacetimes violating weak causality conditions do not allow for an initial
value problem or anything remotely like one. This means we can’t easily apply
a region-based definition of determinism, or any other approach which asks us
to take some physical data and construct the rest of the solution from it.
However, since constraints pick out whole solutions at once, we can certainly
have constraints which contain Humean mosaics corresponding to spacetimes
which violate weak causality conditions, and therefore we can certainly
postulate laws of nature which give rise to such spacetimes, and then check if
those laws exhibit one or more of the forms of holistic determinisms we have
postulated. It is clear that there do exist laws of nature which will produce
such spacetimes, because the Einstein equations can produce them in special
cases; and since the Einstein equations in their standard form are ‘all-at-
once’ laws whose solution is an entire course of history rather than a state
at a time, they are exactly the kind of law for which the constraint
formulation is particularly well-suited.

#### Spacetimes with closed timelike curves

Adlam (2018, 2021b) studied the compatibility of ‘closed loops’ with a concept
of ‘global determinism’ which is similar to the notion of holistic determinism
that we have used in this article. Adlam (2018, 2021b) argues that the values
of variables in these closed loops ‘come out of nowhere,’ which is to say,
they are not determined by anything, so closed causal loops are not consistent
with a universe satisfying holistic determinism. Using the definitions set out
in section 3, we can make this observation more precise. First, clearly
variables which are not determined by anything are incompatible with a
universe that obeys strong holistic determinism, so the argument of Adlam
(2018, 2021b) work perfectly well if ‘global determinism’ is intended to mean
‘strong holistic determinism.’ Moreover, indeterministic closed loops are also
ruled out by delocalised weak holistic determinism, because the arbitrariness
associated with the variables in a closed causal loop is localised to the
events inside the loop. On the other hand indeterministic closed loops are not
ruled out by weak holistic determinism, since the values of variables inside
the loop could be arbitrary rather than chancy.

#### Extendible Spacetimes

An extendible spacetime is a spacetime which is a subset of a larger
spacetime(Minguzzi (2012)). Suppose that a given world has a set of laws such
that the intersection of the constraints associated with the laws contains a
mosaic corresponding to an extendible spacetime. Then, assuming that there is
no law specifically prohibiting extensions, it necessarily follows that the
intersection of the constraints associated with the laws will contain more
than one mosaic, since we can take the mosaic in this intersection which has
an extendible spacetime and obtain another mosaic which will also be in the
intersection by extending it. Therefore a world with laws allowing extendible
mosaics will typically fail to satisfy strong holistic determinism. Moreover,
the extended mosaic will differ from the original mosaic only on a small
subregion of spacetime, in the sense that the new mosaic postulates a small
region of spacetime that does not exist at all in the original mosaic, and
therefore a world with laws allowing extendible mosaics will typically fail to
satisfy delocalised holistic determinism. However a world with such laws may
satisfy weak holistic determinism, since the laws need not define any
probability distribution over the extent of spacetime.

#### Spacetimes with holes

A spacetime is hole-free if ‘ _the Cauchy development of any spacelike surface
is “as large as it can be” ’_(Manchak (2008))- i.e. it is not the case that
the domain of dependence of some partial Cauchy hypersurface can be extended
by imbedding it into a different spacetime (Minguzzi (2012)). Suppose that a
given world has a set of laws such that the intersection of the constraints
associated with the law contains a mosaic corresponding to a spacetime which
is not hole-free. Then, assuming that there is no law specifically prohibiting
extending the domain of dependence of partial Cauchy hypersurfaces, it follows
that the intersection of the constraints associated with the laws will contain
more than one mosaic, since since we can take the mosaic in this intersection
which has an extendible spacetime and obtain another mosaic which will also be
in the intersection by extending the domain of dependence of some partial
Cauchy hypersurface. So a world with laws allowing spacetimes that are not
hole-free will typically fail to satisfy strong holistic determinism.
Moreover, the extended mosaic will differ from the original mosaic only on a
small subregion of spacetime, in the sense that the new mosaic postulates a
small region of spacetime that does not exist at all in the original mosaic,
and therefore such a world will typically not be compatible with delocalised
holistic determinism either. However, a world with such laws may satisfy weak
holistic determinism, since the laws need not define any probability
distribution over the size of the domain of dependence.

##  4 Applications

We have criticized Laplacean determinism for being inappropriately intertwined
with our practical interests (in particular, our desire to predict the future
based on facts about the present); but one might argue that being aligned with
our practical interests is precisely what made Laplacian determinism a useful
concept, and thus one might worry that generalizing determinism as we have
done here serves only to make the concept less useful. Thus in this section we
will explain how this generalization of determinism may have important
consequences for scientific and philosophical thinking. The concept of
Laplacean determinism has played a range of different intellectual roles in
science and philosophy, so we will proceed by examining how a generalised
notion of determinism may contribute to some of these domains of application.

###  4.1 Scientific Explanation

Determinism has played an important role in shaping our expectations for
scientific explanation - for a long time it was standard in science to explain
events by showing how they could be deterministically produced by past
conditions. This is formalised in the ‘deductive-nomological’ model for
scientific explanation (Popper (1959); Hempel and Oppenheim (1948)), which
posits that a valid explanation should be composed of a set of sentences from
which the explanandum logically follows, where at least one of these sentences
must express a law of nature and that law must be an essential element in the
deduction of the explanandum. Most common examples of DN-explanations involve
time-evolution laws or at least temporally directed laws, and thus DN
explanations usually involve postulating some past conditions and showing that
the deterministic forwards time-evolution of these conditions gives rise to
the explanandum. This model is sometimes relaxed to allow for explanations
involving statistical inference rather than exact logical deduction (Hempel et
al. (1965)), but deductive-statistical and inductive-statistical explanations
are simply a generalisation of forwards-evolving deterministic explanations
and thus they typically preserve the expectation that the future will be
explained by lawlike evolution from the past.

Holistic determinism opens up new possibilities for scientific explanation -
for if the definition of Laplacean determinism is largely a function of our
practical interests then there is no reason to expect that all valid
scientific explanations will be predicated on forwards time evolution, since
for scientific realists one of the main purpose of scientific explanation is
to achieve _understanding_ , to which practical concerns are only tangentially
relevant. So when we come upon phenomena for which there seems to be no
satisfactory explanation within a forward time-evolution picture, we should be
open to the possibility of explanations based on global laws using the
constraint framework. A good example of this is the past hypothesis: by
definition nothing evolves into the initial state of the universe, so the
initial state can’t be given a lawlike explanation if we assume that laws
always take the forward time-evolution form. But the past hypothesis can
straightforwardly be written as a constraint - it is simply the set of all
Humean mosaics in which the arrangement of local matters of particular fact at
one end of the mosaic has low entropy (or some other more sophisticated
characterisation of the desired initial state - see Wallace (2011)). The past
hypothesis can therefore be explained in this framework by hypothesizing that
there is some law of nature which induces this particular constraint - perhaps
something like the ‘Mentaculus’ of Loewer (2020) or the quantum ‘Wentaculus’
of Chen (2018)) - thus ensuring that the initial state will necessarily be a
low entropy state whilst the selection of one low entropy initial state in
particular remains arbitrary.

Similar arguments can be made about parameter values: for example,
cosmologists are currently much exercised over the problem of why the
cosmological constant is so much smaller than their models suggest it ought to
be (Weinberg (1989)). It’s hard to see how this feature could be explained
within the kinematical/dynamical picture, because the cosmological constant is
usually considered to be a fundamental constant of nature which has always had
the value that it has, and so there is no option to say that it has been
produced by evolution from some earlier conditions. Thus most current
approaches to explaining the value involve fairly exotic explanatory
strategies, such as anthropic arguments in the context of a putative
multiverse (Banks et al. (2001)). However, within the constraint framework
there is no reason why we can’t simply explain the cosmological constant in a
lawlike manner: all we need to do is suggest that there is some law of nature
which induces the constraint consisting of all the Humean mosaics in which the
cosmological constant is very small. In the constraint framework, this
nomological explanation has exactly the same status as more familiar
nomological explanations, such as explaining that apples fall due to the law
of gravity: the key point is that nomological explanations do not necessarily
have to have a temporal character.

Of course, some caution is required with this strategy. Seeking explanations
for phenomena that we find surprising or conspiratorial is frequently a good
way of making scientific progress, so we don’t want to make explanation too
cheap: answering every scientific question with ‘Because there’s a law which
makes it so,’ isn’t likely to lead to any new insights. But of course, the
same is true with standard DN and inductive-statistical explanation - not
every proposed DN or inductive-statistical explanation is interesting and
informative, so we have a variety of criteria which can be applied to judge
which explanations have merit. These criteria, including simplicity,
unification, the absence of fine-tuning and so on, can equally well be applied
to constraint-based explanations. Thus rather than simply imposing a
constraint consisting of Humean mosaics in which the initial state is simple
or a constraint consisting of Humean mosaics in which the cosmological
constant is small, we might want to come up with some more general feature
from which these constraints could be derived - for example, one could imagine
deriving the past hypothesis from a more general constraint that singles out
Humean mosaics which are sufficiently interesting or varied. This constraint
rules out mosaics where everything is in thermal equilibrium throughout the
whole of history, which does indeed entail that the initial state must have
low entropy, but it may also have _other_ interesting consequences, and if the
consequences turn out to be powerful enough we would have good reason to
accept the proffered explanation. The overarching point is that holistic
determinism offers us a new sort of explanatory framework: there can be good
and bad explanations within that framework, and there’s work to be done to
establish the appropriate criteria for judging these sorts of explanations,
but nonetheless this new approach is a promising route to answering questions
which seem intractable within standard explanatory paradigms.

###  4.2 Assessment of Theories

Determinism has long functioned as a gold standard for an ideal scientific
theory - for example, the idea that quantum mechanics might be indeterministic
was accepted only begrudgingly by many physicists at the time of the theory’s
formulation, as witnessed by Einstein’s complaint that ‘He (God) does not play
dice.’(Einstein (2005)) Over the last century many attempts have been made to
‘complete’ quantum mechanics so that it obeys Laplacean determinism after all
(Genovese (2005)), and indeed two of the most popular interpretations of
quantum mechanics (the de Broglie-Bohm interpretation (Holland (1995)) and the
Everett interpretation (Wallace (2003)) do satisfy Laplacean determinism.

However, if we accept that the standard definition of Laplacean determinism is
to some degree a function of our practical interests, there is less
justification for regarding Laplacean determinism as the ultimate goal of a
scientific theory. Obviously, theories which satisfy Laplacean determinism are
still desirable due to their practical utility for predicting the future, but
from the point of view of the scientific realist with an interest in
understanding how things _really_ are, theories which satisfy weak, strong or
delocalised determinism may be just as well motivated. After all, all three of
these notions of determinism do justice to Einstein’s intuition that ‘He does
not play dice,’ in the sense that none of them allows the existence of genuine
objective chances from the external, ‘god’s-eye’ point of view.

Thus this approach to determinism suggests new ways of thinking about what a
good scientific theory should look like. For example, instead of coming up
with theories which postulate a set of states and a set of differential
equations, perhaps we should be looking more seriously at theories postulating
laws which govern ‘all-at-once’ from an external point of view. As noted in
section 1.1, modern physics does already contain examples of such laws, but in
general these examples have been arrived at by starting from a temporal
evolution picture and subsequently generalizing it or reformulating it; things
might look quite different if we started from the assumption that we are
looking for a holistic, ‘all-at-once’ theory and then proceeded without
insisting on the existence of a time-evolution formulation of the theory.
Adjusting our gold standard to more closely reflect the form of the true laws
of nature is likely to be a good way to stimulate progress toward
understanding those laws, so we have strong practical motivations for
rethinking the gold standard.

###  4.3 Free Will

Laplace’s ideas about determinism gave rise to a spirited debate over the
possibility of free will in a deterministic universe which has continued to
this day (Williams (1980); Inwagen (1975); Mickelson (2019)). The possibility
of holistic determinism certainly raises new questions in this debate. For
example, even if you believe that we don’t have free will in the context of
Laplacean determinism, you might still be willing to say that we could have
free will in the context of some sorts of _holistic_ determinism. After all,
if the course of history is determined by laws which apply to the whole of
history all at once, our actions are determined by other events, but also
those events are partly determined by our actions, since each part of the
history is optimized relative to all the other parts. So there is a degree of
reciprocity which is lacking in the Laplacean context, where the present state
determines our actions and our actions do not act back on the present state.
This is an interesting direction of enquiry, but somewhat outside the scope of
the present paper, so we will leave it as a topic for future research.

###  4.4 Initial Conditions

The dynamic production picture on which Laplacean determinism is predicated
encourages us to take a particular attitude to initial conditions, regarding
them as freely chosen inputs to an otherwise fully determined system. Thus in
particular the Laplacean picture makes a sharp distinction between the freely
chosen initial conditions and the values of any parameters which enter into
the theory (e.g. the masses of fundamental particles, the gravitational
constant and so on), which are normally taken to to be nomic (Wallace (2019)).
But in the constraint framework the case for this different status is
significantly weaker. If the universe satisfies strong holistic determnism
then both the initial conditions and the numerical values of the parameters
must be determined by the laws of nature - that is, we must have state
rigidity and parameter rigidity, in the terms of Wallace (2019). But if the
universe is only weakly deterministic, it could well be the case that across
the set of mosaics in the intersection of the constraints there is some
variation in the initial conditions of the universe and also in the numerical
value of the parameters, so drawing a mosaic from the set entails arbitrarily
selecting both initial conditions and parameter values. In this context both
initial conditions and parameters are simply different facets of the degrees
of freedom left by the laws of nature, and thus they have precisely the same
modal status. Of course, it _could_ also be the case that the parameters are
in fact fixed by the laws of nature while the initial conditions are not, but
likewise it _could_ be the case that the initial conditions are in fact fixed
by the laws of nature while the parameters are not: until we have some
specific evidence for either parameter rigidity or state rigidity, parameters
and initial conditions should be afforded the same modal status.

##  5 Objective Chance

Given the close links between determinism and objective chance, our proposed
alternative approach to determinism will also have consequences for the
concept of objective chance. Famously, objective chance is a murky notion -
though a number of analyses of the topic exist, including frequentist and
propensity approaches, none has so far garnered universal approval and all
seem to have serious problems to overcome (Hájek (2012)). It is tempting to
respond to this situation by simply insisting that there aren’t any objective
chances at the level of fundamental laws, but that strategy is hampered the
fact that the current scientific evidence seems to be pointing away from
Laplacean determinism. It may appear that we are faced with a dichotomic
dilemma - either the world satisfies Laplacean determinism or there exist
objective chances - and thus, since it doesn’t appear to be the case that
quantum mechanics satisfies Laplacean determinism, it may seem that we have no
choice but to accept the existence of objective chances.

But our discussion of holistic determinism demonstrates that this division of
the possibilities is too rigid, for it turns out that we can postulate worlds
which do not satisfy Laplacean determinism but which nonetheless satisfy
holistic determinism, and therefore the probabilistic features of quantum
mechanics do not force us to accept the existence of objective chance from the
holistic, external point of view. There are a variety of ways in which such
apparently probabilistic events might emerge within a world satisfying
holistic determinism - they will appear wherever the world contains some
events which depend on facts about reality that are not accessible to
observers in the local region of those events - but for concreteness, in order
to demonstrate how chances can emerge from a world satisfying holistic
determinism we will henceforth focus on one particularly simple possibility: a
law of the form ‘events of type A have outcome A with probability 0.8’ may be
translated to a law imposing a constraint of the form ‘the set of mosaics in
which eighty percent of events of type E have the outcome A.’ Evidently for
the large majority of mosaics in this set, if we are in that mosaic and we
observe a sufficiently large number of events of type E we will find the
relative frequency of outcome A approaches 0.8, so this frequency-based law
can be expected to give rise to the characteristic behaviour associated with
the original probabilistic law.

The existence of these sorts of global constraints is entirely compatible with
strong, weak or delocalised holistic determinism, so one could equally well
say either that this approach reduces objective chance to frequency
constraints, or that it simply eliminates objective chance altogether. We
could even think of it as a way of reducing objective chances to subjective
probabilities: the relevant events are sampled from a set of events with
prespecified relative frequencies, so the objective chances attached to these
events simply describe a process of sampling without replacement akin to the
common textbook example of selecting a ball from a jar containing a specified
mixture of black and white balls. And of course the probabilities involved in
sampling without replacement can be described in terms of Bayesian credence
functions with partially unknown initial conditions, since we know the initial
proportions of black and white balls but not the order in which they will be
drawn. Thus deriving chances from frequency constraints provides a satisfying
explanation of the close conceptual and mathematical relationship between
objective chance and subjective probability: objective chances behave like
subjective probabilities because they really are just subjective probabilities
over a large and temporally extended domain.

Frequency constraints are similar to a view that has been advanced by Roberts
(2009) under the name of _nomic frequentism_ ; much of Roberts’ discussion is
also relevant here, so we won’t dwell on the points of agreement, but in
appendix A we discuss a few points of difference. Frequency constraints are
also clearly related to finite frequentism, which is which is the view that
objective probabilities are by definition equal to the actual relative
frequencies of the relevant sort of event across all of spacetime(Hájek
(2012)). But the frequency constraint approach avoids several of the main
problems encountered by finite frequentism. For example, it has been objected
that finite frequentism entails that probabilities can’t be defined for
processes which as a matter of fact occur only once or not at all (Hájek
(1996)). But this is not a problem for the frequency constraint view provided
that we take a robust attitude to modality which maintains that constraints
are ontologically prior to the Humean mosaic - in that case a constraint
exists and is well-defined regardless of how many occurrences of the event
type in question actually occur. Another common objection to finite
frequentism is that it has the result that probability depends on a choice of
reference class, and in the case of macroscopic everyday events this can be
very nontrivial, because different descriptions of the event may suggest
different natural reference classes which can lead to very different
probabilities (Hájek (1996)). However, this is not a problem for the frequency
constraint view provided that we take constraints to be objective and mind-
independent, because the reference class is defined within the constraint and
therefore there is always a definite fact about which reference class is
relevant. Of course, there is still an epistemological question about how we
as observers can decide which reference class actually appears in the real
underlying constraint, and it’s always possible that we will get it wrong and
will thus make the wrong predictions, but there is no ambiguity in the
definition of the objective chance itself. Roberts (2009) gives more examples
of the advantages of frequency constraints over finite frequentism.

In order to see if frequency constraints can give rise to chances which behave
in the way we would expect, we will consider the following desiderata for an
account of objective chance:

  1. 1.

_The Principal Principle:_ Objective chances should satisfy the Principal
Principle.

  2. 2.

_Probabilistic confirmation:_ We should be able to confirm facts about
objective chances by means of observing relative frequencies.

  3. 3.

_Exchangeability:_ Given a sequence of events which are not causally related
to one another, the chance for a given sequence should depend only on the
relative frequencies in that sequence and not on the order in which the events
occur.

  4. 4.

_Counterfactual independence:_ Given a sequence of events which are not
causally related to one another, the chance for the outcome of an individual
event should not depend on the frequency of occurrence of other outcomes in
the sequence.

###  5.1 The Principal Principle

Do chances derived from frequency constraints satisfy the Principal Principle?
Well, in accordance with the method of direct inference (McGrew (2001)), if
you know that you are in a Humean mosaic which belongs to the set of mosaics
in which exactly eighty percent of As are Bs, and you have no other
information, you should indeed set your credences for the next observed $A$ to
be $B$ to eighty percent. Now in most real cases we will already have observed
some As and therefore it is not quite the case that we are sampling at random
from the entire class, so the correct probability to assign might actually be
infinitesimally different from eighty percent. But provided that the class is
large enough, this difference will be so small that it will make no practical
difference to the way in which we use this probability in our reasoning
processes and thus for all practical purposes the Principal Principle does
indeed pick out the frequencies that appear in the constraints.

###  5.2 Probabilistic Confirmation

Next let us check for probabilistic confirmation: if we observe a set of
events of a certain type and find that eighty percent of them have the outcome
A, do we have grounds to confirm the hypothesis that we are in a Humean mosaic
in which eighty percent of instances of this event type have outcome A? Of
course, in general we will be able to observe only a very tiny proportion of
the total number of such events occurring across the whole Humean mosaic, so
we can’t confirm this hypothesis by direct observation; rather we must assume
that the As that we have observed are a _representative sample_ of the full
set of As. Thus the legitimacy of this inference depends crucially on the
assumption that the As form a homogenous class which can be expected to
exhibit stable relative frequencies across time - it is not the case that just
any relative frequencies can be extrapolated in this way, as that would allow
us to confirm virtually any hypothesis we like about future relative
frequencies by simply labelling events in the right sort of way. Thus from
observations of a relatively small subclass of instances of a given event
type, we do not have grounds to confirm the hypothesis that eighty percent of
instances of this event type have outcome A, but we _do_ have grounds to
confirm the hypothesis that the laws of nature induce a constraint to the
effect that eighty percent of As must be Bs. That is, the inference must
include the hypothesis that we have identified a special class of events which
appears directly in the laws of nature and that the relative frequencies we
have seen are the consequence of a constraint which is induced by the laws of
nature. Of course we could always be wrong about this hypothesis, but
nonetheless, observing behaviour which is consistent with the hypothesis
provides some degree of confirmation for it.

###  5.3 Exchangeability and Counterfactual Independence

Let us move to the final two desiderata, which are somewhat more mathematical
in character. It is clear that chances obtained from frequency constraints
will satisfy exchangeability, since these constraints govern only relative
frequencies and not order of appearance. But what about counterfactual
independence? Well, we have seen that in the frequency constraint picture,
observing random events must be understood as a form of sampling without
replacement, and so just as we would update our probabilities for the next
event after each random draw when performing sampling without replacement, it
seems we ought to do the same in the frequency constraint case: if I know that
exactly 50 percent of events of a given type must have the outcome A, then if
my first one hundred observations all have the outcome B I will conclude that
the remaining set of events must contain one hundred more As than Bs, so I
will perform the standard sort of inference we see in sampling without
replacement, which involves updating my credences to reflect the fact that the
next outcome is more likely to be A than B. Note that this remains true for
any finite total number of events, so the conclusion does not change if I
don’t know the total number of events. It also doesn’t matter if we have a
constraint which prescribes the probability only approximately; there will
always be some threshold number $n$ of Bs such that if I have observed $n$
more Bs than As, then the set of unobserved events must contain more As than
Bs in order to get final frequencies which are correct to within the limits
allowed by the relevant constraint.

This seems problematic, as the absence of counterfactual independence jars
with many of our intuitions about probability - for example, it has the
consequence that under certain circumstances the gambler’s fallacy is not
really a fallacy at all. Indeed Hajek argues against standard frequentism on
similar grounds, pointing out that defining probabilities as relative
frequencies prevents us from saying that the probability for an individual
event is counterfactually independent of what happens in other similar events
distant in space and time (Hájek (1996)). So we must now ask ourselves whether
we can accept an approach which fails to satisfy counterfactual independence
as an acceptable analysis of objective chance. One possible response for the
proponent of frequency constraints is to point out we will not get failures of
counterfactual independence if the total number of instances is infinite,
since sampling without replacement from an infinite set is mathematically
equivalent to sampling _with_ replacement. And after all, we can never know
for sure that the number of instances of an event type is finite, since that
would require us to have illegitimate knowledge of the future, so it might be
argued that we should never make updates to our probabilities which are
premised on the number of instances of an event type being finite, meaning
that the usual sort of inferences we see in sampling-without-replacement
should not be allowed with regard to sequences of chancy events. However,
although it is true that we can never know for sure that the number of
instances is finite, nonetheless one can imagine cases where our best theories
give us good reason to believe that this is so (for example, if we had a
theory which implied that spacetime is discrete and the universe has a finite
extent and time has both a start and an end), so at least in principle there
are circumstances where counterfactual independence really would be violated
in this picture.

Another possible response is to point out that we would only ever get
violations of counterfactual independence with regard to sequences of chancy
events if we had knowledge of frequency constraints that would trump the
observed relative frequencies, and one might think that this could never come
about. For example, in real life if we performed one hundred measurements and
got the result B every time, we would probably form the working hypothesis
that there is a constraint requiring that all events of this type must have
outcome B, rather than supposing that exactly fifty percent of all events of
this type must have outcome B and that therefore subsequent events must be
more likely to have outcome A than outcome B in order to make up the total.
However, probabilistic laws do not usually exist in isolation, and therefore
one can imagine a case where we decide based on considerations of symmetry or
coherence with the rest of our theory that the objective chance of B is 0.5
even though so far we have seen significantly more Bs than As. It would seem
that under those circumstances, if we accepted the frequency constraint
analysis and we had reason to believe that the total number of instances of
the relevant event type across all of spacetime would be finite, we would have
reason to assign a credence greater than 0.5 to the proposition that the next
such event will have outcome A.

So in fact, probably the appropriate response here is to simply say that
violations of counterfactual independence _should_ sometimes be allowed with
respect to sequences of chancy events, because there is little reason to
expect our intuitions to be a good guide on this point. For a start, most of
our intuitions are based on probabilistic events that involve _subjective_
probabilities rather than objective chances of the kind that appear in the
fundamental laws of nature - due to decoherence, macroscopic events do not
typically depend sensitively on quantum mechanical measurement outcomes, and
therefore paradigmatic probabilistic events like ‘rolling a die’ or ‘flipping
a coin’ can be understood entirely in terms of our ignorance of the specific
initial conditions. Provided that the relevant initial conditions are
independent (or at least independent enough for practical purposes) these
sorts of events do indeed obey counterfactual independence, which is likely
the source of our intuitions on this point. Furthermore, even when we do
observe events that may involve objective chances, such as the results of
quantum measurements, we are presumably observing only a very small subset of
the total number of events of this type, so we should not expect these
observations to tell us very much about the class as a whole. If we know that
exactly fifty percent of instances of a certain event type have the outcome
$A$, and also that $10^{11}$ of these events occur across all of spacetime,
then in principle the fact that we have observed one hundred more $B$ outcomes
than $A$ outcomes does make it more likely that the next outcome will be an A,
but in practice the change in probability is so small that it will not lead to
any observable effects.

Therefore, the fact that counterfactual independence conflicts with our
intuitions is no reason to reject chances obtained from frequency constraints,
since our intuitions have been developed in entirely the wrong context of
application. Indeed, counterfactual independence can be regarded as a way of
expressing the expectation that objective chances should be intrinsic
properties of individual entities which are spatially and temporally local;
but in many ways chances actually look like properties not of individual
entities but rather of large collections of entities. For example, we can’t
ascertain chances during a single observation, but rather we must observe a
large number of events of the same type and then make an inference about what
the chances are. A number of popular approaches to the analysis of objective
chance - particularly the propensity interpretation, which holds that chances
are intrinsic properties of individual entities akin to colours and masses
(Popper (1959)) - struggle to make sense of this feature (Eagle (2004)). But
frequency constraints do justice to the apparently collective nature of
objective chance by turning chances into a form of global coordination across
space and time, where events are constrained to coordinate non-locally amongst
themselves so as to ensure that outcomes occur with (roughly) the right
frequency. Hájek (1996) worries that ‘ _it’s almost as if the frequentist
believes in something like backward causation from future results to current
chances,_ ’ - and indeed, that is exactly what the proponent of frequency
constraints does believe!

##  6 Related Topics

In this section, we discuss the relationship between holistic determinism and
some other relevant research programmes.

###  6.1 Superdeterminism

Since the terms ‘holistic determinism’ and ‘superdeterminism’ are
superficially similar, it’s important to reinforce that these concepts are not
identical. Broadly speaking, ‘superdeterminism’ describes approaches to
quantum mechanics which deny the existence of non-locality in quantum
mechanics by rejecting the assumption of statistical independence which goes
into Bell’s theorem, i.e. the idea that our choice of measurement is
independent of the state of the system which we are measuring (Hossenfelder
and Palmer (2020); Hossenfelder (2020); Palmer (2016)). There are two typical
ways to achieve this. First, given that our choice of measurement is always
determined or at least influenced by facts about the past (e.g. the physical
state of our brain in the time leading up to our choice) one could imagine the
initial state of the universe being arranged such that our measurement choices
are always correlated with the states of the systems we are measuring. Second,
one could imagine there is some sort of retrocausality or ‘future outcome
dependence’ where the state of the system we are measuring is influenced by
our future decisions about what measurement to perform.

It is immediately clear that neither of these approaches actually requires
determinism: a _probabilistic_ dependency between choice of measurement and
state is already enough to vitiate statistical independence and thus restore
locality. Thus it is actually somewhat misleading to use the term
‘superdeterminism’ to refer to the violation of statistical independence. The
term ‘superdeterminism’ contains the word ‘determinism’ because Bell
originally proved a theorem ruling out deterministic local hidden variable
theories (Bell (1990)), and so at that time superdeterminism was suggested as
an approach to restore both locality and determinism; but a later version of
Bell’s theorem ruled out _all_ local hidden variable theories, either
probabilistic or deterministic (Esfeld (2013)), so in fact determinism is not
really the key issue here, and modern proponents of superdeterminism are
usually motivated more by the desire to preserve locality than any particular
interest in preserving determinism.

That said, for the moment let us focus on superdeterministic approaches which
_are_ also deterministic in some sense. What sort of determinism would that
be? Well, if we take the first route where correlations between measurement
choices and states are written into the initial state of the universe, then we
have just standard Laplacean determinism (and thus also weak holistic
determinism). If we take the second route involving retrocausality, the
resulting theory would probably not be compatible with Laplacean determinism,
because events which are determined by something in the future usually can’t
also be fully determined by the past; but such a theory certainly could
exhibit either weak or strong holistic determinism. However, although
superdeterminism can be a form of holistic determinism, the concept we have
defined here is significantly more general than superdeterminism. In
particular, a major motivation for our approach to was to understand what
determinism could look like in a world which includes spatial and/or temporal
non-locality, whereas superdeterminism is typically employed as a way of
_denying_ the existence of either spatial or temporal non-locality, and
therefore holistic determinism also accommodates realist approaches which take
a very different approach from superdeterminism.

###  6.2 Deterministic Probabilities

There is a long-standing philosophical debate around the question of whether
it is possible to have chances in a deterministic world (McCoy (2017); Glynn
(2010); Gallow (forthcoming)), but this debate usually takes place against the
backdrop of Laplacean determinism. Moving to a picture based on holistic
determinism therefore opens up new ways in which we can have ‘chancy’ events
in a deterministic world, since we saw in section 5 that an event can be
described by a non-trivial probability distribution when we condition on all
of the information available to local observers, even if the event is not
objectively chancy when we consider the way in which it is embedded in the
complete objective modal structure of reality. This reinforces the point that
‘chanciness’ is in many cases relative to a perspective: obviously if the
outcome of a process is determined by something in the future it is for all
practical purposes objectively chancy from the point of view of observers like
ourselves who have directly epistemic access only to the past and present, but
this does not necessarily mean that the process must be ‘chancy’ from the
point of view of the underlying modal structure.

##  7 Conclusion

We have argued that the metaphysical picture of determinism captured in
Laplace’s original vision is no longer fit for purpose, since it can’t be
usefully applied to the diverse range of laws outside the time evolution
paradigm that appear in modern physics. However, we contend that even for a
world governed by laws which are not time evolution laws, it is still
meaningful to ask whether or not that world is deterministic. Thus in this
article we have provided several generalized definitions of determinism which
can be applied to a wide variety of nonstandard laws, thus rehabilitating
determinism for the post-time-evolution era.

Using these definitions, we have shown that a world governed by laws outside
the time evolution paradigm may fail to satisfy the standard conception of
Laplacean determinism but may nonetheless exhibit a form of determinism on a
global scale, where we say we have some form of ‘determinism’ provided that we
are never required to assign probability distributions over entire Humean
mosaics or worlds. This has the consequence of dissolving the traditional
dichotomic dilemma where we supposedly have to choose between Laplacean
determinism and intrinsic randomness - a world governed by global aws may
exhibit what appear to be objective chances from the point of view of local
observers whilst requiring no irreducible objective chances from the external,
holistic perspective. We have discussed one possible way of implementing this
vision by means of constraints prescribing relative frequencies across
spacetime, although frequency constraints are only one possible way in which
chancy events could arise within a world satisfying holistic determinism.

Several interesting conceptual points have come up along this journey. In
particular, the possibility of weak holistic determinism requires something of
a conceptual shift, requiring us to distinguish between events which are
chancy and events which are undetermined but nonetheless not chancy. This
distinction is not entirely new - for example, it is common to hold that the
initial conditions of the world are not determined by anything but also not
governed by any objective probabilities - but it is not always well
articulated, and often fails to be taken seriously as an option for any
phenomena other than the initial conditions of the universe. Weak holistic
determinism thus offers a welcome halfway house for those who find it
implausible that everything in the universe is fully determined by the laws of
nature, but who are also suspicious of the ill-defined concept of objective
chance. We have briefly surveyed some of the consequences of this new
perspective, nothing that it offers interesting insight into for long-standing
debates in which determinism plays a role within both physics and philosophy.

##  Appendix A Nomic frequentism

The ‘nomic frequentism’ proposal of Roberts (2009), which suggests that laws
about probabilities should be understood in terms of laws of the form ‘R
percent of the Fs are Gs’ has much in common with the frequency constraint
approach that we have discussed in this article. However, there are a few key
differences. First, Roberts presents nomic frequentism as a general analysis
of all probabilities that appear in laws, including for example the laws of
evolutionary biology, whereas we have suggested it only for the specific case
of objective chances appearing in the fundamental laws of nature, and in
particular quantum mechanics - we would expect that the laws of evolutionary
biology could be satisfactorily understood in terms of subjective
probabilities. Moreoever, we do not even claim that _all_ of the objective
chances appearing in the fundamental laws of nature must be attributed to
frequency constraints; we merely observe that this is one possible way in
which apparently probabilistic events could arise in a deterministic universe.
So our account is in that sense considerably less general and ambitious than
Roberts’.

Second, Roberts suggests that ‘eighty percent of As are Bs’ should be regarded
as conceptually equivalent to a law like ‘All As are Bs,’ (or rather, the
latter should be regarded a special case of the former). But there is one
important conceptual difference: the constraint ‘All As are Bs’ does not seem
to require any ‘communication’ between the As, as it is enough that each $A$
has the intrinsic property of being a $B$, whereas ‘exactly eighty percent of
As are Bs’ or even ‘approximately eighty percent of As are Bs’ does seem to
require some sort of coordination, as in order to ensure that the relative
frequency is exactly or approximately correct, it seems that each A must
‘know’ something about what the other As are doing. Thus frequency constraints
seem to require some form of non-locality (as Roberts himself later observes),
which means we are in quite a different conceptual space from standard laws
like ‘all As are Bs.’

Third, Roberts’ solution to the apparent absence of counterfactual
independence within nomic frequentism is to say that ‘the way a nomic
frequentist will represent independence of distinct fair coin-tosses is by
denying the existence of any law that implies that the conditional frequency
of heads on one toss given the results of another toss is different from (50
percent).’ But as we have seen, it does not seem that this can be entirely
correct: constraint frequentism does allow the violation of counterfactual
independence in at least certain special cases. Roberts justifies his solution
on the basis that knowing facts which violate counterfactual independence
‘would require (us) to have advance intelligence from the future,’ but this
argument seems to depend implicitly on the assumption that we are in a Humean
context where the only information which is relevant to inferences about a
future event is an actual observation of the event itself. But if the laws
which induce the frequency constraints are understood as laws which are
ontologically prior to the Humean mosaic, and if we are able to make correct
inferences about the laws based on observations of a limited subset of the
mosaic, then we would in principle be able to know about violations of
counterfactual independence without having any illegitimate information about
events which have not yet occurred. This sort of ‘knowledge of the future’ is
not really any different from the knowledge of the future that we get from
more familiar sorts of scientific laws: the laws constrain the Humean mosaic,
including the future, and thus by figuring out the laws we can make inferences
about the future.

## References

  * Adlam (2018) Emily Adlam.  Spooky Action at a Temporal Distance.  _Entropy_ , 20(1):41, 2018.
  * Adlam (2018) Emily Adlam.  Quantum Mechanics and Global Determinism.  _Quanta_ , 7:40–53, 2018.  URL http://quanta.ws/ojs/index.php/quanta/article/view/76.
  * Adlam (2021a) Emily Adlam.  Laws of nature as constraints, 2021a.  URL https://arxiv.org/abs/2109.13836.
  * Adlam (2021b) Emily Adlam.  Tsirelson’s bound and the quantum monogamy bound from global determinism, 2021b.  URL https://arxiv.org/abs/2011.08284.
  * Aharonov and Vaidman (2002) Yakir Aharonov and Lev Vaidman.  _The Two-State Vector Formalism of Quantum Mechanics_ , pages 369–412.  Springer Berlin Heidelberg, Berlin, Heidelberg, 2002.  ISBN 978-3-540-45846-3.  doi: 10.1007/3-540-45846-8˙13.  URL https://doi.org/10.1007/3-540-45846-8_13.
  * Armstrong (1983) D. M. Armstrong.  _What is a Law of Nature?_ Cambridge University Press, 1983.
  * Banks et al. (2001) Tom Banks, Michael Dine, and Lubos Motl.  On anthropic solutions of the cosmological constant problem.  _JHEP_ , 01:031, 2001.  doi: 10.1088/1126-6708/2001/01/031.
  * Barbour (2001) Julian Barbour.  On general covariance and best matching.  In Craig Callender and Nick Huggett, editors, _Physics Meets Philosophy at the Planck Scale_. 01 2001.
  * Bell (1990) J. Bell.  Against ’measurement’.  _Physics World_ , August 1990.
  * Berenstain and Ladyman (2012) Nora Berenstain and James Ladyman.  Ontic structural realism and modality.  In Elaine Landry and Dean Rickles, editors, _Structural Realism: Structure, Object, and Causality_. Springer, 2012.
  * Bird (2005) Alexander Bird.  The dispositionalist conception of laws.  _Foundations of Science_ , 10(4):353–70, 2005\.  doi: 10.1007/s10699-004-5259-9.
  * Blau and Theisen (2009) Matthias Blau and Stefan Theisen.  String theory as a theory of quantum gravity: a status report.  _General Relativity and Gravitation_ , 41(4):743–755, 2009.  ISSN 1572-9532.  doi: 10.1007/s10714-008-0752-z.  URL http://dx.doi.org/10.1007/s10714-008-0752-z.
  * Brizard (2008) A.J. Brizard.  _An Introduction to Lagrangian Mechanics_.  World Scientific, 2008.  ISBN 9789812818362.  URL https://books.google.co.uk/books?id=JfQkh-qY8ucC.
  * Butterfield (2005) J. Butterfield.  Defining determinism.  In _Routledge Encyclopedia of Philosophy_. Taylor and Francis Group, 2005.
  * Butterfield (1989) Jeremy Butterfield.  The hole truth.  _The British Journal for the Philosophy of Science_ , 40(1):1–28, 1989.  ISSN 00070882, 14643537.  URL http://www.jstor.org/stable/687461.
  * Chen (2018) Eddy Keming Chen.  Quantum mechanics in a time-asymmetric universe: On the nature of the initial quantum state, 2018.  URL https://arxiv.org/abs/1712.01666.
  * Chen and Goldstein (2021) Eddy Keming Chen and Sheldon Goldstein.  Governing without a fundamental direction of time: Minimal primitivism about laws of nature, 2021.  URL https://arxiv.org/abs/2109.09226.
  * Choquet-Bruhat and Geroch (1969) Yvonne Choquet-Bruhat and Robert Geroch.  Global aspects of the cauchy problem in general relativity.  _Communications in Mathematical Physics_ , 14(4):329–335, 1969.
  * Clark and Butterfield (1987) Peter Clark and Jeremy Butterfield.  Determinism and probability in physics.  _Aristotelian Society Supplementary Volume_ , 61(1):185–244, 1987.  doi: 10.1093/aristoteliansupp/61.1.185.
  * Cramer (1986) John G. Cramer.  The transactional interpretation of quantum mechanics.  _Rev. Mod. Phys._ , 58:647–687, Jul 1986.  doi: 10.1103/RevModPhys.58.647.  URL http://link.aps.org/doi/10.1103/RevModPhys.58.647.
  * Deutsch and Marletto (2015) David Deutsch and Chiara Marletto.  Constructor theory of information.  _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ , 471(2174):20140540, Feb 2015.  ISSN 1471-2946.  doi: 10.1098/rspa.2014.0540.  URL http://dx.doi.org/10.1098/rspa.2014.0540.
  * Dirac (2019) P.A.M. Dirac.  _The Principles of Quantum Mechanics_.  Lulu.com, 2019.  ISBN 9781794767119.  URL https://books.google.ca/books?id=IvrDDwAAQBAJ.
  * Doboszewski (2019) Juliusz Doboszewski.  Relativistic spacetimes and definitions of determinism.  _European Journal for Philosophy of Science_ , 9(2):1–14, 2019.  doi: 10.1007/s13194-019-0248-6.
  * Dowe (2019) Phil Dowe.  Action at a temporal distance in the best systems account.  _European Journal for Philosophy of Science_ , 9, 10 2019.  doi: 10.1007/s13194-019-0258-4.
  * Eagle (2004) Antony Eagle.  Twenty-one arguments against propensity analyses of probability.  _Erkenntnis_ , 60(3):371–416, 2004.  ISSN 0165-0106.  doi: 10.1023/B:ERKE.0000023408.61887.6a.  URL http://dx.doi.org/10.1023/B%3AERKE.0000023408.61887.6a.
  * Earman (1986) J. Earman.  _A Primer on Determinism_.  D. Reidel Publishing, 1986.
  * Earman (2004) John Earman.  Determinism: What we have learned and what we still don’t know.  In Joseph K. Campbell, editor, _Freedom and Determinism_ , pages 21–46. Cambridge Ma: Bradford Book/Mit Press, 2004.
  * Earman (2007) John Earman.  Aspects of determinism in modern physics.  In John Earman and Jeremy Butterfield, editors, _Handbook of the Philosophy of Science. Volume 2: Philosophy of Physics_. 2007.
  * Einstein (2005) Albert Einstein.  Albert Einstein to Max Born 1.  _Physics Today_ , 58(5):16–16, 2005.  doi: 10.1063/1.1995729.  URL https://doi.org/10.1063/1.1995729.
  * Ellis (2001) Brian Ellis.  _Scientific Essentialism_.  Cambridge University Press, 2001.
  * Esfeld (2009) Michael Esfeld.  The modal nature of structures in ontic structural realism.  _International Studies in the Philosophy of Science_ , 23(2):179–194, 2009.  doi: 10.1080/02698590903006917.  URL https://doi.org/10.1080/02698590903006917.
  * Esfeld (2013) Michael Esfeld.  Ontic structural realism and the interpretation of quantum mechanics.  _European Journal for Philosophy of Science_ , 3, 01 2013.  doi: 10.1007/s13194-012-0054-x.
  * Feynman et al. (2010) R.P. Feynman, A.R. Hibbs, and D.F. Styer.  _Quantum Mechanics and Path Integrals_.  Dover Books on Physics. Dover Publications, 2010.  ISBN 9780486477220.  URL https://books.google.co.uk/books?id=JkMuDAAAQBAJ.
  * Gallow (forthcoming) J. Dmitri Gallow.  A subjectivist’s guide to deterministic chance.  _Synthese_ , forthcoming.
  * Genovese (2005) Marco Genovese.  Research on hidden variable theories: A review of recent progresses.  _Physics Reports_ , 413(6):319–396, 2005.
  * Glynn (2010) Luke Glynn.  Deterministic chance.  _British Journal for the Philosophy of Science_ , 61(1):51–80, 2010.  doi: 10.1093/bjps/axp020.
  * Hájek (1996) Alan Hájek.  ”Mises Redux”: Fifteen arguments against finite frequentism.  _Erkenntnis (1975-)_ , 45(2/3):pp. 209–227, 1996\.  ISSN 01650106.  URL http://www.jstor.org/stable/20012727.
  * Hájek (2012) Alan Hájek.  Interpretations of probability.  In Edward N. Zalta, editor, _The Stanford Encyclopedia of Philosophy_. Metaphysics Research Lab, Stanford University, winter 2012 edition, 2012.
  * Hartle (1993) J. B. Hartle.  The spacetime approach to quantum mechanics.  _Vistas in Astronomy_ , 37:569–583, 1993.  doi: 10.1016/0083-6656(93)90097-4.
  * Hempel and Oppenheim (1948) Carl G. Hempel and Paul Oppenheim.  Studies in the logic of explanation.  _Philosophy of Science_ , 15(2):135–175, 1948\.  doi: 10.1086/286983.  URL https://doi.org/10.1086/286983.
  * Hempel et al. (1965) Carl G Hempel et al.  Aspects of scientific explanation.  1965\.
  * Holland (1995) P.R. Holland.  _The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics_.  Cambridge University Press, 1995.  ISBN 9780521485432.  URL https://books.google.co.uk/books?id=BsEfVBzToRMC.
  * Hossenfelder (2020) Sabine Hossenfelder.  Superdeterminism: A guide for the perplexed, 2020.  URL https://arxiv.org/abs/2010.01324.
  * Hossenfelder and Palmer (2020) Sabine Hossenfelder and Tim Palmer.  Rethinking superdeterminism.  _Frontiers in Physics_ , 8:139, 2020.  ISSN 2296-424X.  doi: 10.3389/fphy.2020.00139.  URL https://www.frontiersin.org/article/10.3389/fphy.2020.00139.
  * Inwagen (1975) Peter Van Inwagen.  The incompatibility of free will and determinism.  _Philosophical Studies_ , 27(3):185–199, 1975\.  doi: 10.1007/BF01624156.
  * Kent (2014) A. Kent.  Solution to the Lorentzian quantum reality problem.  _Phys Rev A_ , 90(1):012107, July 2014.  doi: 10.1103/PhysRevA.90.012107.
  * Kuchař (1993) Karel V. Kuchař.  Canonical quantum gravity.  _arXiv: General Relativity and Quantum Cosmology_ , 1993.
  * Ladyman and Ross (2007) James Ladyman and Don Ross.  _Every Thing Must Go: Metaphysics Naturalized_.  Oxford University Press, 2007.
  * Laplace (1820) Pierre Simon Laplace.  _Théorie analytique des probabilités_.  Courcier, 1820.
  * Lewis (1980) David Lewis.  A subjectivist’s guide to objective chance.  In Richard C. Jeffrey, editor, _Studies in Inductive Logic and Probability_ , pages 83–132. University of California Press, 1980.
  * Lewis (1986) David Lewis.  _On the Plurality of Worlds_.  Wiley-Blackwell, 1986.
  * Lewis (1994) David Lewis.  Humean supervenience debugged.  _Mind_ , 103(412):473–490, 1994.  ISSN 00264423, 14602113.  URL http://www.jstor.org/stable/2254396.
  * Loewer (2020) Barry Loewer.  _The Mentaculus Vision_ , chapter Chapter 1, pages 3–29.  2020\.  doi: 10.1142/9789811211720˙0001.  URL https://www.worldscientific.com/doi/abs/10.1142/9789811211720_0001.
  * Maar (2019) Alexander Maar.  Kinds of determinism in science.  _Principia: An International Journal of Epistemology_ , 23(3):503–528, 2019.
  * Manchak (2008) John Manchak.  Is spacetime hole-free?, December 2008.  URL http://philsci-archive.pitt.edu/4380/.  Forthcoming in ”General Relativity and Gravitation”.
  * Masanes et al. (2014) L. Masanes, R. Renner, M. Christandl, A. Winter, and J. Barrett.  Full security of quantum key distribution from no-signaling constraints.  _Information Theory, IEEE Transactions on_ , 60(8):4973–4986, Aug 2014.  ISSN 0018-9448.  doi: 10.1109/TIT.2014.2329417.
  * Maudlin (2002) Tim Maudlin.  _Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics_.  Blackwell, 2002.
  * McCoy (2017) C. D. McCoy.  No chances in a deterministic world.  2017\.  URL http://philsci-archive.pitt.edu/13166/.
  * McGrew (2001) Timothy McGrew.  Direct inference and the problem of induction.  _The Monist_ , 84:153–178, 04 2001.  doi: 10.2307/27903722.
  * Meacham (2010) Christopher J. G. Meacham.  Contemporary approaches to statistical mechanical probabilities: A critical commentary - part i: The indifference approach.  _Philosophy Compass_ , 5(12):1116–1126, 2010\.  doi: 10.1111/j.1747-9991.2010.00356.x.
  * Mickelson (2019) Kristin M. Mickelson.  The problem of free will and determinism: An abductive approach.  _Social Philosophy and Policy_ , 36(1):154–172, 2019.  doi: 10.1017/s0265052519000207.
  * Minguzzi (2012) E. Minguzzi.  Causally simple inextendible spacetimes are hole-free.  _Journal of Mathematical Physics_ , 53(6):062501, Jun 2012.  ISSN 1089-7658.  doi: 10.1063/1.4721271.  URL http://dx.doi.org/10.1063/1.4721271.
  * Montague (1974) Richard Montague.  Deterministic theories.  In Richmond H. Thomason, editor, _Formal Philosophy_. Yale University Press, 1974.
  * Müller (2013) Thomas Müller.  _ELIMINATING MODALITY FROM THE DETERMINISM DEBATE? MODELS VS. EQUATIONS OF PHYSICAL THEORIES_ , pages 47–62.  De Gruyter, 2013.  doi: doi:10.1515/9783110328875.47.  URL https://doi.org/10.1515/9783110328875.47.
  * Müller and Placek (2018) Thomas Müller and Tomasz Placek.  Defining determinism.  _The British Journal for the Philosophy of Science_ , 69(1):215–252, 2018.  doi: 10.1093/bjps/axv049.  URL https://doi.org/10.1093/bjps/axv049.
  * Palmer (2016) T. N. Palmer.  Invariant set theory, 2016.  URL https://arxiv.org/abs/1605.01051.
  * Pawlowski et al. (2009) M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Żukowski.  Information causality as a physical principle.  _Nature_ , 461:1101–1104, October 2009.  doi: 10.1038/nature08400.
  * Penrose (2016) R. Penrose.  _The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics_.  Oxford Landmark Science. OUP Oxford, 2016.  ISBN 9780192550071.  URL https://books.google.co.uk/books?id=JF4vDwAAQBAJ.
  * Popper (1959) Karl Popper.  _The logic of scientific discovery_.  Routledge, 1959.
  * Popper (1992) Karl Popper.  _The Open Universe: An Argument for Indeterminism From the Postscript to the Logic of Scientific Discovery_.  Routledge, 1992.
  * Roberts (2009) John T. Roberts.  Laws about frequencies.  2009\.  URL http://philsci-archive.pitt.edu/4785/.
  * Rousseaux (2005) Germain Rousseaux.  The Gauge non-invariance of classical electromagnetism.  _Annales Fond. Broglie_ , 30:387–397, 2005.
  * Rovelli and Vidotto (2015) C. Rovelli and F. Vidotto.  _Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory_.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2015.  ISBN 9781107069626.  URL https://books.google.ca/books?id=4VjeBAAAQBAJ.
  * Santo (2021) Flavio Del Santo.  Indeterminism, causality and information: Has physics ever been deterministic?, 2021.
  * Scarani et al. (2005) V. Scarani, S. Iblisdir, N. Gisin, and A. Acín.  Quantum cloning.  _Reviews of Modern Physics_ , 77:1225–1256, October 2005\.  doi: 10.1103/RevModPhys.77.1225.
  * Smolin (2009) Lee Smolin.  The unique universe.  _Physics World_ , 22(06):21, 2009.  URL http://stacks.iop.org/2058-7058/22/i=06/a=36.
  * Sorkin (2007) R. D. Sorkin.  Quantum dynamics without the wavefunction.  _Journal of Physics A Mathematical General_ , 40:3207–3221, March 2007.  doi: 10.1088/1751-8113/40/12/S20.
  * Sutherland (2006) R. Sutherland.  Causally Symmetric Bohm Model.  _eprint arXiv:quant-ph/0601095_ , January 2006.
  * Wallace (2003) David Wallace.  Everett and structure.  _Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics_ , 34(1):87–105, 2003\.
  * Wallace (2011) David Wallace.  The logic of the past hypothesis, November 2011.  URL http://philsci-archive.pitt.edu/8894/.  To appear in B. Loewer, E. Winsberg and B. Weslake (ed.), currently-untitled volume discussing David Albert’s ”Time and Chance”.
  * Wallace (2019) David Wallace.  Naturalness and emergence, February 2019.  URL http://philsci-archive.pitt.edu/15757/.
  * Weinberg (1989) Steven Weinberg.  The cosmological constant problem.  _Rev. Mod. Phys._ , 61:1–23, Jan 1989.  doi: 10.1103/RevModPhys.61.1.  URL https://link.aps.org/doi/10.1103/RevModPhys.61.1.
  * Wharton (2015) K. Wharton.  The Universe is not a Computer.  In Foster B. Aguirre, A. and G. Merali, editors, _Questioning the Foundations of Physics_ , pages 177–190. Springer, November 2015.
  * Wharton (2018) Ken Wharton.  A new class of retrocausal models.  _Entropy_ , 20(6):410, May 2018.  ISSN 1099-4300.  doi: 10.3390/e20060410.  URL http://dx.doi.org/10.3390/e20060410.
  * Williams (1980) Clifford Williams.  _Free Will and Determinism: A Dialogue_.  Hackett Publishing Company, 1980.