Dynamic and stochastic systems as a framework for metaphysics and the philosophy of science


Synthese (2021) 198:2551–2612
https://doi.org/10.1007/s11229-019-02231-8

Dynamic and stochastic systems as a framework
for metaphysics and the philosophy of science

Christian List1 · Marcus Pivato2

Received: 3 November 2016 / Accepted: 23 April 2019 / Published online: 3 September 2019
© The Author(s) 2019

Abstract
Scientists often think of the world (or some part of it) as a dynamical system, a stochas-
tic process, or a generalization of such a system. Prominent examples of systems are
(i) the system of planets orbiting the sun or any other classical mechanical system,
(ii) a hydrogen atom or any other quantum–mechanical system, and (iii) the earth’s
atmosphere or any other statistical mechanical system. We introduce a general and
unified framework for describing such systems and show how it can be used to exam-
ine some familiar philosophical questions, including the following: how can we define
nomological possibility, necessity, determinism, and indeterminism; what are symme-
tries and laws; what regularities must a system display to make scientific inference
possible; how might principles of parsimony such as Occam’s Razor help when we
make such inferences; what is the role of space and time in a system; and might they
be emergent features? Our framework is intended to serve as a toolbox for the formal
analysis of systems that is applicable in several areas of philosophy.

Keywords Dynamical systems · Stochastic processes · Nomological possibility and
necessity · Determinism · Indeterminism · Laws · Symmetries · Regularities ·
Scientific inference · Ergodicity · Occam’s Razor · Space · Time · Formal
metaphysics

Part of this paper was written when Pivato was at the Department of Mathematics at Trent University,
Canada.

B Christian List
c.list@lse.ac.uk

1 Department of Philosophy, Logic, and Scientific Method, London School of Economics, London,
UK

2 THEMA, Université de Cergy-Pontoise, Cergy-Pontoise, France

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http://orcid.org/0000-0003-1627-800X


2552 Synthese (2021) 198:2551–2612

1 Introduction

For both scientific and philosophical purposes, we often find it useful to think of the
world (or some part of it that we are studying) as a system evolving over time: a
dynamical system, a stochastic process, or a suitable generalization of such a system.
In both science and philosophy, many theories represent the world (or the part they
are concerned with) in terms of such systems, with various structures and properties.
Metaphysical commitments often take the form of claims about the nature of those
structures and properties: which of them are real and not just artefacts of our models,
which are fundamental as opposed to derivative, and which are necessary as opposed
to contingent.

In this paper, we introduce a general and unified framework for describing systems,
based on the theory of dynamical systems and stochastic processes, and show how
this framework can be used to examine and illuminate some familiar philosophical
questions. Here are some examples:

• What does it mean for a system to be deterministic or indeterministic, and which
features of the system, if any, determine which others?

• Does the present determine the future? Does it determine the past? What is the
smallest set of facts encoding the system’s entire history? Could there be non-
temporal forms of determinism?

• How can we define nomological possibility and necessity for a system?
• What are the laws governing a particular system, and is there a distinction between
“laws” and “brute necessities”? How do laws depend on symmetries?

• What structure must a system have in order to permit generalizations from local
observations to global regularities?

• How might we use principles of parsimony such as Occam’s Razor when we make
such generalizations? And can we formulate a version of Occam’s Razor in terms
of symmetries?

• What is the role of space and time in a system? What is the relationship between
the geometry of space and time and the system’s behaviour?

• Is this spatiotemporal geometry exogenous, or is it determined by the dynamics? In
other words, are space and time more fundamental than the system’s dynamics, or
the other way around? Might space and time be “emergent”?

• How should we individuate systems? Should two structurally indistinguishable sys-
tems count as “the same”, or might they count as different?

For each of these questions, our framework allows us to identify in clear and precise
terms what is at stake. We illustrate the generality of the framework by sketching
how it can accommodate, schematically, the systems described by some standard
physical theories, such as classical mechanics, electrodynamics, quantum mechanics,
and special and general relativity. In principle, our framework can also be used to
describe many systems studied in the special sciences, such as biological, social, and
economic systems, though we do not have the space to develop these applications
here. We make a few remarks about special-science systems at the end of the paper

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and hope that our framework will serve as a basis for future work in some of those
areas.1

The paper is structured as follows. We discuss three classes of systems, in increasing
order of generality. We call the first temporally evolving systems (Sect. 2), the sec-
ond spatially extended systems (Sect. 3), and the third amorphous systems (Sect. 4).
We offer a conceptual toolbox for describing and analysing each class of systems,
covering notions such as states and histories, determinism and indeterminism, nomo-
logical possibility and necessity, modal and probabilistic properties, symmetries and
laws, ergodicity and its significance in making scientific inference possible, Occam’s
Razor, and the role of time and/or space. We first explain all of these notions in the
context of the simplest class of systems (in Sect. 2) and then generalize from there (in
Sects. 3 and 4). The paper also includes some more technical appendices, on factor
systems (relevant to the analysis of systems at different levels of abstraction), on par-
tial and local symmetries (relevant to “local” laws and the analysis of systems with
special initial or boundary conditions), on criteria of parsimony in relation to which
symmetries to postulate (relevant to Occam’s Razor), and on the definition of spatial
distance in quantum–mechanical systems (which raises special challenges).

Although the paper presupposes a willingness to engage with technical materi-
al—and a basic familiarity with science will be helpful—our goal is to keep the
exposition as simple and self-contained as possible. Our intended contribution is
twofold: methodological and substantive. On the methodological side, we aim to offer
a unified and yet accessible framework for the philosophical analysis of many of the
systems studied in the sciences. While the basic ideas originate from the theory of
dynamical systems and stochastic processes in mathematics and physics, and partially
overlapping formalisms can be found in earlier works (e.g., by Earman 1986; van
Fraassen 1989; Frigg et al. 2011; Werndl 2009a, b; Bishop 2011; Butterfield 2012;
Yoshimi 2012), the key ideas remain underappreciated in philosophy, and to our knowl-
edge, an equally unified (and, we think, accessible) framework is not yet available in
the philosophical literature.

On the substantive side, we aim to offer a number of novel insights, for example
concerning (i) the nature of nomological possibility and necessity in a system and
the definition of determinism and indeterminism, (ii) the role of symmetries in distin-
guishing between “laws” and “brute necessities” in a system, (iii) the significance of
symmetries and ergodicity as prerequisites for scientific inference, (iv) the relation-
ship between Occam’s Razor and the symmetries of a system, and (v) the possibility
that the topology and geometry of space and time may be emergent properties result-
ing from a system’s correlation structure. These, we hope, will be useful substantive
contributions, over and above the paper’s unificatory contribution.

1 Ideas from the theory of dynamical systems and stochastic processes have already begun to be applied
to fields such as economics, biology, and cognitive science. See, e.g., van Gelder (1995), Auyang (1998),
Juarrero (1999) and Silberstein and Chemero (2012).

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2 Temporally evolving systems

2.1 Basic definitions

We begin with the simplest class of systems whose states evolve over time.2 To define
a system in this class, we need to specify what time is, what the system’s states are, and
how these states may evolve over time. Time is represented by a set of points T that is
linearly ordered; we write<for the “before” relation. The state of the system at each
point in time is given by an element of some state space X. For the moment, we make
no assumptions about the internal structure of the states in X; they are uninterpreted
primitives. A history of the system, capturing “state evolution”, is a path through the
state space, represented by a function h from T into X. For each time t in T, h(t) is
the system’s state at time t. In a physical system, each state might be a completely
specified microphysical state in which the system could be at a particular point in time,
and histories would be possible trajectories of the system through its state space over
time.

We write � to denote the set of all histories deemed possible. Histories play the
role of possible worlds. Thus, the structure of � reflects the notion of possibility we
wish to capture. If we are interested in logical possibility, then � is simply the set of
all logically possible functions from T into X, which we call H. If we are interested in
some form of nomological possibility, such as physical possibility, � will often be a
proper subset of H. Our intended interpretation of possibility throughout this paper is
the nomological one, since we want to distinguish between histories that are permitted
by the laws governing our system and histories that are not.

Subsets of � are called events. We can apply logical operations to events. The con-
junction of two events E and E ′ is given by their intersection E ∩ E′. Their disjunction
is given by their union E ∪ E′. The negation of an event E is given by its complement
~ E � �\E. Later we introduce possibility and necessity operators.

To complete the definition of a temporally evolving system, we must define prob-
abilities on �. Formally, we introduce a conditional probability structure.3 This is
a family of conditional probability functions {PrE}E⊆�, consisting of one PrE for
each event E in �, where PrE assigns to any event in � the conditional probability of
that event, given E.4 The family must satisfy certain consistency conditions, such as
compatibility with Bayesian conditionalization.5 Now, a temporally evolving system
is the pair consisting of the set � of possible histories and the conditional probability
structure {PrE}E⊆�.

For example, in a weather system, X would be the set of all possible weather states
and � the set of all possible weather histories. For each particular weather event E,

2 We build on the formalism in List (2014) and List and Pivato (2015).
3 Conditionalprobabilitystructureshavepreviouslybeenconsideredbyseveralauthors,e.g.,Popper(1968),
Rényi (1955), van Fraassen (1976), as reviewed in Halpern (2010).
4 Each PrE is defined on a suitable σ-algebra A(�) on �; we set the technicalities aside. For any E �� ∅,
PrE has the standard properties of a probability measure. But, for technical reasons, Pr∅(D) � 1 for all D.
5 For any subsets C ⊆ D ⊆ E ⊆ � we have Pr E (C) � Pr E (D) × Pr D(C). Also, PrD(E) � 1 for all
D ⊆ E ⊆ �.

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say a hot temperature on Monday, the function PrE assigns to every weather event D,
say a thunderstorm on Tuesday, the conditional probability of its occurrence, given E.

In principle, the probability structure admits two interpretations. Under an objec-
tivist interpretation, it is a feature of the system itself and thus represents objective
chance (see, e.g., Lewis 1986; Schaffer 2007; List and Pivato 2015). Of course, objec-
tivechancecouldbe degenerate, i.e.,restrictedtotheextremalvalues0or1.Degenerate
objective chance is a much-discussed feature of deterministic systems; we return to
this point later. Under a subjectivist interpretation, the probability structure is not a
feature of the system itself, but represents an observer’s beliefs about the system, as
in subjective Bayesianism (e.g., de Finetti 1972). The most natural way to read this
paper is to assume the objectivist interpretation, though our formalism itself is neutral.

Familiar examples of temporally evolving systems are the system of planets orbiting
the sun or any other classical mechanical system, a hydrogen atom or any other quan-
tum–mechanical system, the earth’s climate system or any other statistical mechanical
system, and (arguably) the global economy or some other closed macro-economic
system. Generally, any classical dynamical system is a special case of a temporally
evolving system, as is any stochastic process under the standard definition.6

For theoretical simplicity, we focus on closed systems, which are not subject to any
external perturbations. However, one could also represent open systems in our frame-
work, by encoding any external perturbations as additional sources of randomness in
the system’s conditional probability structure (“random forcings”).7

2.2 Determinism and indeterminism

Conventionally, a system is called deterministic if, in that system, the past always
determines the future. Formally, for any history h and any point in time t, let ht be the
initial segment of that history up to t. This is the function h restricted to the points

6 A classical dynamical system consists of a set X (the state space) and a function φ from X into itself
that determines how the state changes over time (the dynamics). Let T � {0,1,2,3,…}. Given any state
x in X (the initial conditions), the orbit of x is the history h defined by h(0) � x, h(1) � φ(x), h(2) �
φ(φ(x)), and so on. Let � be the set of all orbits determined by (X, φ) in this way. Let {Pr ′E }E⊆X be
any conditional probability structure on X. For any events E and D in �, we define PrE (D) � Pr ′E ′ (D′),
where E′ and D′ are the sets of all states in X whose orbits lie in E and D, respectively. Then {PrE }E⊆�
is a conditional probability structure on �. Thus, � and {PrE }E⊆� together form a temporally evolving
system. However, not every temporally evolving system arises in this way. While classical dynamical
systems are deterministic, temporally evolving systems also subsume stochastic processes. Formally, a
stochastic process is a temporally indexed collection of random variables {X t: t ∈ T} (with T as before)
on some probability space (�, A(�), Pr), where � is some underlying (abstract) set of possible worlds,
A(�) is a σ-algebra on �, and Pr is a probability measure on A(�). For each time t, we can think of the
random variable X t as expressing the state of the stochastic process at t. To see that this gives rise to a
temporally evolving system, note that each world ω in � induces a history h in our sense, where, for each
t, h(t) is the realization of X t at world ω. Most of the conditional probability structure {PrE }E⊆� can then
be derived from Pr via Bayes rule, but {PrE }E⊆� may contain some additional information not encoded
in Pr (namely, conditional probabilities arising from zero-probability events). In Sects. 3 and 4, we extend
our framework to even more general classes of systems.
7 The use of such random forcings does not imply that certain features of the world are genuinely random.
Instead, the “randomness” of such forcings is best understood epistemically—as a shortcut for an explicit
and detailed description of the part of the world which lies outside the model. (This is true whether we
adopt an objectivist or subjectivist interpretation of the probability structure overall.).

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in time up to t. History h is deterministic if, at any time t in T, the initial segment ht
admits only one possible continuation in �, where a continuation of ht is a history h

′
such that h′t � ht. History h is indeterministic if, for some time t, ht has more than one
possible continuation in �. The system as a whole is called deterministic if all histories
in � are deterministic, and indeterministic if some histories in � are indeterministic.8

For example, classical mechanical systems, such as the solar system on the New-
tonian picture, are deterministic. By contrast, quantum–mechanical systems, such as
a decaying uranium atom, are indeterministic (assuming no hidden variables). If the
wave function, which encodes the state of the quantum system, collapses at time t, the
initial segment ht of the system’s history h can admit multiple continuations.

Indeterministic systems allow non-degenerate chance as we move along a given
history, while deterministic systems do not.9 To see this, note that the chance of any
event E in history h at time t is the conditional probability of E, given that the initial
segment ht has occurred. Since the event that the initial segment ht has occurred is
given by the set of all continuations of ht—call this set [ht]—the probability in question
is Pr[ht ](E). If history h is deterministic, the entire conditional probability function
Pr[ht ] is degenerate, i.e., it assigns probability 0 or 1 to every event E. This is because
the initial segment ht has only one continuation, namely h itself, and so the specified
event [ht] contains only a single history, h. Then Pr[ht ](E) is 1 if h belongs to E and 0
otherwise. In contrast, if history h is indeterministic, Pr[ht ] may be non-degenerate,
assigning probabilities strictly between 0 and 1 to some events E. This is because [ht]
need not be singleton here, and so Pr[ht ] is less constrained. (For the moment, we set
aside phenomena such as “higher-level” indeterminism and chance, as discussed in
List and Pivato 2015. We briefly consider such phenomena at the end of this paper.)

Our framework also allows us to formulate some more general, less familiar notions
of determinism. For any subset T ′ of T—not just the set of time points up to a particular
time t—we can ask whether the restriction of a given history to the points in T ′ uniquely
determines the rest of that history. Let hT ′ denote the restriction of the function h to
T ′. Our question, then, is whether hT ′ has a unique extension to all of T in �, where
an extension of hT ′ is a history h

′ such that h′T ′ � h T ′. If there is a unique extension,
history h may be called T ′-deterministic.10

We might ask, for instance, whether the entire history of a system, both past and
future, is determined by its present state alone. Similarly, we might ask whether, given

8 On these definitions, see also List (2014) and List and Pivato (2015). Related definitions of determinism
(broadly, in terms of a history’s unique extendibility based on an initial state or segment) can be found,
for instance, in van Fraassen (1989, section 10.4) and Butterfield (2012). See also the classic overview in
Earman (1986) and the discussion of varieties of determinism in Sobel (1998). Earman shares our focus on
local-to-global determination, as noted below. While Sobel discusses more than 90 variants of determinism,
he frames the question differently than us. First of all, his focus, unlike ours in this paper, is on the free-
will debate. Secondly, for us, the central question is (roughly): does the complete specification of the
world (history) for a particular space–time region (e.g., a history’s initial segment) determine a complete
specification of that world (history) for other space–time regions? By contrast, for Sobel, the central question
is (roughly): does an event that occurs at a particular space–time location have a cause at some antecedent
space–time location? Since we do not explicitly discuss the topic of causation here, Sobel’s analysis and
ours are not immediately inter-translatable.
9 For discussion, see, e.g., Schaffer (2007) and List and Pivato (2015).
10 For related ideas, see also van Fraassen (1989, section 10.4).

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the states of the system at two points in time, there is a unique history connecting
them. So, one can in principle consider not only the familiar idea of “past-to-future”
determinism, but also other forms of “local-to-global” determinism. In Sect. 3.2, we
develop these ideas further and consider, among other things, spatial rather than tem-
poral forms of determinism as well as locally restricted ones.

2.3 Nomological possibility and necessity

We can explicitly define the notions of nomological necessity and possibility in our
framework.11 Intuitively, an event E is nomologically possible in history h at time t if
the initial segment of that history up to t admits at least one continuation in � that lies
in E; and E is nomologically necessary in h at t if every continuation of the history’s
initial segment up to t lies in E.

More formally, we say that one history, h′, is accessible from another, h, at time t
if the initial segments of h and h′ up to time t coincide, i.e., ht � h′t . We then write
hRt h

′. The binary relation Rt on possible histories is in fact an equivalence relation
(reflexive, symmetric, and transitive). Now, an event E ⊆ � is nomologically possible
in history h at time t if some history h′ in � that is accessible from h at t is contained
in E. Similarly, an event E ⊆ � is nomologically necessary in history h at time t if
every history h′ in � that is accessible from h at t is contained in E.

We can thus define two modal operators, ◆t and �t, to represent possibility and
necessity at time t. We define each of them as a mapping from events to events. For
any event E ⊆ �,

◆t E � {h ∈ �: for some h′ ∈ � with hRt h′, we have h′ ∈ E},
�t E � {h ∈ �: for all h′ ∈ � with hRt h′, we have h′ ∈ E}.
So, ◆t E is the set of all histories in which E is possible at time t, and �t E is the

set of all histories in which E is necessary at time t. Accordingly, we say that ◆t E
holds in history h if h is an element of ◆t E, and �t E holds in h if h is an element of
�t E. As one would expect, the two modal operators are duals of each other: for any
event E ⊆ �, we have �t E � ~ ◆t ~ E and ◆t E � ~ �t ~ E.

Two remarks are due. First, although we have here defined nomological possibility
and necessity, we can analogously define logical possibility and necessity. To do this,
we must simply replace every occurrence of the set � of nomologically possible
histories in our definitions with the set H of logically possible histories. Second, by
defining the operators ◆t and �t as functions from events to events, we have adopted
a semantic definition of these modal notions. However, one could also describe them
syntactically, by introducing an explicit modal logic. For each point in time t, the logic
corresponding to the operators ◆t and �t would then be an instance of a standard S5
modal logic (on S5, see, e.g., Priest 2001).

Our analysis shows how nomological possibility and necessity depend on the
dynamics of the system, as evident from the time-indexed nature of the relevant modal
operators. In particular, as time progresses, the notion of possibility becomes more
demanding: fewer events remain possible at each time. And the notion of necessity

11 We here employ, and subsequently generalize, a construction from List (2014).

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becomes less demanding: more events become necessary at each time, for instance
due to having been “settled” in the past. Formally, for any t and t′ in T with t < t′ and
any event E ⊆ �,
if ◆t′ E then ◆t E,
if �t E then �t′ E.

Furthermore, in a deterministic system, for every event E and any time t, we have
◆t E � �t E. In other words, an event is possible in any history h at time t if and
only if it is necessary in h at t. In an indeterministic system, by contrast, necessity and
possibility come apart.

Just as we previously discussed different notions of determinism—not just “past to
future” but also “local to global”—so we can generalize the notions of possibility and
necessity in a similar way. Let us say that one history, h′, is accessible from another,
h, relative to a set T ′ of time points, if the restrictions of h and h′ to T ′ coincide, i.e.,
h′T ′ � hT ′. We then write hRT ′ h′. Accessibility at time t is the special case where T ′
is the set of points in time up to time t. We can define nomological possibility and
necessity relative to T ′ as follows. For any event E ⊆ �,

◆T ′ E � {h ∈ �: for some h′ ∈ � with hRT ′ h′, we have h′ ∈ E},
�T ′ E � {h ∈ �: for all h′ ∈ � with hRT ′ h′, we have h′ ∈ E}.
Although these modal notions are much less familiar than the standard ones (pos-

sibility and necessity at time t), they are useful for some purposes. In particular, they
allow us to express the fact that the states of a system during a particular period of
time, T ′ ⊆ T, render some events E possible or necessary.

Finally, our definitions of possibility and necessity relative to some general subset
T ′ of T allow us to define completely “atemporal” notions of possibility and necessity.
If we take T ′ to be the empty set, then the accessibility relation RT ′ becomes the
universal relation, under which every history is related to every other. An event E is
possible in this atemporal sense (i.e., ◆∅E) if and only if E is a non-empty subset of
�, and it is necessary in this atemporal sense (i.e., �∅E) if E coincides with all of �.
These notions might be viewed as possibility and necessity from the perspective of
some observer who has no temporal or historical location within the system and looks
at it from the outside.

2.4 Modal and probabilistic properties

Ultimately, all modal properties of a temporally evolving system are encoded by the
set � of nomologically possible histories, and all probabilistic properties are encoded
by the conditional probability structure {PrE}E⊆�. This raises the question: which, if
any, of these properties qualify as “laws” of the system, and what does this mean?

One possible view is that:

• any property that is satisfied by all histories in � counts as a law of the system,
specifically a “modal law”; and

• any property of the conditional probability structure {PrE}E⊆� counts as a law of
the system, specifically a “probabilistic law”.

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Indeed, since the system is fully specified by � and {PrE}E⊆�, one might interpret
anything that is globally true of its possible histories or its probability structure as a
law of that system. A view along these lines is expressed in a classic paper by Sellars
(1948, p. 309): “A natural law is a universal proposition, implicative in form, which
holds of all histories of a family of possible histories; as such it is distinguished from
‘accidental’ formal implications which hold of one or more possible histories of the
family, but do not hold of all.” So, the notions “being a law” and “being nomologically
necessary” essentially coincide.

Against this view, however, we want to argue that even among nomologically neces-
sary properties of a system—those that are not contingent on particular histories—one
can distinguish between “laws” on the one hand and “brute necessities”, which are not
law-like, on the other. Laws, we suggest, have a testable and generalizable character
which brute necessities lack. To explain this, we introduce two preliminary notions,
properties of histories and probabilistic properties, and then provide a criterion for
identifying which of them qualify as laws.

A property of histories, P, is a binary feature that a history may or may not have.
Formally,itcanbeassociatedwithsomesubset,denoted[P],ofthesetH ofalllogically
possible histories. A history h satisfies P if h belongs to [P]. We call [P] the extension
of P. A property satisfied by every history in � can be called nomologically necessary
for the system. Newton’s three laws of motion are examples of such properties in the
case of a classical mechanical system.

A probabilistic property, P, is a binary feature that a conditional probability struc-
ture may or may not have. Formally, it is associated with a subset, denoted [P], of
the set � of all logically possible conditional probability structures on �. A condi-
tional probability structure {PrE}E⊆� satisfies P if it belongs to [P]. We call [P]
the extension of P. An example of a probabilistic property is the one that says: “The
unconditional probability of event F is ½.” Its extension is the set of all conditional
probability structures {PrE}E⊆� for which Pr�(F) � ½. Another example is the sec-
ond law of thermodynamics. This is a probabilistic property that is satisfied by the
conditional probability structure of a statistical mechanical system.

Our goal is to distinguish between those properties that qualify as “laws” of the
system and those that do not. We capture that distinction through the notion of symme-
tries. Informally, a symmetry is a transformation that acts on either the state space X or
the set of time points T, or both, and which can capture certain admissible changes in
perspective on the system. Laws, we suggest, are those nomologically necessary prop-
erties which are invariant under symmetries and which therefore hold across changes
in perspective. We now make this formally precise.

2.5 Symmetries

We first consider symmetries acting on the state space; we then turn to symmetries
acting on time; and we finally consider more general symmetries. To introduce state
symmetries, we begin with some preliminary definitions. Let φ be any function from
X into itself, i.e., a transformation on the state space. We use this transformation to
define a function from histories to other histories. For reasons that will become clear,

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we do not restrict the function to nomologically possible histories, but define it as a
function on H, the set of all logically possible histories. Specifically, for any history
h in H, we define the transformed history

φ(h) � h′, where, for all t in T , h′(t) � φ(h(t)).

For example, if X � {a, b, c, d,…, z}, the function φ might shift every letter in the
alphabet one place to the right, i.e., a to b, b to c, and so on, and z back to a. If we
represent histories as sequences of elements in X, interpreted as the system’s states at
times 1, 2, 3, …, then applying φ to the history h � (b, a, c, f , z,…) yields the history
h′ � (c, b, d, g, a,…). For convenience, we use the letter φ to denote both the original
function on the state space and the induced function on the set H of histories. Note
that since the set of nomologically possible histories may be a proper subset of the set
of all logically possible histories, the image of a history in � need not be in �.

To define what it means for φ to be a symmetry, we need one further preliminary
definition. For any collection of histories E in H, the inverse image of E under φ is
the set of all histories h in H such that φ(h) lies in E.12 For example, if E is the set
of all histories whose state at time 3 is c, and φ is the letter-shifting transformation,
then the inverse image of E under φ is the set of all histories whose state at time 3 is
b. Now, the function φ is a symmetry of our system if

• φ(�) � �, i.e., (i) φ(h) is in �, for all h in �, and (ii) for any h in �, there is some
h′ in � such that φ(h′) � h; and

• for any events E and D in �, if E′ and D′ are the inverse images of E and D under
φ, then PrE′(D

′) � PrE(D).13
Intuitively, a symmetry is a transformation that preserves the system’s modal and

probabilistic structure. In our example, where X � {a, b, c, d,…, z} and φ is the letter-
shifting function, the first part of this definition implies that the set of nomologically
possible histories is preserved under shifting of letters. For instance, if (b, a, c, f , z,…)
is a nomologically possible history, then so is (c, b, d, g, a,…).14 To illustrate the second
part, let E be the set of all histories in � whose state at time 3 is c, and let D be the set
of all histories in � whose state at time 5 is a (so that E′ is a suitable set of histories
whose state at time 3 is b, and D′ is a suitable set of histories whose state at time 5 is
z).15 The conditional probability that the state of a history at time 5 is a, given that at
time 3 it is c, must then equal the conditional probability that the state at time 5 is z,
given that at time 3 it is b.

12 Formally, we write φ−1(E) � {h in H: φ(h) is in E}. The use of this notation for inverse images of sets
does not imply that the function φ is invertible. Moreover, the inverse image of an event E could be empty,
namely if none of the histories in E can be “reached” as transformations of other histories.
13 Strictly speaking, {PrE }E⊆� is only defined for subsets of �. But we can extend it as follows: for any
D, E ⊆ H, if E ∩ � �� ∅, let Pr E (D) � Pr E ∩�(D ∩ �).Cases where E ∩ � � ∅ are not relevant here,
because of the first bullet point in the definition of symmetry.
14 And further, there is a nomologically possible history, namely (a, z, b, e, y,…), such that applying the
letter-shifting function to it yields the history (b, a, c, f , z,…).
15 Formally, E′ consists of all the histories h in H such that φ(h) is in � and h(3) � b. Similarly, D′ consists
of all the histories h in H such that φ(h) is in � and h(5) � z.

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Obviously, not all state transformations are symmetries. Whether there are any
non-trivial state symmetries depends on the system in question, i.e., it depends on
� and {PrE}E⊆�. In classical mechanical systems, state symmetries include spatial
translations, which shift everything in a certain direction by a certain distance, rotations
and reflections, and permutations of particles with equal mass. Those transformations
preserve the modal and probabilistic structure of the relevant systems.

Similarly, we can define time symmetries. Again, we begin with some preliminary
definitions. Let ψ be any function on T, i.e., a transformation on time. For any history
h, we define the transformed history16

ψ(h) � h′, where, for all t in T , h′(t) � h(ψ(t)).

For example, if T � {1,2,3,…}, the function ψ might be given by ψ(t) � t + 5 for
all t in T. It maps the history (x1, x2, x3, …) (a sequence of states across time) to the
history (x6, x7, x8, …). As in the case of state symmetries, ψ induces a function from
the set H to itself. In analogy to the earlier definition, ψ is a symmetry if
• ψ(�) � �;
• for any events E and D in �, if E′ and D′ are the inverse images of E and D under

ψ, then PrE′ (D
′) � PrE(D).

In our example, where T � {1,2,3,…} and ψ(t) � t + 5, the first part of this definition
implies that if h � (x1, x2, x3,…) is a nomologically possible history of the system,
then so is h′ � (x6, x7, x8,…).17 To illustrate the second part, suppose that E is the set
of all histories in � whose state at time 3 is c, while D is the set of all histories in �
whose state at time 4 is a (so that E′ is a suitable set of histories whose state at time 8
is c, while D′ is a suitable set of histories whose state at time 9 is a). The conditional
probability that the state at time 9 is a, given that at time 8 it was c, must then equal
the conditional probability that the state at time 4 is a, given that at time 3 it was c.18

Just as not all state transformations are symmetries, so not all time transformations
aresymmetrieseither.Inmostclassicalphysicalsystems,timesymmetriesinclude time
translations, such as ψ(t) � t + 5, but exclude non-linear transformations, such as ψ(t)
� t2. In systems where the state does not encode explicitly “kinetic” properties (such
as momentum), simple time reversals, such as ψ(t) � − t, can also be time symmetries.
16 Typically, we require ψ to be order-preserving, i.e., for all t and t′ in T, if t < t′, then ψ(t)< ψ(t′). For
example, if T � {1, 2, 3,…} with the standard ordering, the functions ψ(t) � t + 5 and ψ(t) � 5t are
order-preserving. But we do not build this requirement into our definition of a time symmetry. Note that
some time symmetries, such as time reversals in classical physical systems, are not order-preserving.
17 And further, there is some nomologically possible history h′′ � (u, v, w, y, z, x1, x2, x3,…) such that
shifting the system’s state in h′′ five time periods into the future yields the history h. (Here, the exact values
of u, v, w, y, z are irrelevant, as long as h′′ is nomologically possible.).
18 Note that classical dynamical systems have a particularly rich set of time symmetries. Let (X, φ) be
a dynamical system, as defined in footnote 6. Suppose the function φ (which maps from X into itself) is
surjective, i.e., for all x in X, there is some y in X such that φ(y) � x. Then the set � of orbits is invariant
under all time-shifts. Let {Pr′E }E⊆X be a conditional probability structure on X, and let {PrE }E⊆� be
the conditional probability structure it induces on �. Suppose that {Pr′E }E⊆X is φ-invariant, i.e., for any
subsets E and D of X, if E′ � φ−1(E) and D′ � φ−1(D), then Pr ′

E ′ (D
′) � Pr ′E (D). Then every time

shift is a temporal symmetry of the resulting temporally evolving system. The study of dynamical systems
equipped with invariant probability measures is the purview of ergodic theory.

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For example, the partial differential equations describing wave propagation in an ideal
medium are invariant under simple time reversals. But many other systems, such as
thermodynamic ones and diffusion processes, do not admit such simple time reversals.

More general symmetries include composite functions resulting from the combi-
nation of transformations of X and transformations of T. These are best understood as
functions acting on the set H of logically possible histories directly, with the properties
introduced above. A familiar example in classical mechanical systems is a time rever-
sal, which involves both a negation of the time index and a negation of all momentum
vectors in the system (not to be confused with a simple time reversal, as mentioned
earlier).19 A more complex example is a Galilean transformation, which adds a con-
stant vector to the momentum vectors of all particles and also a time-varying sequence
of spatial shifts to the particle positions, thereby converting the system to a different
inertial reference frame. See footnote 48 below for details.

We can think of symmetries—whether they act on the state space, on time, or on
both—as transformations that encode admissible changes in perspective on a system,
insofar as they preserve the system’s modal and probabilistic structure. We write � to
denote the set of all symmetries of our temporally evolving system. This set has the
algebraic structure of a monoid. Formally, a monoid is a set of transformations (here
of H) which (i) contains the identity transformation (mapping every history to itself)
and (ii) is closed under composition (i.e., for any two transformations in the set, the
transformation obtained by applying first one of the two transformations and then the
other is also in the set). An example of a symmetry monoid is the set of all rotations of a
classical mechanical system around a fixed axis: the identity transformation obviously
belongs to this set, being a rotation by an angle of zero, and the composition of any
two rotations is still a rotation.20

2.6 Laws and their significance

As anticipated, the laws of a system are those nomologically necessary properties
within it that are invariant under symmetries. This, we show, makes laws open to
testing and generalization. Laws, one might say, have a “scrutable” and “projectable”
character. The close relationship between symmetries and laws has been recognized
before.21 For instance, Wigner (1967) takes symmetries to be “a prerequisite for the

19 As Roberts (2013) has argued, in general, a time reversal must not only map the time coordinate
t to − t, but also apply an appropriate transformation to the system’s state at each point in time (in special
cases, this could be the identity transformation, as in the case of a simple time reversal). Generally, we
can think of the state of the system at each point in time as encoding not only some “static” properties
(such as each particle’s position), but also some “kinetic” properties (such as each particle’s momentum).
While static properties are preserved under time reversals, kinetic properties are not generally preserved.
Similarly, in quantum mechanics, time reversals involve not only a reversal of the time coordinate, but also
taking the conjugate of the wave function’s values.
20 To see that the set � of all symmetries of a temporally evolving system forms a monoid, note that (i) the
identity transformation is trivially a symmetry, and (ii) if two transformations each qualify as symmetries,
by preserving the modal and probabilistic structure of the system, then so does their composition. If all
symmetries are invertible, then � becomes a group, but we need not assume this.
21 See, e.g., Wigner (1967), van Fraassen (1989, Part III), Mainzer (1996, section 5.3), Brading and Castel-
lani (2003, 2013) and Baker (2010).

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very possibility of discovering the laws of nature” (as Brading and Castellani 2013 put
it; see also French 2014). And van Fraassen (1989), in his classic study of symmetries
in science, considers defining laws as “facts which are invariant under symmetries”,
though ultimately does not endorse that definition. But none of the existing accounts
clarifies the relationship between laws and symmetries in a way that we consider fully
satisfactory.22 We develop this relationship in detail in the case of modal laws. We
subsequently consider probabilistic laws too, but, due to space constraints, discuss
those more briefly.

To define the notion of a modal law, consider a property of histories, P. Recall that
P is nomologically necessary for the given system if its extension, [P], includes all
histories in �. For any symmetry ψ, we say that P is invariant under ψ if the set [P] is
equal to its inverse image under ψ. Property P is a law if it is nomologically necessary
for the system and invariant under all symmetries in �.

For example, suppose T � {1,2,3,…}, and suppose that, for any non-negative
integer r, the system has the time symmetry ψr defined by ψr(t) � t + r for all t in T;
for simplicity, the system has no other symmetries. So, � � {ψr: r � 0,1,2,…}. Now,
suppose all histories in � satisfy property P which says: “If the state at time 5 is x, then
at time 6 it is y.” Despite being nomologically necessary for the present system, this
property falls short of being a law. The inverse image of [P] under any symmetry ψr
corresponds to the property P′ which says: “If the state of the system at time 5 + r is x,
then at time 6 + r it is y.” Clearly, unless r � 0, [P′] is not the same as [P], and so P is
not invariant under the system’s symmetries. We call such a property—nomologically
necessary but not invariant under symmetries—a brute necessity.

By contrast, suppose all histories in � have the property P which says: “For any t
in T, if the state of the system at time t is x, then at time t + 1 it is y.” It is easy to see
that this property is invariant under all symmetries of the system: the inverse image of
[P] under any symmetry ψr is the same as [P]. Thus, P is a law.

For another example, consider the kinds of temporally evolving systems that arise
in classical mechanics. These satisfy the law of conservation of energy, which says
that the total energy (kinetic plus potential) remains constant over time. This can be
formulated as a property P of the form: “For any times t and t′ in T, the total energy of
the state at time t′ equals the total energy of the state at time t.” Clearly, this property
is invariant under the time symmetries {ψr} introduced above. As already mentioned,

22 For example, Wigner (1967) recognizes the centrality of laws to scientific discovery, but does not define
laws as facts that are invariant under symmetry. Instead, he seems to regard symmetry invariance principles
as “second-order laws”, which relate laws to other laws (or which relate a law to itself). So, for him, it
appears that laws establish relationships between events, and symmetries establish relationships between
laws. Van Fraassen’s analysis (1989) is in many ways a precursor to ours (see, e.g., ibid., Section 11.2).
But surprisingly he seems to reject the definition of laws as “facts invariant under symmetries” (ibid.,
Section 11.5). His proposed counterexamples (inspired by a passage from Weyl 1952) involve symmetry-
invariant facts which are not nomologically necessary (such as the number of planets in the solar system). In
contrast, we define laws as nomologically necessary facts which are invariant under symmetries. Mainzer
(1996) alludes to the connection between symmetry and simplicity (esp. on p. 580), but does not offer
a precise formal analysis. Brading and Castellani (2013) discuss the importance of symmetry in modern
physics, but do not propose symmetry as a criterion for scientific law. Finally, Baker (2010) discusses the
importance of symmetry arguments in metaphysics, especially in relation to the “identity of indiscernables”,
but again does not propose symmetry as a criterion for scientific law. That said, our analysis clearly lies in
the vicinity of what others have had in mind in their accounts of laws and symmetries.

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classical mechanical systems also have certain state symmetries, such as spatial trans-
lations, rotations, reflections, and the permutation of (equal-mass) particles. The total
energy of a state is unchanged by such symmetries too, so the property P will also
be invariant under spatial translations and (equal-mass) particle permutations. Indeed,
total energy is unchanged by every symmetry of the system, and for this reason, prop-
erty P is a law.23

As we will now see, laws are testable and generalizable in a way in which properties
that fall short of being laws are not, even if they are nomologically necessary. Suppose
we are trying to figure out the status of some property P. Is it nomologically necessary?
Does it capture a general regularity of our system? Is it a law? The first thing to note is
that when we investigate a system, we are seldom able to observe all its nomologically
possible histories. Conducting many “runs” of the same experiment is an attempt to
observe as many histories as possible, but even the best experimental design rarely
allows us to observe all possible histories. Furthermore, this strategy works only for
smaller systems that we can isolate in laboratory conditions. When the system is the
economy, the global ecosystem, or the universe as a whole, we are stuck in a single
history. We cannot step outside that history and look at alternative histories. The
observed history is the only evidence we have. Can we still say anything useful about
the status of property P? It is at this point that symmetries come into play.

Let us return to our simply example of a system with T � {1,2,3,…} and time
symmetries of the form ψr, where ψr(t) � t+r. Consider again the property P that
says “if the state at time 5 is x, then at time 6 it is y”, and suppose, as before, that P
is nomologically necessary, i.e., every history in � satisfies P. If we could observe
many nomologically possible histories of the system, we would be able to verify the
satisfaction of P in each case. But, as noted, we may be trapped in a single history,
h. All we can do is watch this history unfold. We first see h(1), then h(2), then h(3),
and so on. Importantly, we get to observe h(5) and h(6) only once, so we get only one
chance to observe whether h satisfies property P. Furthermore, even if h does satisfy
P, this is only a single data point, which tells us very little about the broader status
of P. Property P might as well be a contingent feature of the actual history we have
observed.

However, we do get to observe h(7), h(8), h(9), and so on. So, we can consider
properties such as P′: “if the state at time 7 is x, then at time 8 it is y”; and P′′: “if the
state at time 9 is x, then at time 10 it is y”; and so on. Note that P′ corresponds to the
inverse image of [P′] under ψ2; and P′ corresponds to the inverse image of [P] under
ψ4; and so forth. In other words, if we are patient, we can observe whether history
h satisfies the properties corresponding to the inverse images of the original property
under a lot of elements of the system’s symmetry monoid. Similarly, in a system with
spatial symmetries (of the sort we introduce in Sect. 3.5), we can in principle observe
whether h satisfies the properties corresponding to many of the relevant inverse images
simply by traveling in space.

Now, if property P was not itself invariant under symmetries, as in the case of our
example, we would not learn much from this exercise. We would learn that h satisfies
P (“if h(5) � x, then h(6) � y”), that it satisfies P′ (“if h(7) � x, then h(8) � y”), and
23 Similarly, one can define the laws of conservation of momentum, of angular momentum, and so on.

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that it satisfies P′′ (“if h(9) � x, then h(10) � y”), and so on. But, strictly speaking,
these are distinct properties, and on the face of it, they do not have all that much in
common. By contrast, if P is symmetry-invariant, as in the case of the property which
says “for all t, if h(t) � x, then h(t + 1) � y”, then P, P′, P′′, … are all the same
property, and thus the present exercise yields a whole series of experimental tests of
the same law.

Moreover, in this case, the single property P picks up a general pattern, of which
we can observe many instances even within a single history, h, and which lends itself
to extrapolation into the future. As h unfolds, we can observe that state x is followed
by state y not just once but many times. Furthermore, once we have observed this
regularity a sufficient number of times, we may feel confident in hypothesizing that P
is indeed a law and then predicting that, in the future, state x will also be followed by
state y.

Contrast this with the case of a property that is not symmetry-invariant, such as
“if h(5) � x, then h(6) � y”. Here, there is no such general pattern, and we have no
basis for making any predictions. This is the sense in which laws have a testable and
generalizable character that non-symmetry-invariant properties lack, even when they
are nomologically necessary.

There is another way of making the same points. Let P be some property, and let
P′, P′′, P′′′, and so on, be all of its inverse images under the various time (and other)
symmetries of the system. Let h be the history that we observe. Suppose that, by
exhaustive testing, we verify that h satisfies P, P′, P′′, P′′′, and so on. (Or perhaps
we only verify some subcollection of these properties, but then infer the rest of them
through some form of “empirical induction”, which is ubiquitous in science.) At this
point, we have actually verified that h satisfies an entire conjunction of properties,
informally P ∧ P′ ∧ P′′ ∧ P′′′ ∧ …, or more formally, the property P* with extension

[P ∗] �
⋂

ψ∈�
ψ

−1([P]).

Note that, by construction, property P* is invariant under all symmetries in �.
Thus, although we get to test the initial property P only once, by testing a bunch of
“P-like” properties at various points in time (and/or positions in space etc.), we have
tested not only P, but something much stronger, namely P*. But note that P* is not
just any arbitrary property: it is symmetry-invariant by construction and thus qualifies
as a law (provided it is also nomologically necessary). Moreover, by entailing all the
various instances of P-like properties, i.e., P, P′, P′′, P′′′, and so on, the hypothesis that
property P* is a law allows us to make predictions as to what will happen at different
points in time (or in space, or after making other admissible changes corresponding
to symmetries of the system).

This argument suggests that any property that we think we have corroborated by
performing a large number of empirical tests at different times (or locations in space,
or different orientations of the experimental apparatus, or different collections of oth-

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erwise identical atoms, and so on) is ipso facto a symmetry-invariant law, and not
merely a brute necessity.24

One can give a similar account of probabilistic laws. Let {Pr ′E }E⊆� be any con-
ditional probability structure, and let ψ be a symmetry of the system. We define
ψ({Pr ′E }E⊆�) to be the conditional probability structure {Pr

′′
E }E⊆� such that, for

any events E and D, we have Pr ′′E (D) � Pr ′E ′(D′), where E′ and D′ are, respectively,
the inverse images of E and D under ψ. Let P be a probabilistic property. Recall
that its extension, [P], is a subset of the set � of all possible conditional probability
structures on �. We say that P is invariant under ψ if [P] is equal to its inverse
image under ψ. A property P that is satisfied by the system’s conditional probability
structure {PrE}E⊆� is a law of the system if it is invariant under all symmetries in �.

For example, suppose T � {1,2,3,…}, and let the time symmetries ψr be as defined
before. Let Y and Z be two subsets of the state space X, and suppose the system’s
conditional probability structure satisfies the probabilistic property P which says:
“Conditional on the state being in Y at time 5, there is a 50% probability that the
state will be in Z at time 6.” The inverse image of [P] under ψ2 corresponds to the
property P′ which says: “Conditional on the state being in Y at time 7, there is a 50%
probability that the state will be in Z at time 8.” Clearly, [P′] is not the same as [P].
Thus, [P] is not invariant under ψ2, and so P is not a probabilistic law of the system.

However, suppose the conditional probability structure satisfies the property P
which says: “For any time t in T, conditional on the state being in Y at time t, there
is a 50% probability that the state will be in Z at time t + 1.” Then it is easy to see
that [P] is invariant under ψr for all positive integers r. If � consists only of the time
symmetries {ψr: r � 0,1,2,3,…}, then P is invariant under all elements of �, and
so P is a probabilistic law.

As in the case of modal laws, probabilistic laws capture general and repeatable
patterns. Consider again the probabilistic property P which says: “Conditional on the
state being in Y at time 5, there is a 50% probability that the state will be in Z at time
6.” Recall that this property is not invariant under our system’s time symmetries. Even
if the system’s conditional probability structure satisfies this property, the property
does not capture a general pattern. It concerns only the probabilistic transition from
time 5 to time 6. If, however, the system has all the time symmetries in �, then we
can expect the system to satisfy the properties corresponding to the inverse images of
[P] under the various time symmetries, for instance: P′, which says: “conditional on
the state being in Y at time 7, there is a 50% probability that the state will be in Z at
time 8”; and P′′, which says: “conditional on the state being in Y at time 9, there is
a 50% probability that the state will be in Z at time 10”; and so forth. By conjoining
those properties, we can deduce the more general property P∗, which says: “For any
t in T, if the state of the system is in the set Y at time t, there is a 50% probability that

24 One might raise the following concern. If P is nomologically necessary, and h is a possible history, then
h satisfies not only P, but also all inverse images of P under all symmetries. To put it more simply, if P is
nomologically necessary, then all its inverse images under all symmetries are also nomologically necessary.
So, whenever P is nomologically necessary, we should be able to corroborate this fact via repeated testing,
even if P is merely a brute necessity, rather than a symmetry-invariant law. At first sight, this may seem to
challenge our claim that laws stand out in their testability. However, what have we really corroborated via
the present exercise of repeated testing is not property P itself, but the much stronger (symmetry-invariant)
property P* defined above. And that property P* is not merely nomologically necessary, but a law.

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it will be in Z at time t + 1.” This property is invariant under all the time symmetries,
and it does indeed qualify as a law.

The foregoing considerations show that symmetries are central to the testable and
generalizable character of laws. Without suitable symmetries, generalizing from local
observations to global laws or testing hypothesized laws would not be possible, espe-
cially if we can observe only a single history of a given system. Nor would it be
possible to make predictions about the future based on regularities observed in the
past. In a slogan, for scientific inference and prediction to work, the system must have
sufficient symmetries. In effect, when we engage in scientific reasoning about some
system, or even about the world at large, we rely on the auxiliary hypothesis that this
system, or the world, is sufficiently symmetrical. If our system, or the world, were
what Cartwright (1999) calls “dappled”, then presumably we would not be able to
presuppose such symmetries, and our ability to make scientific generalizations would
be compromised.25

In “Appendix A”, we extend the present analysis to factor systems, which are
obtained by abstracting away from certain details of the original system. In “Appendix
B”, we extend it to partial and local symmetries, which are often found in systems
with special initial conditions and/or boundary conditions.26

2.7 Ergodicity and its significance

We have noted that, when we scientifically investigate a system, we rely heavily on
symmetries. As we may be able to observe just a single history, it is only thanks to
symmetries that we can learn general features of the system from local observations.
We have seen, for instance, that if we can observe that “if h(5) � x, then h(6) � y”,
and the system has the time symmetries of the form ψr(t) � t+r, then we can infer
the general law that says: “for all t, if h(t) � x, then h(t + 1) � y”. Similarly, if we can
observe that “conditional on the state being in Y at time 5, there is a 50% probability
that it will be in Z at time 6”, then we can infer the general law that says: “For any t

25 In a system without symmetries (aside from the identity transformation, which is trivially a symmetry),
the distinction between laws and brute necessities could not be drawn. Every nomologically necessary
property would then vacuously qualify as a “law”: it could not fail to be invariant under any symmetries.
But this is clearly a degenerate case. As just argued, in such a system our ability to perform science would
be seriously limited. Terminologically, one might distinguish between trivial and non-trivial laws. Trivial
laws are ones that are vacuously symmetry-invariant (because there are no non-trivial symmetries). Our
interest is of course in non-trivial laws.
26 Finally, it is worth emphasizing that, in our analysis, laws are always defined relative to a given system.
When some property qualifies as a law of the system given by � and {PrE }E⊆�, this does not imply that it
will also qualify as a law of a different system, say �′ and { Pr ′

E ′ }E ′⊆�′ . Different systems may be governed
by different laws. Indeed, it is widely held that while our universe has certain laws of nature, other universes
with distinct laws are logically possible. One might wonder whether our analysis only captures a notion of
“system laws” rather than “laws of nature”. However, if we take the system given by � and {PrE }E⊆� to
represent the universe as a whole, then our account can be interpreted as an account of the laws of nature.
Further, as implied by “Appendix A”, the laws of a larger system, such as the universe, will constrain the
laws of any subsystem that can be derived from it as a “factor system” (via constraining its symmetries).
Thus, the laws of the universe will constrain the laws of any smaller subsystems in it. In Sect. 4.4, we show
that structurally equivalent systems may share the same laws.

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2568 Synthese (2021) 198:2551–2612

in T, if the state is in Y at time t, there is a 50% probability that it will be in Z at time
t + 1.”

However, while the first, non-probabilistic example (where we observe that one
state at time 5 is followed by another at time 6) seems unproblematic, the second,
probabilistic example is trickier. If we are trapped in a single history, it is unclear
how we could ever make an observation such as: “Conditional on the state being in
Y at time 5, there is a 50% probability that it will be in Z at time 6.” Making this
observation would seem to require looking at many repetitions of states 5 and 6. So,
even if probabilistic properties could be generalized via symmetries once we have
observed them, it is unclear how we could observe such properties in the first place.

The solution to this problem lies in the property of ergodicity. This is a property
that a system may or may not have and that, if present, serves as a prerequisite for
inferring probabilistic information from single histories. Indeed, it may be considered a
prerequisite for scientific inference more generally. To explain this notion, let us begin
with a simple example of how we learn probabilistic information from observing just
a single history. Consider a system whose state at any time is the outcome of an
independent coin toss, where T � {1,2,3,…}. So, the state space is X � {Heads,
Tails}, and each possible history in � is one possible Heads/Tails sequence.

Suppose the true conditional probability structure on � is induced by the single
parameter p, the probability of Heads. In this example, the Law of Large Numbers
guarantees that, with probability 1, the limiting frequency of Heads in a given history
(as time goes to infinity) will match p. This means that the subset of � consisting of
“well-behaved” histories has probability 1, where a history is well-behaved if (i) there
exists a limiting frequency of Heads for it (i.e., the proportion of Heads converges
to a well-defined limit as time goes to infinity) and (ii) that limiting frequency is p.
For this reason, we will almost certainly (with probability 1) get arbitrarily close to
the true conditional probability structure on � just by observing a single history and
counting the number of Heads and Tails in it.

Now why does this inference work in the present example? As we will see, the
system is an example of an ergodic system. Its ergodicity manifests itself in the fact
that “almost all” histories of the system are “well-behaved”, in the sense that we can
read off the desired probability parameter p from the limiting frequency of Heads.

To define ergodicity more precisely, consider again a system with T � {1,2,3,…}
which has all the time symmetries in the set � � {ψr: r � 0,1,2,3,…} (and perhaps
other symmetries as well, though we set these aside for now). Heuristically, the sym-
metries in � can be interpreted as describing the evolution of the system over time.27

Suppose each time-step corresponds to a day. Then the history h � (a, b, c, d, e,…)
describes a situation where today’s state is a, tomorrow’s is b, the next day’s is c, and
so on. Suppose today is Monday. The transformed history ψ1(h) � (b, c, d, e, f ,…)
describes a situation where today’s state is b, tomorrow’s is c, the following day’s
is d, and so on. Thus, ψ1(h) describes the same “world” as h, but as seen from the

27 Mathematically, the pair (�, �) can be interpreted as a classical dynamical system, as defined in footnote
6, with � playing the role of a state space (from an outside observer’s perspective) and the transformations
in � playing the role of state transformation rules.

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perspective of Tuesday. Likewise, ψ2(h) � (c, d, e, f , g,…) describes the same “world”
as h, but as seen from the perspective of Wednesday, and so on.28

Given the set � of symmetries, an event E (a subset of �) is �-invariant if the
inverse image of E inside of � under ψ is E itself, for all ψ in �. Formally, ψ−1(E) ∩ �
� E for all such ψ. Thus, for any history h in �, h is an element of E if and only if
ψ(h) is an element of E. For example, suppose again that the elements of T represent
days, and E is the event that some property P holds today. If ψ1, ψ2, ψ3, … are the
symmetries that shift time by 1 day, by 2 days, by 3 days, and so on, then the �-
invariance of E implies that property P holds today if and only if it holds tomorrow,
the day after tomorrow, and so on. Thus, E is a “persistent” event: an event one cannot
escape from by moving forward in time. In a coin-tossing system, where � is still
the set of time translations, examples of �-invariant events are “all Heads”, where E
contains only the history (Heads, Heads, Heads, …), and “all Tails”, where E contains
only the history (Tails, Tails, Tails, …).

Recall that symmetries preserve the unconditional probabilities of any event E. The
system is ergodic (with respect to �) if, for any �-invariant event E, the unconditional
probability of E, i.e., Pr�(E), is either 0 or 1.

29 In other words, the only persistent
events are those which occur in almost no history (i.e., Pr�(E) � 0) and those which
occur in almost every history (i.e., Pr�(E) � 1).30 The ergodicity of our coin-tossing
system is exemplified by the fact that the �-invariant events “all Heads” and “all Tails”
occur with probability 0.

In an ergodic system, it is possible to estimate the probability of any event “em-
pirically”, by counting the frequency with which that event occurs, much like the
probability of Heads in the coin-tossing example.31 Frequencies are thus evidence for
probabilities. The formal statement of this is the following important result from the
theory of dynamical systems and stochastic processes.

Ergodic Theorem: Suppose the system is ergodic. Let E be any event and let h
be any history. For all times t in T, let N t be the number of elements r in the set

28 Note that, under this heuristic interpretation, the world “forgets” its past history: from the perspective
of Tuesday, it is as if Monday never happened. This is just an artefact of the formal mathematical model
we are using in this example and has no deeper significance. If we used the set Z of all integers instead of
the natural numbers to model time, it would obviate this issue.
29 See, e.g., Petersen (1989, Section 2.4), Walters (2000, Section 1.5, Definition 1.4), or Krengel (1985,
Section 1.1.3, Definition 1.7) for precise definitions and further discussion. Ergodicity is usually defined
for a symmetry monoid generated by a single transformation ψ, in which case the pair (X, ψ) is called an
ergodic dynamical system. But the definition generalizes immediately to arbitrary symmetry monoids. See,
e.g., Krengel (1985, p. 203). At first sight, this seems unrelated to the use of the term in probability theory,
where a stochastic process (such as a Markov process or random field) is called ergodic if it has a unique
stationary measure. But the two definitions are related because any stochastic process can be represented
as a dynamical system (Petersen 1989, Sections 1.2B to 1.2D). Finally, in statistical physics, Boltzmann’s
Ergodic Hypothesis conjectures that the fraction of the time that a physical system spends in a particular
region of its state space is proportional to the size of that region. This was the inspiration for the Birkhoff
Ergodic Theorem (see below).
30 If � is infinite, there is a subtle distinction between almost no history (Pr�(E) � 0) and no history (E
� ∅). Likewise, almost every history (Pr�(E) � 1) is subtly distinct from every history (E � �).
31 This insight is the basis for Reichenbach’s (1949) “straight rule”, which is to take observed frequencies
as the best estimates of “true” probabilities. See, e.g., Eberhardt and Glymour (2009).

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2570 Synthese (2021) 198:2551–2612

{1, 2,…, t} such that ψr(h) is in E. Then, with probability 1, the ratio N t/t will
converge to Pr�(E) as t increases towards infinity.

32

Intuitively, N t is the number of times the event E has “occurred” in history h from
time 1 up to time t. The ratio N t/t is therefore the frequency of occurrence of event E (up
to time t) in history h. This frequency might be measured, for example, by performing a
sequence of experiments or observations at times 1, 2,…, t. The Ergodic Theorem says
that, almost certainly (i.e., with probability 1), the empirical frequency will converge
to the true probability of E, Pr�(E), as the number of observations becomes large.
The estimation of the probability of Heads via the Law of Large Numbers in our
coin-tossing example is a special case of this.

To understand the significance of the Ergodic Theorem, let Y and Z be two sub-
sets of X, and suppose E is the event “h(1) is in Y” and D is the event “h(2) is in
Z”. Then the intersection E ∩ D is the event “h(1) is in Y, and h(2) is in Z”. The
theorem says that, by performing a sequence of observations over time, we can esti-
mate Pr�(E) and Pr�(E ∩ D) with arbitrarily high precision. Thus, we can compute
the ratio Pr�(E ∩ D)/Pr�(E) (provided Pr�(E) �� ∅). But this ratio is the conditional
probability PrE(D). And so we are able to estimate the conditional probability that
the state at time 2 will be in Z, given that at time 1 it is in Y. This illustrates that, by
allowing us to estimate unconditional probabilities, the Ergodic Theorem also allows
us to estimate conditional probabilities, and thereby to infer the conditional probabil-
ity structure {PrE}E⊆�. Clearly, the system’s symmetries were indispensable for this
exercise. Without symmetries, the frequentist reasoning to which the Ergodic Theorem
appeals would not make sense.

2.8 Occam’s Razor

We have seen that a system must possess a sufficiently rich set of symmetries to permit
general inferences from local observations. Up to now, we have taken for granted that
we know, or are justified in hypothesizing, that the system has these symmetries. But
what justifies this hypothesis?

This question is crucial for the success of science. Why are we justified in assuming
that the system’s laws are the same at different times or in different places? Why should
replicability of other scientists’ experimental results be considered the norm, rather
than a miraculous exception? Why is it normally safe to assume that the outcomes of
experiments will be insensitive to irrelevant details such as the height of the laboratory
bench, or the orientation of the apparatus relative to the planet Jupiter?

In effect, we are assuming that the phenomena under investigation are invariant
under certain symmetries—both temporal, as discussed earlier, and spatial, as dis-
cussed later, including translations, rotations, and so on. But where do we get this
assumption from? The answer lies in Occam’s Razor.

32 This result is often called the Birkhoff (or Pointwise) Ergodic Theorem. For simplicity, we have stated it
relatively informally. For formal statements, see Krengel (1985, Section 1.2), Petersen (1989, Section 2.2),
Walters (2000, Theorem 1.14), or Berkovitz et al. (2006). Further, we have defined ergodicity for a discrete
set of time symmetries. More generally, the statements in this section hold for any set of symmetries that
forms an amenable monoid. See Krengel (1985, Section 6.4).

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Occam’s Razor is generally a principle of parsimony. One of its best-known versions
says that, when we try to explain some phenomenon, we should not postulate more
entities than strictly explanatorily necessary.33 While this version of Occam’s Razor
deals with the question of which entities to postulate, we are here focusing on another
version, which concerns the question of which regularities to postulate. Roughly, it
says that, if two hypotheses about the regularities in the world are equally consistent
with our total evidence, we should prefer the simpler hypothesis.

Now the key point is that the hypothesis of a symmetry-rich system is simpler than
the hypothesis of a symmetry-poor system, other things being equal.34 To see why this
is the case, contrast two cases. If you hypothesize that the universe has a very large
set of symmetries, you are thereby postulating a very simple universe. By contrast,
if you hypothesize that the universe has very few symmetries, you are postulating a
very complex universe. The first universe admits a parsimonious description in light
of its symmetry-induced regularity, the second does not. This suggests the following
provisional formulation of Occam’s Razor principle:

Occam’s Razor: Always assume that a system has the largest possible set of
symmetries consistent with all facts about the system that we believe to be
nomologically necessary.

We must now make this more precise. We begin by explaining what we mean by
“facts about the system that we believe to be nomologically necessary”. We represent
this by a collection of those histories among the logically possible ones that we have
not ruled out as nomologically impossible. We call this collection of histories our
total nomological evidence about the system. Formally, it is a subset E of H. It could
capture the “hard” constraints that we take the system to satisfy, such that, to the best
of our knowledge, any history outside E is not permitted by the laws of the system. Of
course, we do not strictly know that � is a subset of E. When we empirically study a
system, we do not normally know what � is. We can at most be certain that E overlaps
with �. We will suppose, however, that we are ready to make the auxiliary assumption
that E includes, but may be logically weaker than, �.

Given this assumption, we are in a position to test the hypothesis that any given
transformation of H is a symmetry of our system. Let ψ be such a transformation, and
for any n, let ψn be the transformation obtained by applying ψ repeatedly, n times in a
row. For example, if ψ is a rotation about some axis by angle θ, then ψn is the rotation
by the angle nθ.35 For any such transformation ψn, we write ψ−n(E) to denote the
inverse image in H of E under ψn. We say that the transformation ψ is consistent with
the nomological evidence E if the intersection

33 The literature contains many proposals on how to formalize Occam’s Razor. See, e.g., Baker (2013) and
Fitzpatrick (2015). For an efficiency argument for Occam’s Razor, see Kelly (2007).
34 Mainzer (1996, Section 5.3, p. 580) also relates symmetry to simplicity and notes that scientists generally
prefer simpler theories. Likewise, van Fraassen (1989, Section 10.2) notes that when constructing a scientific
model, we generally assume that the model satisfies a given symmetry unless we have good reason to believe
the contrary. His slogan is: “an asymmetry must always come from an asymmetry”, with some caveats. But
neither of these authors connects these ideas to Occam’s Razor.
35 In the present terms, rotations must be represented as transformations of the state space X. In Sect. 3.5,
we represent rotations more explicitly, relying on a formal representation of space.

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2572 Synthese (2021) 198:2551–2612

E ∩ ψ−1(E) ∩ ψ−2(E) ∩ ψ−3(E) ∩ . . .

is non-empty. This means that E does not falsify the hypothesis that ψ is a symmetry
of the system.

For example, suppose we are interested in whether electrostatic forces work the
same way at all times. We can test this hypothesis by means of Coulomb’s famous
“torsion balance” experiment, which measures the electrostatic attraction or repulsion
between two charged objects. Suppose we perform the experiment at time t1 and
obtain evidence E1, and we perform the same experiment again at time t2 and obtain
evidence E2. Thus, our evidence is summarized by the event E � E1 ∩ E2. Let ψ be
a time symmetry that shifts t1 to t2. Then, focusing for simplicity just on the first two
terms of the infinite intersection above, we have

E ∩ ψ−1(E) � E1 ∩ E2 ∩ ψ−1(E1) ∩ ψ−1(E2).

If the experimental results are the same at times t1 and t2, then E1 � ψ−1(E2),
and the expression for E ∩ ψ−1(E) simplifies to E1 ∩ E2 ∩ ψ−1(E1). Under reason-
able assumptions, this is non-empty, meaning that the evidence has not falsified time
invariance of electrostatic forces. But if the experimental results at times t1 and t2 were
different, then E1 and ψ

−1(E2) would be disjoint, and so the intersection E ∩ ψ−1(E)
would be empty, which would mean that the evidence is inconsistent with time invari-
ance. As it happens, many thousands of repetitions of Coulomb’s experiment strongly
suggest that the intersection is non-empty, and so ψ is a symmetry.

Now our version of Occam’s Razor says that we should postulate as symmetries
of our system a maximal monoid of transformations consistent with our evidence.
Formally, a monoid � of transformations (where each ψ in � is a function from H
into itself) is consistent with our total nomological evidence E if the intersection

⋂

ψ∈�
ψ

−1(E)

is non-empty. This is the generalization of the infinite intersection that appeared in our
definition of an individual transformation’s consistency with the evidence. Further, a
monoid � that is consistent with E is maximal if no proper superset of � forms a
monoid that is also consistent with E.

Occam’s Razor (formal): Given our total nomological evidence E about a
temporally evolving system, always assume that the set of symmetries of the
system is a maximal monoid � consistent with E.

What is the significance of this principle? Recall that we earlier defined � to be the
set of all symmetries of our temporally evolving system. In practice, we do not know
�. A monoid � that passes the test of Occam’s Razor, however, can be viewed as our
best guess as to what the true symmetry monoid is. To disambiguate, let �true denote
the true symmetry monoid, and let �hyp denote the hypothesized one.

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If �hyp is the hypothesized symmetry monoid, and E is our total nomological
evidence, the intersection

⋂

ψ∈�hyp
ψ

−1(E)

can be viewed as our best guess as to what the set of nomologically possible histories
is. It consists of all those histories among the logically possible ones that are not ruled
out by the hypothesized symmetry monoid �hyp and the nomological evidence E. We
call this intersection our nomological hypothesis and label it �(�hyp, E).

To see that this construction makes sense, note that, under certain conditions, our
nomologicalhypothesis �(�hyp, E)willreflectthetruthaboutnomologicalpossibility.

Remark: If (i) the hypothesized symmetry monoid �hyp is a subset of the true
symmetry monoid �true, and (ii) E is a superset of �, then the true set � of
nomologically possible histories is a subset of �(�hyp, E).

Condition (i) says that we have not postulated any incorrect symmetries, which is
compatible with having overlooked some correct symmetries. Condition (ii) says that
we have not mistakenly ruled out any nomologically possible histories, which was our
auxiliary assumption about our total nomological evidence. If these conditions hold,
our nomological hypothesis will indeed be consistent with the truth and will, at most,
be logically weaker than the truth.

It is worth explaining the significance of the auxiliary assumption that we have not
mistakenly ruled out any nomologically possible histories (i.e., E ⊇ �). Consider the
simple coin-tossing system from Sect. 2.7, where histories are sequences of Heads
and Tails, and time shifts are symmetries. Now consider the event E of getting Heads
at time 1 and Tails at time 2. If we treated E as our total nomological evidence, this
would exclude time shifts as symmetries: the event of getting Heads at time 1 and Tails
at time 2 is not invariant under time shifts. The problem is that E, in this case, is not a
superset of �: it excludes histories that are in fact nomologically possible. The notion
of “total nomological evidence” that we require is a “cautious” one. The set E should
exclude only histories that we are confident in deeming nomologically impossible.
This is a subtle issue, and a full treatment is beyond the scope of this paper.

In “Appendix C”, we extend the present analysis by offering criteria for choosing
a maximal symmetry monoid � consistent with the evidence E in case more than one
such monoid can be constructed. We suggest that criteria of inferential modesty and
informational parsimony should guide that choice in cases of non-uniqueness.

2.9 The role of time

What is the significance of the linear order of the set T of times? Why is time ordered
in one way, and not in another? Do the laws of a given system “care” about the ordering
of time? To put it another way: what does it mean to say that today comes between
yesterday and tomorrow? Intuitively, it means this: the events that happened yesterday
cannot “directly influence” the events that will happen tomorrow; their influence must

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be “mediated” by the events that happen today. We now make this claim precise using
a standard notion from probability theory: the Markov property.36

To explain this property, we first introduce the notion of conditional independence.
Let{PrE}E⊆� beaconditionalprobabilitystructure,andlet D and E betwoevents(i.e.,
subsets of �). We say that D and E are independent if PrD(E) � Pr�(E) and PrE(D)
� Pr�(D).37 Informally, if we interpret probabilities as encoding “information”, this
means that learning whether or not D has occurred provides no information about
whether or not E will occur, and vice versa.

To illustrate, recall the simple coin-tossing system from Sect. 2.7. Let E and D be
the events “the outcome at time 1 is Heads” and “the outcome at time 2 is Tails”.
Then Pr�(E) � ½ and Pr�(D) � ½, assuming for simplicity that p � 0.5. Here, the
outcome at time 1 has no effect on the outcome at time 2. So, even if we tossed Heads
at time 1, this would not change the probability of obtaining Tails at time 2, and so
PrE(D) � ½. Likewise, the outcome at time 2 tells us nothing about what happened at
time 1. If we had not observed the outcome at time 1 but obtained the outcome Tails
at time 2, we would still assign probability ½ to Heads at time 1. So, PrD(E) � ½.
Thus, the events E and D are independent.

Now let C, D, and E be three events. We say that C and E are conditionally inde-
pendent, given D, if PrC∩D(E) � PrD(E) and PrE∩D(C) � PrD(C). Again, if we
interpret probabilities as encoding “information”, this means the following. Suppose
you already know that D has occurred. Then learning whether or not C has occurred
provides no further information about whether or not E will occur, and vice versa.

To illustrate, return again to the coin-tossing example (where T � {1,2,3,…}) with
p � 0.5, but suppose we use the tosses of the fair coin to determine the position of a
token on an infinite line. We move the token after each coin toss: if we toss Heads,
we move the token one space to the right, and if we toss Tails, we move it one space
to the left. Let us represent the position of the token by an integer (either positive
or negative); in other words, X � {…,− 3, − 2, − 1,0,1,2,3,…}. Let xt denote the
position of the token at time t. Then the rule becomes the following: “If you toss
Heads at time t, then xt+1 � xt + 1; if you toss Tails at time t, then xt+1 � xt − 1.” For
simplicity, suppose the coin always starts at position 0 (i.e., x1 � 0).38

If D is an event describing the position of the token at time t, and E is an event
describing its position at time t + 1, then these two events are not independent. For
example, suppose E is the event “x6 � 3”. Then a simple calculation shows that
Pr�(E) � 5/16. If D is the event “x5 � 2”, then PrD(E) � ½, because the token now
has a 50% probability of moving from position 2 to position 3 in one time step. Thus,
PrD(E) �� Pr�(E). The location of the token at time 5 tells us a great deal about its
probable location at time 6.

However, once we know the position at time 5, learning the position at time 4 tells
us nothing further about the position at time 6. Continuing the previous example, let

36 The importance of Markov properties in understanding causality has been emphasized by Pearl (2000)
and Spirtes et al. (2000).
37 If Pr�(D)>0, the first equation is equivalent to Pr�(E ∩ D) � Pr�(D) Pr�(E). If Pr�(E)>0, the
second equation is equivalent to Pr�(E ∩ D) � Pr�(D) Pr�(E). Thus, if Pr�(D)>0 and Pr�(E)>0, the
two equations are equivalent. But if Pr�(D) � 0 or Pr�(E) � 0, the equations must be stated separately.
38 Technically, the system just described is a simple random walk.

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C be the event “x4 � 1”. Then straightforward calculations show that PrC∩D(E) � ½
� PrD(E) and PrE∩D(C) � ½ � PrD(C). In other words, if we already knew that the
token’s position was 2 at time 5 (so that it had a 50% probability of moving to position
3 at time 6), then learning its position at time 4 tells us nothing further about where
it might be at time 6. Likewise, if we already knew that the token’s position was 2 at
time 5 (so that it has a 50% probability of having been at position 1 at time 4), then
learning its position at time 6 tells us nothing further about where it might have been
at time 4.

In this example, the conditional independence of the events C and E, given D, is
due to the fact that D concerns the state of the system at a point in time between the
times described by C and E and that D provides us with complete information about
the state of the system at this intermediate time. If D provided only partial information
about that state, we would not get the same result. For example, suppose D′ is the
event, “x5 � 0, 2, or 4”, which does not fully specify the state at time 5. Then it can
be shown that PrC∩D′ (E)> PrD′ (E). Here, learning additional information about the
state at time 4 can still tell us something about where the coin is likely to be at time 6.

Now let us generalize this example. Let T be any linearly ordered set, let X be any
set of states, and consider a temporally evolving system given by a collection � of
possible histories (i.e., functions from T into X) and a conditional probability structure
{PrE}E⊆�. For any time t in T, and any state x in X, let Etx denote the event “the state
of the system at time t is x”. More generally, for any subset Y of X, let EtY denote the
event “the state of the system at time t is an element of Y”. We say that the system
satisfies the Markov property if, for any times r < s < t in T, any subsets Y and Z of
X, and any state x in X, the events ErY and E

t
Z are conditionally independent, given

the event Esx. In other words, if you have complete information about the state of the
system at some time s (you know that the state is x), then learning something about its
state at some earlier time (e.g., that it was an element of Y at time r) tells you nothing
further about its probable state at some later time (e.g., about how probable it is that
it will fall into the set Z at time t). Roughly speaking, this means that the state of the
system at time r cannot “directly influence” the state of the system at time t. It can
only influence that state “indirectly”, via influencing the state at the intermediate time
s. Any system with this property is called Markovian.

Note that the Markov property does not say that the system’s future evolution is
unconditionally independent of its past. It just says that the dependency of the future
on the past is mediated through the present. This property is fundamental to the way we
normally think about time. To see this, imagine a universe where the Markov property
was not true. Then there would exist some times r < s < t in T, some subsets Y and Z
of X, and some state x in X, such that the conditional probability Pr(EtZ | E

r
Y ∩Esx) is

distinct from Pr(EtZ | E
s
x).

39 In other words, even with a complete specification of the
present state x, the probability of some future event Z would depend on whether or not
some past event Y had occurred. This would suggest that the state specification x does
not, in fact, contain all the information about the system’s present state; somehow,
information about the past is bypassing the present and “leaking” directly into the

39 Here, to avoid cumbersome subscripts, we are using the notation Pr(A | B) to denote the conditional
probability PrB(A).

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2576 Synthese (2021) 198:2551–2612

future. This, in turn, suggests that this so-called “past” is not really in the past at all;
our model of the system’s time structure is incorrect.

We take the Markov property to be a necessary condition for the “correct” ordering
of time. To be “well-behaved”, a temporally evolving system must be Markovian. What
the present must do at any point in time in order to count as the present is “separate”
the past from the future. If this property is violated, the set T does not properly play
the role of time.

Three points are worth noting. First, some systems may admit multiple time order-
ings with respect to which they are Markovian. An extreme limiting case is given by
our original coin-tossing system without the moving token, which is Markovian with
respect to every ordering of T. Here, the precise order of time is irrelevant. By contrast,
in the modified coin-tossing system with the token, the order of time matters, as we
have seen. In fact, the temporal order with respect to which the system satisfies the
Markov property is essentially unique; it is unique up to time reversals. This brings us
to our second point. Although the Markov property says something about the linear
“topology” of time, it tells us nothing about the direction of time. As illustrated by the
modified coin-tossing system, the Markov property is completely invariant under time
reversals. In other words, the Markov property only says that the present separates the
past from the future. But it does not tell us on which side of the present lies the past,
and on which side lies the future. And third, just as the Markov property says nothing
about the direction of time, so it says nothing about its duration. There is no purely
Markovian way of measuring the “length” of a time interval or saying when one time
interval is longer than another.

What, then, can we say about the directionality and length of time? It turns out
that symmetries are crucial for the analysis of both. In the case of length, we offer a
detailed analysis in Sect. 3.9, showing that there is a natural way of measuring time
duration, as long as the system has sufficiently rich symmetries. And in the case of
directionality, we can say that a condition for time to have a direction is that time
reversals are not symmetries of the system. Since time reversals are symmetries of
classical mechanical systems (in the sense explained in footnote 19), it follows that,
in those systems, there is no real direction of time: temporal orders are unique at most
up to time reversal. By contrast, in thermodynamic systems, time reversals are not
symmetries, and hence these systems meet the condition for time to have a direction.
To the extent that the world, as seen from our perspective, is best understood as a
system in which time reversals are not symmetries, there is then a coherent basis for
the directionality of time (for further discussion, see Roberts 2013).

3 Spatially extended systems

3.1 Basic definitions

We now turn to a more richly described class of systems whose states evolve over
time. To define a system in this class, we still represent time by a linearly ordered set
T, but also incorporate an explicit notion of space, represented by a set S of spatial
locations. Let S × T be the set of all ordered pairs of the form (s, t), where s is an

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element of S, and t is an element of T. We refer to S × T as space–time. Again, let
X denote a set of possible states, called the state space. Unlike before, the elements
of X are no longer “global” states, in which the system can be at specific points
in time, but “local” states, in which the system can be at specific points in space
and time. Again, we treat the elements of X as primitives of our model. Histories
are now functions from space–time (rather than merely time) into the state space.
Formally, a spatially extended history is a function h from S × T into X. For each point
(s, t) in S × T, h(s, t) is the state of the system in spatial location s at time t.

In analogy to our earlier model, we write � to denote the set of all spatially extended
histories deemed possible, which, as before, play the role of possible worlds. Again,
this is a subset—often a proper one—of the set H of all logically possible histories
(here, all functions from S × T into X). So, membership in � is best interpreted as
nomological possibility. Subsets of � are called events.

Finally, we define a conditional probability structure on �. As before, this is a
family of conditional probability functions {PrE}E⊆�, containing one PrE for each
event E in �, with standard properties. Recall that PrE assigns to any event in � the
conditional probability of that event, given E. A spatially extended system is the pair
consisting of the set � of possible spatially extended histories and the conditional
probability structure {PrE}E⊆�.

For example, in a classical mechanical system, T is the set R of real numbers, S is
the three-dimensional Euclidean space (i.e., S � R3), and each state h(s, t) in X is given
by the set of particles present at spatial location s at time t, along with their physically
relevant properties (e.g., masses and momenta) and the values of any force fields (e.g.,
gravity) acting on these particles.40 In a classical electrodynamical system, the state
h(s, t) must also specify the particles’ charges, along with the electric and magnetic
field vectors at (s, t). In that sense, electrodynamics relies on a richer ontology than
classical mechanics.

In a quantum–mechanical system, it might be tempting to suppose that S � R3,
and to suppose that h(s, t) is given by the values of the wave functions of each of the
particles in the system at space–time location (s, t). But this is not correct, because
the wave functions of interacting particles in a quantum system cannot generally be
defined independently of each other. Instead, we must define a joint wave function for
the entire multi-particle system. So, in a quantum–mechanical system with n particles,
we would define space to be S � (R3)n, with three coordinates representing the spatial
“position” of each of the n particles in an underlying ordinary Euclidean space41; and
we would define the set X of possible states of the system to be the set of complex
numbers, capturing amplitudes, whose squared absolute values behave formally like
probabilities. Thus a spatially extended history h is a function from (R3)n × T into the
set of complex numbers, representing the joint wave function of the whole ensemble
of particles.

40 We are not saying that this is the most parsimonious or computationally convenient way to represent a
classical mechanical system. It is only one way of representing such a system in our framework.
41 Strictly speaking, particles in quantum systems do not have “positions”, so we are using this term rather
loosely. Also, there is a dual representation of the wave function (obtained via Fourier transform), where the
coordinates in (R3)n represent the “momenta” (again, loosely) of the n particles. These two representations
are equally valid.

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For instance, if there are two particles, labelled 1 and 2, then h(x1, y1, z1, x2, y2,
z2, t) represents the joint state at time t of particles 1 and 2 at positions x1, y1, z1 and
x2, y2, z2 in the underlying three-dimensional Euclidean space. This joint state of the
two particles is a complex number whose squared absolute value can be interpreted,
under some assumptions, as the probability of particles 1 and 2 being observable at
positions x1, y1, z1 and x2, y2, z2, respectively, at time t.

3.2 Determinism and indeterminism

As in the case of temporally evolving systems, we can define a family of notions of
determinism and indeterminism for spatially extended systems. For any subset L of
locations in S × T, we write hL to denote the restriction of the function h to the points
in L. We can then ask for which proper subsets L of S × T, if any, hL has a unique
extension to all of S × T in �. Again, an extension of hL is a history h′ such that
h′L � h L . When hL is uniquely extendible to all of S × T, we say that history h is
L-deterministic.

For example, the histories of classical mechanical systems are L-deterministic for
any subset L of S × T that has the form S × T ′ where T ′ is any non-empty subset of T.
Information about the system for even a single “time slice” of space–time, i.e., a set
of the form S × {t} for some t in T, suffices to determine the full spatially extended
history. In contrast, the histories of quantum–mechanical systems (if wave-function
collapses are allowed) are not generally L-deterministic when L consists of time slices.

The present definitions allow us to explore some interesting possibilities not cap-
tured by standard definitions that focus exclusively on past-to-future determination.42

For example, some systems might encode their entire spatially extended history in each
individual space–time location. Histories would then be L-deterministic for every sin-
gleton set L � {(s, t)}, where (s, t) is in S × T. Here, we would have an extreme
form of local-to-global determinism. Alternatively, some systems might encode their
entire spatially extended history in some collection of “spatial slices of time”, i.e.,
some subset L of S × T which has the form S′ × T, where S′ is a non-empty subset of
S, possibly singleton. This would be a kind of spatial, not temporal, determinism.43

Other systems might never be L-deterministic for any proper subset L of S × T.
There may also be some more limited, non-global forms of determination, for

instance when a history restricted to some set L of locations is uniquely extendible to
a history restricted to some superset L* of L, which is still smaller than S × T in its
entirety.44 To capture this idea, we can say that a history h is L-to-L*-deterministic if,
for any history h′ in �, if h L � h′L , then h L ∗ � h′L ∗.

We might imagine, for instance, systems that are deterministic “across space” but
not “across time”. In such a system, a history restricted to some set L of the form
S′ × {t}, where S′ is a non-empty subset of S and t a point in time, might determine the
42 For an earlier discussion of local-to-global forms of determinism, see Earman (1986, pp. 33–35).
43 This sort of determinism occurs in expansive cellular automata, a class of spatially extended systems
discussed in the theory of dynamical systems. See Pivato (2009).
44 For example, this phenomenon arises frequently in the solution of boundary value problems in mathe-
matical physics. See Pivato (2010).

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entire “time slice” of that history across L* � S × {t}, but not the rest of the history.
Some crystals and other chemical or physical systems involving highly regular spatial
structures might have this feature. Similarly, for suitable specifications of L and L*, we
can represent the phenomenon that, in some systems in which “information” travels
with finite speed, events at particular space–time locations at time t1 are entirely
determined by the events occurring within their “backwards light cones” at some time
t0 < t1. Such systems may be L-to-L

*-deterministic, but not deterministic in a more
global sense.

3.3 Nomological possibility and necessity

In analogy to the case of temporally evolving systems, we can define two modal
operators for each set L of space–time locations, namely nomological possibility and
necessity relative to L. For each set L ⊆ S × T, call one history, h′, accessible from
another, h, relative to L, if the restrictions of h and h′ to L coincide, i.e., h′L � h L . We
then write hRL h

′. For any event E ⊆ �, we define
◆L E � {h ∈ �: for some h′ ∈ � with hRL h′, we have h′ ∈ E},
�L E � {h ∈ �: for all h′ ∈ � with hRL h′, we have h′ ∈ E}.
Here, ◆L E and �L E are, respectively, the sets of all histories in which E is

nomologically possible and nomologically necessary once the history in space–time
region L is given. Important special cases are (i) L � S × T ′, where S is all of space
and T ′ is a particular set of time points, such as those up to time t, (ii) L � S′ × T,
where T is all of time and S′ is some spatial region, and (iii) L � ∅ for possibility
and necessity in the “atemporal” sense. Since the present definitions are completely
analogous to their earlier counterparts in Sect. 2.3, we will not say more about them
here.

3.4 Modal and probabilistic properties

We now turn again to the question of how to distinguish between those properties of
a system that qualify as “laws” and those that fall short of being laws. As before, our
analysis is based on the notion of symmetry, but now with the additional ingredient
that these symmetries can involve space as well as time.

In analogy to our earlier definition, a property of histories, P, is a binary feature that
a spatially extended history may or may not have. Its extension is some subset [P] of
the set H of all logically possible histories. A spatially extended history h satisfies P
if h belongs to [P]. Again, if [P] includes all of �, then P can be called nomologically
necessary. Similarly, a probabilistic property, P, is a binary feature that a conditional
probability structure may or may not have, and its extension, [P], is the set of all those
conditional probability structures on � that satisfy P.

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3.5 Symmetries

The notion of a state symmetry for spatially extended systems is virtually identical to
the one defined in Sect. 2.5 for temporally evolving systems, so we do not discuss it
further.45 Instead, we turn directly to symmetries acting on space–time. Let ψ be a
function from S × T into itself (i.e., a transformation of space–time). Again, ψ induces
a function from the set H of logically possible histories into itself. For any spatially
extended history h, we define the transformed history

ψ(h) � h′, where, for all (s, t) in S × T , h′(s, t) � h(ψ(s, t)).

As before, given any set E of histories in H, the inverse image of E under ψ, written
ψ−1(E), is the set of all histories h in H such that ψ(h) lies in E. The function ψ is a
symmetry if

• ψ(�) � �; and
• for any events E and D in �, if E′ and D′ are the inverse images of E and D under

ψ, then PrE′ (D
′) � PrE(D).46

For example, if T is the set of real numbers (i.e., T � R) and S is the three-
dimensional Euclidean space (i.e., S � R3), we can consider a spatially extended
system in classical mechanics. The following transformations of S × T are space–time
symmetries of such a system, each defined for all (s, t) in S × T:
• Time translation: ψ(s, t) � (s, t + r), where r is a fixed real number;
• Spatial translation: ψ(s, t) � (s + v, t), where v is a fixed three-dimensional vector
(an element of R3); and

• Space–time rescaling: ψ(s, t) � (r s, r t), where r >0 is a fixed real number.
More general symmetries include composite functions resulting from the combina-

tion of a transformation φ of the state space (X) with a transformation ψ of space–time
(S × T).47 Examples in classical mechanics are spatial rotations, spatial reflections,
spatial rescalings, and Galilean transformations.48 Crucially, it is possible that neither

45 For an example, take an n-particle quantum system, where S � (R3)n, X is the set of complex numbers,
and a spatially extended history h is a wave function. Let φ be a phase rotation map on the complex plane;
formally, there is some angle θ such that, for all x in X, φ(x) � eiθx. Then φ is a state symmetry.
46 As before, for any subsets D, E of H, we define PrE (D) � PrE∩�(D ∩ �), provided E ∩ � �� ∅.
47 An additional property we might require of a space–time transformation ψ is time preservation: for any
points (s1, t1) and (s2, t2) in S × T, with ψ(s1, t1) � (s′1, t ′1) and ψ(s2, t2) � (s′2, t ′2), if t1 ≤ t2, then
t ′1 ≤ t ′2. This implies that if t1 � t2, then t ′1 � t ′2. A time-preserving transformation acts on S × T such that
the set of all space–time points at time t1 gets moved en bloc to the set of all space–time points at time t

′
1.

The transformations described above have this property, but we do not need to include it in our definition.
48 These are defined as follows. Spatial rotation: Fix a line L in S and an angle θ. For any point s in S, let s′
be the point obtained by rotating s by an angle of − θ around L. For all (s, t) in S × T, define ψ(s, t) � (s′, t).
Let L′ be the line parallel to L, but passing through the origin. For all x in X, define φ(x) by rotating all
the momentum vectors and force field vectors in x by the angle θ around L′. Spatial reflection: Fix a plane
P in S. For any point s in S, let s′ be the point obtained by reflecting s across P. For all (s, t) in S × T,
define ψ(s, t) � (s′, t). Let P′ be the plane parallel to P, but passing through the origin. For all x in X,
define φ(x) by reflecting all the momentum vectors and force field vectors in x across P′. Spatial rescaling:

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the transformation φ of the state space nor the transformation ψ of space–time alone
is a symmetry, and yet, when combined, they form a symmetry.49

Of course, any combination of symmetries is also a symmetry. An example is a
spatiotemporal translation, which is a combination of a time translation and a spatial
translation.Inaclassicalelectrodynamicalsystem, only thespatiotemporaltranslations
and rotations are space–time symmetries. Galilean transformations are not space–time
symmetries of classical electrodynamics; indeed, this was the original impetus for the
development of special relativity theory.

3.6 Laws and their significance

As in the earlier case of temporally evolving systems, a modal law of a spatially
extended system is a property of histories, P, that is nomologically necessary for the
system and invariant under all of the system’s symmetries. A probabilistic law is
a probabilistic property, P, that is satisfied by the system’s conditional probability
structure and invariant under all of its symmetries.

For example, let S � R3 and T � R, and suppose the symmetry monoid � contains
all the spatiotemporal translations defined in the previous section. Suppose all histories
of the system satisfy the property P which says: “If the state at space–time position
(3,7,2,14) is x, then at position (4,8,1,17) it is y.” If ψ is a spatial translation by
the vector (1,2,3), then the inverse image of [P] under ψ corresponds to the property
P′ which says: “If the state at (4,9,5,14) is x, then at position (5,10,4,17) it is y.”
Clearly, [P′] is not the same as [P], and so property P falls short of being a law.

However, suppose all histories satisfy the property P which says: “For any location
(s1, s2, s3) in S and any time t in T, if the state at space–time position (s1, s2, s3, t)
is x, then at position (s1 + 1, s2 + 1, s3 − 1, t + 3) it is y.” It is easy to see that [P] is
invariant under all spatiotemporal translations. If � consists only of the spatiotemporal
translations, then P is invariant under all symmetries, and so P is a law.

An illustration is Gauss’s Law in an electrodynamical system. This asserts, roughly,
that the net “flux” of the electric field passing through the walls of any closed com-
partment is proportional to the net charge contained inside that compartment. This
property is invariant under spatiotemporal translations, because the net flux and the
net charge are unchanged by such transformations. Indeed, Gauss’s Law is preserved
by every symmetry of an electrodynamical system; that is why it is a law.

As before, the significance of laws, as opposed to properties that fall short of being
laws, lies in their openness to testing and generalization. Consider again the property:

Footnote 48 continued
Fix some real number r >0, and define ψ(s, t) � (s/r , t) for all (s, t) in S × T. Meanwhile, let φ be a
transformation of X that multiplies the momentum vector of every particle by r, and also multiplies all force
field vectors by r. Galilean transformation: For all (s, t) in S × T, define ψ(s, t) � (s − t v, t), where v is
a fixed three-dimensional vector (an element of R3). Meanwhile, for all x in X, define φ(x) by adding the
vector v to all momentum vectors in x.
49 In all four examples in footnote 48, neither ψ nor φ is itself a symmetry. But when combined, they do
form a symmetry. For another example, in classical electrodynamics, let ψ be a spatial reflection acting on
S ×T , and let φ be a transformation of X which applies the corresponding reflection to all momentum vectors
and field vectors, and which further negates the magnetic field vector. Neither one of these transformations
is a symmetry by itself, but when combined, they do form a symmetry.

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2582 Synthese (2021) 198:2551–2612

“If the state at space–time position (3,7,2,14) is x, then at position (4,8,1,17) it is
y.” This property is observable exactly once in any history, namely at space–time
position (3,7,2,14) alone. Taken in isolation, the observation that some history has
this property tells us very little. It leaves open whether there is some broader regularity.
By contrast, consider the property: “For any location (s1, s2, s3) in S and any time t in
T, if the state at position (s1, s2, s3, t) is x, then at position (s1 + 1, s2 + 1, s3 − 1, t + 3)
it is y.” Recall that, if the system’s symmetry monoid consists of all spatiotemporal
translations, then this property is a law. Indeed, it has many observable manifestations
in each history, both at different times and in different places, and it thus picks up a
pattern that we can in principle test and use as a basis for predictions, even within a
single history.

3.7 Spatiotemporal ergodicity and its significance

Before turning to a more detailed analysis of the role of space in a spatially extended
system, it is worth sketching how the property of ergodicity can be extended to such
a system and discussing the significance of this. In the present case, too, ergodicity is
the key to learning a system’s conditional probability structure, even if we are able to
observe only a single history of the system.

Recall that, for some set � of symmetries, an event E (a subset of �) is �-invariant
if, for every ψ in �, the inverse image of E inside of � under ψ is E itself. For
illustrative purposes, suppose � consists of all spatiotemporal translations by four-
dimensional vectors of integers (applying the definition from Sect. 3.5).50 The system
is spatiotemporally ergodic if the unconditional probability of any �-invariant event
E, Pr�(E), is either 0 or 1.

Since � consists of spatiotemporal translations, �-invariant events are events
from which one cannot escape by travelling through space, or by travelling for-
wards or backwards through time. In our example, let ψ be a spatiotemporal
translation in � such that, for all (s1, s2, s3, t) in S × T, we have ψ(s1, s2, s3, t) �
(s1+5, s2−7, s3+10, t +3). If we interpret the spatially extended history h as describing
a possible world “from the perspective of position (0,0,0,0)”, then, heuristically, the
transformed history ψ(h) describes the same world “from the perspective of position
(5, − 7,10,3)”. Here a �-invariant event E has the property that whenever a history
h is in E, then so is ψ(h). Roughly speaking, this means that the world described by
h appears to be in the set E “from the perspective of position (0,0,0,0)” if and only
if it appears to be in E “from the perspective of position (5, − 7,10,3)”, and so on.
Ergodicity requires any such event to occur either almost always (with probability 1)
or almost never (with probability 0).

In a spatiotemporally ergodic system, we can estimate the probability of any event
by counting the spatiotemporal frequency with which that event occurs.

Spatiotemporal Ergodic Theorem: Suppose the system is spatiotemporally
ergodic. Let E be any event and let h be any history. For all r >0, let �r be the
set of all spatiotemporal translations by any vector (v1, v2, v3, v4) with integer

50 The system might also have other spatiotemporal symmetries, but this is irrelevant here.

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coordinates between 1 and r. Let N r be the number of translations ψ in �r
such that ψ(h) is in E. Then, with probability 1, the ratio N r/r

4 will converge to
Pr�(E) as r increases towards infinity.

51

Intuitively, N r is the number of times the event E has “occurred” in the spatially
extended history h from time 1 to time r and inside a three-dimensional box with
side-length r. The ratio N r/r

4 is therefore the frequency of occurrence of event E, up
to time r inside this box, in the spatially extended history h. This frequency might
be measured, for example, by performing a sequence of experiments or observations
inside this box. The Spatiotemporal Ergodic Theorem says that, with probability 1,
the empirical frequency will converge to the true probability of E as the number of
observations becomes large.52 As explained in Sect. 2.7, we can use this procedure to
estimate not only unconditional probabilities but also conditional ones, and thereby to
learn the properties of the conditional probability structure {PrE}E⊆�.

A broader lesson is that whether a system is ergodic in the first place depends on the
system’s symmetries. If a system is rich in symmetries, then ergodicity becomes easier
to achieve than if the system has only few symmetries. To see this, note that the notion
of �-invariance is logically more demanding for a larger set � of symmetries than
for a smaller one, since an event E will need to be preserved under more symmetries
in order to qualify as �-invariant. As a result, there will be fewer �-invariant events
if � is large, and hence the property of ergodicity, which constrains the probability of
�-invariant events, becomes less demanding. Conversely, if the set � of symmetries
is small, more events may qualify as �-invariant. In the limit, if � contains only the
(trivial) identity symmetry, then every event E will be �-invariant, and so no system
with a non-degenerate conditional probability structure will qualify as ergodic. (Recall
that, in an ergodic system, the unconditional probability of every �-invariant event
must be either 0 or 1. If all events are �-invariant, this rules out non-degenerate
probabilities.) Thus, we must conclude not only that ergodicity is a key prerequisite
for inferring a system’s conditional probability structure from local observations, but
also that without enough symmetries this inference would not get off the ground.53

51 For simplicity, we here assume that the symmetry monoid � is isomorphic to Z4, where Z is the monoid
of integers. The theorem, which we have stated somewhat informally, also holds if � is isomorphic to R4,
or if � is any amenable monoid. For a more formal statement, see Krengel (1985, Chapter 6).
52 It is not necessary to average over a sequence of “boxes”; the same argument works for any sequence
of sets which increase in size and thickness in an appropriate sense, technically any Følner sequence.
53 One complication is that not all systems are ergodic. For example, in systems with conservation laws,
such as conservation of energy or momentum, each value of the conserved variables determines a non-trivial
invariant subset of �. But the “Ergodic Decomposition Theorem” shows that any non-ergodic system can
be split up into “ergodic components”—informally, minimal invariant subsets of �, each of which (except
possibly a set of measure zero) supports its own ergodic probability function (see, e.g., Glasner 2003,
p. 72, Theorem 3.22). If we are part of the system, then we are already confined to one such component.
Furthermore, even if the system as a whole is not ergodic, many of its factor systems may be ergodic
(see “Appendix A”). This suggests that, by choosing the right level of description for the system (e.g.,
by adopting a sufficiently coarse-grained, higher-level description, as discussed in List and Pivato 2015),
we may be able to reap the benefits of ergodicity. For the applicability of ergodic methods to non-ergodic
Hamiltonian systems, see Berkovitz et al. (2006, Section 4).

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3.8 The role of space

What is the role of “space” in a spatially extended system? As we will now see, the
structure of space affects the way the system evolves over time. To make this precise,
we first introduce a formal representation of the topology of space and then discuss
its role in the system’s dynamics.

The topology of space can be represented by a binary relation → between subsets of
S. Heuristically, if R and R′ are two subsets of S, such as two “regions” of space, then R
→ R′ means that R and R′ are “adjacent” in that information from R can flow “directly”
into R′, without needing to pass through some intervening points “between” R and R′.
Later, we explain exactly what we mean by “information flow”, but for our initial
discussion, we leave it unexamined. We call → the adjacency structure of space.54

Adjacency structures arise naturally in many systems. For example, suppose S
is ordinary three-dimensional Euclidean space, and suppose information can flow
only “continuously” through this space. This would be the case, for instance, in a
system consisting of particles travelling along continuous trajectories and interacting
via continuous force fields, such as those found in classical mechanics, or in a system
described by partial differential equations, such as those found in quantum mechanics,
classical electrodynamics, or hydrodynamics. In such systems, for any subsets R and
R′ of S, we have R → R′ if there exists a point s in R such that, for any radius r >0,
the ball of radius r centred at s intersects R′.55

For another example, suppose S is the three-dimensional integer lattice: the set of
all ordered triples s � (s1, s2, s3), where s1, s2, and s3 are integers. Say that two points
s and s′ in S are neighbours if they differ in only one coordinate and that difference
is 1. Thus (3,7,5) and (3,6,5) are neighbours. Suppose information can flow only
directly between neighbours in the lattice. Then, for any subsets R and R′ of S, we
have R → R′ if some point in R is a neighbour of some point in R′.56 Discrete spatial
geometries of this kind can be found in a class of systems called cellular automata.57

For a final example, consider a directed graph, which consists of a set of “vertices”,
along with a set of “arrows” which connect pairs of vertices. Directed graphs can be
used to model electric circuits, communication networks (e.g., the internet), economic
and transportation networks, and biological systems (e.g., neural networks, gene reg-
ulatory networks, and epidemiological networks). Suppose S is the set of vertices.
Then, for any subsets R and R′ in S, we have R → R′ if there is an arrow from some
vertex in R to some vertex in R′.

If the sets R and R′ overlap (i.e., R ∩ R′ �� ∅), then clearly we have both R → R′ and
R′ → R. However, the examples above show that we can have R → R′ even if R and R′
do not overlap, as long as the two sets “touch” each other in some sense. Intuitively, R
→ R′ means that it is not possible to interpose any “barrier” between R and R′; there
is no “gap” between them.

54 Adjacency structures are similar to proximity relations, which have been studied extensively in general
topology (e.g., Willard 1970, Sections 40 and 41). But we do not assume that our adjacency structures
satisfy the axioms of a proximity relation, so they are more versatile.
55 Generally, an adjacency structure can be defined in a similar way on any metric or topological space.
56 Generally, an adjacency structure can be defined in a similar way on any Cayley graph of any group.
57 See Ilachinski (2001) and Moore and Mertens (2011).

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What role does the adjacency structure play in a spatially extended system? Why
does space have one adjacency structure rather than another? Just as we argued earlier
in the case of time, we will now argue that a “correct” adjacency structure on space is
onethatsatisfiesaMarkovpropertywithrespecttotheconditionalprobabilitystructure
{PrE}E⊆�. This Markov property is defined by considering conditional probabilities
based on “partial information” about a spatially extended history.

We therefore need a precise way to talk about such “partial information”. Let R be
a subset of S, and let R × T be the set of all ordered pairs (s, t), where s is an element of
R, and t is an element of T. So, R × T is the set of all time-slices restricted to the spatial
region R. For any history h in �, recall that h R×T denotes the restriction of h to the
set R × T. This restriction records only the part of the history h which “happens inside
R”. Let us then define the event [h R×T ] to be the set of all extensions of h R×T to full
histories in �, i.e., the set of all h′ in � such that h R×T � h′R×T . These are precisely
the histories that are accessible from h relative to the space–time region R × T. The
Markov property for adjacency structures will be based on conditional independence
with respect to such events, in the following way.

For any event E (i.e., a subset of �), we say that E happens inside R if, for all
histories h and h′ such that h R×T � h′R×T , history h is an element of E if and only if
history h′ is an element of E. In other words, the question of whether or not a particular
history is an element of E is completely determined by the restriction of that history
to spatial “region” R.

A tripartition of S is a triple (R, R′, R′′), where R, R′, and R′′ are three disjoint
subsets of S which together cover S (i.e., R ∪ R′ ∪ R′′ � S), such that it is not the case
that R → R′′ or R′′ → R. Heuristically, this means that the set R′ “separates” R from
R′′. For example, suppose S is three-dimensional Euclidean space, with the adjacency
structure described above. Let R be the set of all points whose distance to the origin
is less than 1: the unit ball. Let R′ be the set of all points whose distance to the origin
is between 1 and 2, so R′ is a sort of thick spherical “shell” around R. Finally, let R′′
be the set of all points whose distance to the origin is greater than 2. Then (R, R′, R′′)
is a tripartition of S.

We say that the adjacency structure → satisfies the Markov property with respect to
the conditional probability structure {PrE}E⊆� if, for any tripartition (R, R′, R′′) and
any history h in �, any event which happens inside R is conditionally independent from
any event which happens inside R′′, given everything that happens in R′ (i.e., given
[h R′×T ]). Heuristically, this means that there is no way for information to propagate
from R into R′′, or vice versa, without first passing through R′. For example, suppose
S is three-dimensional Euclidean space, and (R, R′, R′′) is the “concentric sphere”
tripartition described above. In this case, the spherical shell R′ acts as a barrier that
isolates the ball-shaped compartment R from any influences coming from the “outer
region” R′′. If we have complete information about the history inside R′ (i.e., we
know [h R′×T ]), then we have complete control over the boundary conditions for any
experiment we conduct inside R, and thus we do not need to control or even know
what happens in the outer region R′′.

Scientists implicitly assume that space satisfies the Markov property every time they
construct a laboratory apparatus that “isolates” some experiment from the surrounding
environment. Indeed, people also implicitly assume the Markov property every time

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they close the doors and windows of their houses to keep out the cold. Thus, the
Markov property is central to the way we ordinarily think of space. It underpins the
adjacency structure of space in the same way it underpins the order structure of time.

Just as with time, however, the Markov property does not completely determine the
structure of space. First, there may be more than one adjacency structure on S which
satisfies the Markov property with respect to {PrE}E⊆�, just as there may be more
than one Markovian order on T. Second, the adjacency structure alone leaves many
important geometric properties of S unspecified. For example, in many contexts, we
would like to define a metric on S, which determines a notion of “distance” between
points. This is obviously crucial in classical mechanics, for example. The adjacency
structure does not determine a unique metric. We therefore now turn to the question
of how we might arrive at such a metric.

3.9 Duration and distance

Recall that the set T of times is linearly ordered. In many contexts, we would like to
define a notion of duration on T. That is, given four moments t1, t2, t3, and t4 in T,
with t1 < t2 and t3 < t4, we would like to determine whether the time interval between
t1 and t2 is greater or smaller than that between t3 and t4. To do this, we suppose
that the monoid of temporal symmetries, �, acts freely and transitively on T, and all
symmetries in � are order-preserving. This means that, for any times t1 and t2 in T,
there is a unique symmetry ψ in � such that ψ(t1) � t2, and, for any symmetry ψ in
�, t1 < t2 implies ψ(t1)< ψ(t2). We can then define a formal “subtraction” operation
on T as follows. Fix some reference time t0. Now, for any times t1 and t2 in T, we
define

t2 − t1 � ψ(t2), where ψ is the unique temporal symmetry in � such that ψ(t1) � t0.

In particular, this implies that t − t0 � t, for any t in T. For any four points t1, t2,
t3, and t4 in T, we say that the time interval from t1 to t2 is greater than the one from
t3 to t4 if t2 − t1 > t4 − t3. Similarly, we can define a formal “addition” operation on
T. For any times t1 and t2 in T, we define

t1 + t2 � ψ(t2), where ψ is the unique temporal symmetry in � such that ψ(t0) � t1.

The set T, with the ordering < and the operation +, forms a left-linearly ordered
group.58

In many contexts, we would also like to define a metric on S, which determines
a notion of “distance” between points in space. As we have noted, the adjacency
structure does not determine a unique metric. But we can define a concept of distance

58 Formally, (i) the operation + is associative, i.e., (t1 + t2) + t3 � t1 + (t2 + t3) for all t1, t2, t3 in T;
(ii) there is an identity element, namely t0, such that t0 + t � t � t + t0 for all t in T; (iii) every element
t in T has an inverse − t such that t + (− t) � t0 � (− t) + t; and (iv) the ordering<is left-homogeneous,
meaning that, for all t1, t2, t3 in T, we have (t1 + t2 < t1 + t3) ⇔ (t2 < t3). Left-linearly ordered groups are
not generally commutative (“abelian”), i.e., we could have t1 + t2 �� t2 + t1 for some t1, t2 in T. See, e.g.,
Fuchs (2011) for an introduction to ordered groups.

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on S by measuring how long it takes for information to travel from one point to the
other. To do this, we need to use the concept of duration we have just introduced.

Given any two regions R and R′ of S, and a time t in T, we define what it means for
region R′ to be “not reachable” from region R in time t. We begin with some preliminary
definitions. For any subset R of S, and any time t in T, let R × {t} denote the set {(s, t):
s∈R}. Adapting our earlier definition, we say that an event E happens inside R at time
t if, for all histories h and h′ such that h R×{t } � h′R×{t }, history h is an element of E
if and only if history h′ is an element of E. In other words, whether or not a particular
history is an element of E is completely determined by the restriction of that history
to the space–time region R × {t}. Further, let RC denote the complement of R in S,
i.e., RC � {s ∈ S: s /∈ R}. Given any two subsets R and R′ of S, and a time t in T with t
> t0, we now say that R

′ is not reachable from R in time t if, for any history h in �, any
event which happens in R′ at time t is conditionally independent of any event which
happens in R at t0, given [h RC ×{t0}]. Informally, once we have complete information
about the state of the system outside the set R at time t0, learning something about
the state of the system inside R at time t0 gives us no further information about the
eventual state inside R′ at the later time t.59

We now define the distance d(R, R′) between R and R′ to be the maximum time t in
T such that R′ is not reachable from R in time t, if this maximum exists.60 This can be
interpreted as the minimum time required for information to “propagate” from R to R′.
It would be natural to suppose that this notion of distance satisfies three properties:

Symmetry: For all subsets R, R′ of S,

d(R, R′) � d(R′, R).

Triangle inequality: For all subsets R, R′, R′′ of S,

d(R, R′′) ≤ d(R, R′) + d(R′, R′′).

Non-complementarity: For all subsets R1, R2, R3 of S,

d(R1 ∪ R2, R3) � min{d(R1, R3), d(R2, R3)}.

However, none of these properties can be guaranteed, unless the conditional prob-
ability structure {PrE}E⊆� has the right underlying properties. For example, if the
information flow between different spatial locations is asymmetrical, such as in many
communications networks, then Symmetry might not be satisfied; it might take longer
for information to propagate from R to R′ than vice versa. If information can be

59 In our definition of “non-reachability”, we have referred to the reference time t0. However, because �
acts freely and transitively on T, the reference time does not matter. When region R′ is not reachable from
region R in time t, this implies that, for any times t1 and t2 with t2 − t1 � t, any event which happens in
R′ at time t2 is conditionally independent of any event which happens in R at t1, given [h RC ×{t1}].
60 If the maximum does not exist, we can use the supremum, provided the order of time is supremum-
complete (i.e., any subset of T has a supremum), as it is if T is the set of real numbers. If the order of time
is not supremum-complete, then the precise distance between R and R′ may not be well-defined.

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“forgotten” or “erased” at some spatial locations in the system, then the Triangle
Inequality might not be satisfied; some information propagating from R to R′ might
be forgotten before it reaches R′′. Turning to Non-Complementarity: it is always true
that d(R1 ∪ R2, R3) ≤ min{d(R1, R3), d(R2, R3)}. However, this inequality could
be strict; i.e., we could have d(R1 ∪ R2, R3) < min{d(R1, R3), d(R2, R3)}. For
example, what happens in regions R1 and R2 at time t1 could be like two pieces of a
puzzle, which reveal little about what happens in region R3 at time t2 when considered
separately, but determine it completely when put together.61

Note that our definition of distance between regions of space immediately entails a
definition of distance between points in space: the distance between any two points s1
and s2 in S is simply the distance between the singleton regions consisting of them, i.e.,
d(s1, s2) � d({s1},{s2}). Clearly, d(s, s) � 0 for any point s in S. Thus, if our distance
measure satisfies Symmetry and the Triangle Inequality, it determines a metric on the
space S (or a pseudo-metric if d(s1, s2) � 0 for some s1 �� s2). Furthermore, if it satisfies
Non-Complementarity, this metric completely determines the distance between any
two regions R and R′ in S.62 However, as we have pointed out, the distance measure
need not generally satisfy these properties.

One notable feature of the present approach is that it measures the distance between
spatial locations in units of time. This is, of course, consistent with the practice in
modern physics of measuring distance in units such as light seconds or light years.
However, the approach works only if the maximum speed of information propaga-
tion in our system is finite. In classical physics, information can propagate through
space at arbitrarily high speeds. Therefore, in a classical physical system, the effec-
tive “distance” between any two spatial locations collapses to zero, according to our
definition. To recover a non-trivial definition of “distance” in such a system, we must
impose some restriction on the sort of “information transmission” we can use. For
instance, we could consider information transmission via some messenger or signal
travelling at a fixed velocity. Similarly, in Maxwell’s theory of electrodynamics, which
is complementary to classical mechanics, electromagnetic waves propagate at a fixed
and finite speed, namely the speed of light, even if classical-mechanical particles can
exceed this speed. Thus, in the world of classical physics, we could define a non-
trivial concept of “electromagnetic distance”, even if there is no non-trivial concept
of “mechanical distance”. We discuss the issue of distance in quantum mechanics in
“Appendix D”.

61 Technically, this means that there exist events E1, E2, and E3 in � which happen, respectively, in region
R1 at time t1, in region R2 at time t1, and in region R3 at time t2 such that E1, E2, and E3 are pairwise
independent, but not jointly independent. This situation is common in probability theory.
62 To be precise, d(R, R′) � min{d(s, s′) : s ∈ R and s′ ∈ R′}. Strictly speaking, this only works if R and
R′ are finite setsofpoints.For infinite sets,wewouldneedaslightlystrongerversionofnon-complementarity,
which says that d(R, R′) � inf{d(s, s′) : s ∈ R and s′ ∈ R′} (and this infimum exists).

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4 Amorphous systems: space–time as an emergent property

4.1 Basic definitions

So far, we have defined histories as functions from a set of points in time or
space–time into some state space, where histories play the role of possible worlds.
Time or space–time, in turn, had an exogenously given structure. In a temporally
evolving system, time was some linearly ordered set (T), and in a spatially extended
system, space–time was explicitly decomposed into space (S) and time (T), consistent
with some fixed geometry. This picture can, and for many purposes must, be general-
ized. Both special and general relativity theory, for example, go against the idea that
there exists a fixed temporal dimension (for a classic discussion, see Putnam 1967).

A more general approach is to define a history as a function from some “index set”,
which we call a set of loci, into a state space. The set of loci could be a linearly ordered
set of points in time, thereby accommodating temporally evolving systems, or a set
of space–time locations with an explicit decomposition into space and time, thereby
accommodating spatially extended systems. But it could also be a more general four-
dimensional space–time manifold without any exogenous decomposition, or even a
completely abstract index set.

Formally, let I (for “index set”) be the set of loci, and let X denote the state space.
A generalized history is a function h from I into X, where, for each locus i in I, h(i)
is the state of the system at locus i. As in the case of spatially extended systems, the
state h(i) is best interpreted, not as a “global” state in which the system is at some
specific point in time (indeed, there is no exogenous notion of time), but as a “local”
state in which the system is at a specific locus. We write � to denote the set of all
generalized histories deemed possible, which can again be viewed as nomologically
possible worlds, and subsets of � are called events.63

To complete the definition of what we call an amorphous system, we must, once
more, introduce a conditional probability structure on �. As should be clear by now,
this is a family of conditional probability functions {PrE}E⊆�, consisting of one PrE
for each event E in �. Now an amorphous system is the pair consisting of the set � of
nomologically possible generalized histories and the conditional probability structure
{PrE}E⊆�.

How much of our earlier framework can be extended to amorphous systems? We
might ask, for instance, whether an abstract index set, despite not being endowed with
any exogenous structure, can attain some spatial and/or temporal structure as an emer-
gent property, for instance as a byproduct of the correlations encoded in {PrE}E⊆�.
We might also ask whether, and to what extent, the geometry of the set of loci is
unique, or whether there might be multiple, equally admissible geometries.

63 Note that, in the literature on general relativity theory, the word “event” is used to refer to the objects
we call “loci”. Our use of the word “event” is consistent with its use in probability theory.

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4.2 Adjacency structure and the Markov property

Just as in Sect. 3.8, the topology of the set I of loci can be represented by an adjacency
structure: a binary relation → defined between subsets of I. For example, suppose I
is a set of times, as in Sect. 2, i.e., I � T. For any subsets R and R′ of I, define R
→ R′ if there does not exist any time t such that r < t < r′ for all r in R and all r′ in R′.
For another example, let I be the four-dimensional space–time manifold of a general
relativistic system. Then, for any subsets R and R′ of I, we might define R → R′ if
there is a locus i in R such that every open neighbourhood around i intersects R′.

In Sect. 2.9, we related the order structure of the set T of times to the conditional
probability structure {PrE}E⊆� by means of a temporal Markov property. Likewise,
in Sect. 3.8, we related the adjacency structure of the set S of spatial locations to the
conditional probability structure {PrE}E⊆� by means of a spatial Markov property.
We now discuss a similar idea concerning a general set of loci. This will allow us to
view the adjacency structure among loci, and thereby its topology, as an “emergent
property”: something that emerges from the correlations between events encoded in
{PrE}E⊆�.

Let R be a subset of I (i.e., a collection of loci). As before, for any generalized
history h in �, we define hR to be the restriction of that history to the set R. We then
define the event [hR] to be the set of all histories h

′ in � such that h R � h′R. For
any event E (i.e., a subset of �), we say that E happens inside R if, for all histories
h and h′ such that h R � h′R, history h is an element of E if and only if history h′
is an element of E. That is, whether or not a particular history is an element of E is
completely determined by the restriction of that history to R.

As in Sect. 3.8, we define a tripartition of the set I of loci as a triple (R, R′, R′′),
where R, R′, and R′′ are disjoint subsets of I which together cover I (i.e., R ∪ R′ ∪ R′′
� I), such that it is not the case that R → R′′ or R′′ → R. Again, this means that the
set R′ “separates” R from R′′.

For example, let I be a set of times (I � T) with the adjacency structure introduced
at the start of this section. Fix two times t0 and t1 with t0 ≤ t1. Let R be the set of all
times strictly before t0, let R

′ be the set of all times between t0 and t1 (including t0 and
t1), and let R

′′ be the set of all times strictly after t1. Then (R, R′, R′’) is a tripartition
of I.

For another example, let I be the four-dimensional Minkowski space–time of spe-
cial relativity, with the “open neighbourhood” adjacency structure introduced above.
Let λ be a linear time-like trajectory through I, for instance the trajectory of an “ob-
server” traveling through space–time at a constant velocity, and let p be a point on this
trajectory. In special relativity theory, there is a unique three-dimensional simultaneity
hyperplane R′ passing through p, such that all events that happen inside R′ seem to
occur simultaneously from the perspective of the λ-observer at p. Let R be the set of
all points in I which have some part of R′ in their future light-cone, and let R′′ be the set
of all points in I which have some part of R′ in their past light-cone. Then (R, R′, R′′)

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is a tripartition of I.64 More generally, let R and R′′ be any disjoint open subsets of I,65
and let R′ be the complement of the union R ∪ R′′. Then (R, R′, R′′) is a tripartition
of I.

We say that the adjacency structure → satisfies the amorphous Markov property
with respect to the conditional probability structure {PrE}E⊆� if, for any tripartition
(R, R′, R′′) and any generalized history h in �, any event which happens inside R
is conditionally independent from any event which happens inside R′′, given [hR′].
Again, this means, roughly, that there is no way for information to propagate from
R into R′′, or vice versa, without first passing through R′. For example, suppose I is
four-dimensional Minkowski space–time, and (R, R′, R′′) is the tripartition described
above. In this case, the simultaneity hyperplane R′ plays the role of the “present”,
which isolates the “past” R from the “future” R′′. If we have complete information
about the history inside R′ (i.e., we know [hR′]), then we have complete information
about the “present state” of the world. Thus, we can predict its future evolution (in
R′′) without needing to know anything about its past history (in R).

In Sect. 2.9, we argued that the temporal Markov property was the key property of
time; a “correct” ordering of the set T was any ordering that satisfied this property.
Likewise, in Sect. 3.8, we argued that the spatial Markov property was the key property
of space; a “correct” adjacency structure on the set S was any adjacency structure
that satisfied this property. Now we make a parallel claim for amorphous systems: a
“correct” adjacency structure on I is one that satisfies the amorphous Markov property.
This Markov property subsumes both the temporal Markov property of Sect. 2 and
the spatial Markov property of Sect. 3.

Thishasanimportantconsequence.Thetopologyoftheindexset I, intheformofthe
adjacency structure, does not need to be imposed exogenously. Instead, this topology
can emerge endogenously from the conditional probability structure {PrE}E⊆�. We
say that an adjacency structure → between subsets of I is {PrE}E⊆�-admissible if it
satisfies the amorphous Markov property with respect to {PrE}E⊆�. If we think of I
as a sort of generalized space–time, this means that the topology of space–time is an
emergent property of the amorphous system.66

64 In a model of general relativity, a similar construction works if R′ is a Cauchy surface in the four-
dimensional space–time manifold.
65 A subset R of I is open if, for any s in R, there is some r >0 such that the ball of radius r around s is
contained in R.
66 We are not the first to suggest that the geometry and/or topology of space–time could be an emergent
property of more fundamental causal structures. Brown and Pooley (2001, 2006) have argued that the
geometry of relativistic space–time should be seen as a consequence of the symmetries (i.e., Lorentz
covariance) of the dynamical laws governing matter and electromagnetism. In their words (2006, Section 5):
“space–time’s Minkowskian structure cannot be taken to explain the Lorentz covariance of the dynamical
laws.Fromourperspective,ofcourse,thedirectionofexplanationgoestheotherwayaround.ItistheLorentz
covariance of the laws that underwrites the fact that the geometry of space–time is Minkowskian.” See also
Brown (2005). However, Brown and Pooley’s approach is very different from ours. The idea of emergent
space–time geometry also appears in the literature on high-energy physics and quantum cosmology. See,
e.g., Konopka et al. (2008) and Hamma et al. (2010).

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4.3 Time and predictability

Both temporally evolving systems and spatially extended systems come with a set T
which plays the role of “time”. What plays the role of time in an amorphous system?
The adjacency structure described in the previous section tells us whether two subsets
of the index set I are in “informational contact” or are “informationally separated”
from one another, but it does not tell us which subset comes “before” and which comes
“after”, or even whether this question makes sense. We now explain how time itself
can be an emergent property of an amorphous system.

Let → be an adjacency structure on the index set I. Let T be a linearly ordered set.
A possible time structure on I is a function τ from I onto T (i.e., with T � τ(I)) such
that, for any t in T, if (i) R is the set of all points i in I such that τ(i)< t, (ii) R′ is the
set of all points i in I such that τ(i) � t, and (iii) R′′ is the set of all points i in I such
that τ(i)> t, then (R, R′, R′′) is a tripartition of I. Heuristically, the function τ specifies,
for each locus in I, the time at which that locus occurs, according to the given time
structure.

For example, let I be four-dimensional Minkowski space–time as described in
Sect. 4.2, and let λ be a linear time-like trajectory through I. Fix some point p0 on the
trajectory λ. Let T be the set of real numbers. Then, for every t in T, there is a unique
point pt along the trajectory λ which appears to be t seconds in the future of p0 (or
in the past, if t <0), with respect to the subjective time (i.e., proper time) experienced
by an observer traveling along the trajectory λ. Let Rt be the simultaneity hyperplane
passing through pt. If we define τ(i) � t for all points i in Rt, then τ is a possible time
structure on I.

As this example illustrates, an amorphous system may admit many possible time
structures. In special relativity, there is a distinct time-structure for every inertial
reference frame. All of these time structures are equally “correct”. Indeed, this is one
of the key insights of special relativity theory. However, unless we impose further
constraints, a system may also admit many “absurd” time structures. For example,
suppose I is four-dimensional Newtonian space–time (i.e., I � R3 × R), with the “open
neighbourhood” adjacency structure described in Sect. 4.2. For all points (s1, s2, s3, t)
in I, define τ(s1, s2, s3, t) � s3. Then τ is a possible time structure on I. But if the
“true” time coordinate is t, not s3, it seems that this time structure is not correct. So,
what property of the system determines which time structures are the correct ones?

Clearly, a “correct” time structure should satisfy something like the temporal
Markov property from Sect. 2. However, if the adjacency structure → satisfies the
amorphous Markov property with respect to the conditional probability structure
{PrE}E⊆�, then it is easy to see that any possible time structure will satisfy the
temporal Markov property.67 So, the Markov property alone is not enough to pick out
the “correct” time structures.

Arguably, what picks out the correct time structures is predictability. To understand
this, suppose we took a classical mechanical system with Newtonian space–time I �
R3 × R, and applied the “absurd” time structure τ(s1, s2, s3, t) � s3, as defined above.
67 To be more precise: given a possible time structure on I, we can represent the amorphous system as
a temporally evolving system, and this system, in turn, will satisfy the temporal Markov property. The
construction is straightforward, but to avoid too many technicalities, we set aside the details here.

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How would the system appear with respect to this time structure? It would appear
very strange and unpredictable. Particles would randomly pop in and out of existence.
Energy and momentum would not be conserved from one moment to the next. Events
would seem to unfold over time without any rhyme or reason. This total lack of
predictability would be an indication that we had picked the wrong time structure for
the system.

On the other hand, if we had picked the “correct” time structure, namely
τ(s1, s2, s3, t) � t, then the system would appear completely deterministic; its state
at one “moment” in time, as defined by τ, would completely determine its “past” and
“future” behaviour, as defined by τ. This total predictability is an indication that this
is the correct time structure for the system.

In this example, there was a particularly stark contrast between an “incorrect” time
structure, which renders the system totally unpredictable, and a “correct” one, which
renders it totally predictable. This is because classical mechanical systems are deter-
ministic. In an indeterministic system, there will not generally be such a stark contrast.
Nevertheless, some time structures will render the system more predictable than oth-
ers, and among these, we claim, the ones that render the system most predictable are
the correct time structures for that system.

To make this idea more precise, we need a way to measure the “predictability” of
a system under a given time structure. One way to do this is to use the information-
theoretic notion of entropy.68 For any subset R of I, let �R be the set of all R-restricted
histories hR obtained from any h in �. For simplicity, let us assume that the underlying
state space X is finite. If R′ is some other finite subset of I, then �R′ is also finite.69
Suppose we know hR, and we want to predict hR′. For any hR in �R, there is a quantity
called the conditional entropy of R′ given hR, denoted by η(R′, hR), which measures
how “unpredictable” the restricted history hR′ is, given the restricted history hR.

70 For
example, if hR′ is entirely determined by hR, then η(R

′, hR) � 0. At the other extreme,
if hR′ is effectively as unpredictable as a collection of independent coin-tosses, even
after conditioning on hR, then η(R

′, hR) � 1. Intermediate levels of entropy represent
intermediate degrees of unpredictability.

Now, let τ be a time structure, mapping I into T. Let t be some time in T; let R
be the set of all points i in I such that τ(i) � t; and let RC be the set of all points i in
I such that τ(i) �� t. We define η(τ, t), the unpredictability of the system under τ at t,
to be the maximum value of η(R′, hR), where hR can be any element of �R and R′ is
allowed to be any finite subset of RC.71 If η(τ, t) � 0, then this means roughly that
any generalized history h in � is almost entirely predictable, based on its restriction

68 This is not the same as thermodynamic entropy, although it is loosely related. Thus, the discussion that
follows should not be interpreted thermodynamically.
69 If |X| and |R′| are the cardinalities of X and R′, then the cardinality of �R′ is at most |X ||R

′|.
70 Formally, η(R′, h R) is the sum, over all possible R′-restrictions h R′ in �R′ , of

− Pr([h R′ ]|[h R]) log2[Pr([h R′ ]|[h R])]/|R′|log2(|X |).

However, the precise formula is not important for this discussion.
71 To be more precise, it is the supremum of this set. The maximum is not always well-defined.

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2594 Synthese (2021) 198:2551–2612

hR.
72 If η(τ, t)>0, then histories in � are not, in general, fully predictable from their

restrictions to R. The larger η(τ, t) is, the less predictable these histories are. We then
define η(τ), the unpredictability of the system under the time structure τ, to be the
maximum value of η(τ, t) over all times t in T.73

For example, suppose I is the four-dimensional Newtonian space–time of a classical
mechanical system (i.e., I � R3 × R), and τ is the “correct” time structure for this
system, namely τ(s1, s2, s3, t) � t. Then η(τ) � 0, because classical mechanics is
entirely deterministic. However, if τ was an “incorrect” time structure, such as τ(s1,
s2, s3, t) � s3, then we would have η(τ)>0, because the ascription of this incorrect
time structure would render the system unpredictable, as we have explained.

We now come to the key point of this section. A correct time structure for an
amorphous system is one that minimizes unpredictability and thereby maximizes reg-
ularity. This definition allows that there may be many correct time structures, all of
which render the system equally predictable, as in the case in special or general relativ-
ity. This has an important consequence. The correct time structure does not need to be
imposed exogenously. Instead, the correct time structure (or structures) could emerge
endogenously from the conditional probability structure {PrE}E⊆�. In other words,
the structure of time itself could be an emergent property of the amorphous system.
Using a more metaphysical language, it might be that space and time are grounded in
the dynamics of the system, rather than the other way around.

4.4 Which features of a system are real?

A final philosophical question on which we wish to comment briefly is the following.
Suppose we have described a given system using our formal framework. Should we
treat all features of that system as “real”, or should we treat some features as mere
artefacts of our formal description?

The debates between relational and substantival views about space and time, and
between structuralist and full-blown realist views in science more generally, can be
seenasattemptstoanswerthisquestion.74 Letusbeginwitha relational or structuralist
view, which may be about space and time in particular or about the properties of a
system more generally. On such a view (of which there can be several variants), only
some “relational” or “structural” properties of a system count as real, while “intrinsic”,
“non-structural” properties do not. It does not matter, for instance, what the nature of

72 Even if η(τ, t) � 0, there may be some “residual” unpredictability, in that � may contain more than
one extension of hR to all of I. However, the conditional probability structure {PrE }E⊆� concentrates all
probability on one of these possible extensions; the rest of the extensions get probability zero.
73 Again, strictly speaking, we require the supremum of η(τ, t) across all t. This is not the only possible
measure of the system’s unpredictability under time structure τ. We could also take the average or some
other aggregate measure. For example, suppose that I is an N-dimensional integer lattice (i.e., I � ZN ).
Then we could measure the system’s unpredictability under different time structures using the theory of
entropy geometry and expansive subdynamics first developed for multidimensional cellular automata by
Milnor (1988) and later extended to arbitrary multidimensional symbolic dynamical systems by Boyle and
Lind (1997). See the section on “Entropy” in Pivato (2009) for a summary.
74 On a broadly “structuralist” or “relational” approach to metaphysics, see, e.g., Ladyman and Ross (2009).
On “absolute” versus “relational” accounts of space and time, see, e.g., Earman (1989). On “substantivalism”
and its discontents, see, e.g., Nerlich (2003).

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the system’s spatiotemporal loci in the set I is, nor what the nature of the system’s
possible states in the set X is. All that matters is how these loci and/or states are related
to one another and what dynamics they display. Two formally distinct systems, with
formally distinct index sets I and I′ and/or formally distinct state spaces X and X′, will
count as the same if their nomologically possible histories and conditional probability
structures are structurally indistinguishable.

By contrast, on a substantival or full-blown realist view, which may also be about
space and time in particular or about the properties of a system more generally, even
intrinsic, non-structural properties of a system can be real, over and above the sys-
tem’s relational or structural properties. So, the system’s spatiotemporal index set I
and its state space X may be significant in ways that go beyond the structures and
relations in which they stand. (Again, there can be several variants of such a view.)
An example of a non-structural property is the exact index of time. One can imagine
two structurally identical temporally evolving systems, indexed by T � {0,1,2,3,…}
and T ′ � {1,2,3,4,…}, respectively. The only difference is that in one system history
“starts at time zero”, whereas in the other it “starts at time 1”. For a relationalist or
structuralist, these are “the same” system. But a substantivalist or full-blown realist
might insist that there is a genuine difference between them.

The debates between these different views occur in several places in philosophy and
take a variety of forms, so we cannot do justice to them here. We wish to note, however,
that our formal framework can be used to express some salient positions within those
debates. Specifically, different answers to the question of which features of a system
are real can be expressed in terms of different criteria for individuating systems. If we
begin with a very large class of systems that are formally described in our framework,
there are a number of ways in which one might partition this class of systems into
equivalence classes that are each taken to represent the same system. Different such
partitions then correspond to different answers to the question of which features of a
system are real, rather than mere artefacts of our formal description. In particular, only
those features that are present among all members of any given equivalence class count
as real. Features on which there can be differences even within the same equivalence
class count as artefacts of our formal description.

A relational or structuralist view would entail that any two systems that do not differ
in any relational or structural properties count as the same and thereby fall into the
same equivalence class. A substantival or full-blown realist view, by contrast, would
entail that two such systems could still count as different; thus, the equivalence classes
would be more fine-grained according to such a view, and might even be singleton (in
which case all features of any given system would count as real).

Here is one way of formalizing this idea. Consider two amorphous systems, given by
the pairs (�, { Pr E }E ⊆�) and (�, { Pr ′E }E ⊆�′), where the histories in � are functions
from the set I of loci into the state space X, and the histories in �′ are functions from
the set I′ of loci into the state space X′. Let H and H′ denote the sets of logically
possible functions from I into X and from I′ into X′, respectively.

Suppose there is a bijection θ from I into I′, and also a bijection ξ from X into
X′ (recall that a bijection is a one-to-one, onto function). Using θ and ξ, we can then
define a bijection σ from H into H′ which maps each history h in H to the history h′
in H′ defined as follows: for each i′ in I′,

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2596 Synthese (2021) 198:2551–2612

h′(i ′) � ξ[h(i)], where i � θ−1(i ′) (with θ−1 defined as the inverse of θ).
The bijection σ is an isomorphism between the two systems if

• σ(�) � �′; and
• for any events E′ and D′ in �′, if E and D are the inverse images of E′ and D′ under

σ, then Pr ′E ′(D
′) � Pr E (D).

We call two systems isomorphic if there exists an isomorphism between them.
Isomorphicsystemsdisplaythesamedynamics,andtheyarerelationallyorstructurally
indistinguishable.75 Moreover, any topology of space and time that is admissible for
one such system can be mapped, in a structure-preserving way, onto a topology that
is admissible for the other.

Thus, on a relational or structuralist view, any two isomorphic systems should
be considered the same. On a substantival or full-blown realist view, they may still
differ. A view of the first kind would therefore take systems to be unique only up
to isomorphism, so that our initial large class of systems would be partitioned into
equivalence classes of isomorphic systems. A view of the second kind would opt for
a more fine-grained partition, acknowledging that even isomorphic systems may be
distinct in reality.

The properties of systems on which we have focused in this paper are mainly
structural and are preserved by all isomorphisms. This includes the symmetries and
ergodic properties of a system, the distinction between laws and “brute necessities”,
and the topology (or topologies) and geometry (or geometries) of space and time that
are compatible with the system’s correlation structure (in the sense that they satisfy the
relevant Markov conditions). Thus, even a relationalist or structuralist would accept
that all these properties are “real” features of the system, and not mere artefacts.

5 Concluding discussion

We have introduced a framework for describing three general classes of systems and
shown how it can be used to address a number of philosophical questions. We began
with the class of temporally evolving systems, of which classical dynamical systems
are a special case, and then moved on to the class of spatially extended systems and
the class of amorphous systems. As noted, the framework can accommodate systems
as diverse as the solar system, quantum–mechanical systems, special and general
relativistic systems, and the earth’s climate system.

We have discussed questions such as: how can we define nomological possibility,
necessity, determinism, and indeterminism? What is special about laws, and how
are laws related to symmetries? What regularities must a system display to permit
global generalizations from local observations? How can we formulate principles of
parsimony such as Occam’s Razor? What is the role of space and time in a system?
And what is at stake in the debate between relational and substantival views about
space and time, and between structuralist and full-blown realist views about systems
more generally?

75 In fact, any bijective symmetry of a system constitutes an isomorphism from a system into itself.

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While our framework and what it says about these questions should already be of
sufficient interest to make it worth studying, a further payoff lies arguably in the variety
of applications to which the framework lends itself. Developing these is beyond the
scope of this paper, but we conclude by mentioning a few.

5.1 Higher-level versus lower-level properties

Our framework can be used to explore the relationship between lower-level (“mi-
cro”) and higher-level (“macro”) properties of a system. By partitioning the system’s
state space X into suitable equivalence classes, we can capture the idea that “higher-
level” or “macro” states are more coarse-grained than “lower-level” or “micro” states,
so that each “macro” state can be realized by different “micro” states: the phe-
nomenon of multiple realizability. Consider, for example, all the different possible
micro-level trajectories of a tossed coin that each correspond to the macro-property
of “landing heads”. Or consider all the different possible micro-states of individual
water molecules that each correspond to a macro-state such as “frozen”, “liquid”, or
“gaseous”.

Suppose X istheoriginalstatespace,and istherelevantsetofequivalenceclasses,
which we interpret as the higher-level state space. We can then write σ to denote the
function that maps each lower-level state x in X to the corresponding higher-level state
in . Note the outlined font for higher-level objects. This function can be interpreted

as the supervenience relation connecting the two levels. We can then use σ to specify
the resulting higher-level histories.76 For each lower-level history h in the original set
�, the corresponding higher-level history is the function from T into , where, for
each t in T, (t) � σ(h(t)). (If we are dealing with a spatially extended or amorphous
system instead of a temporally evolving one, we must replace T in this definition with
S × T or I.) The set of higher-level histories is therefore � σ(�). Similarly, we can
use σ to arrive at a conditional probability structure defined over higher-level events,
formally written . See “Appendix A” for details. The pair can be
viewed as our system, re-described at a higher level. In the terminology of “Appendix
A”, the higher-level system is a factor system of the original, lower-level system.

This construction allows us to study the dynamics of the higher-level system and
to compare its properties with those of the lower-level system. Interestingly, the
higher-level dynamics may be different from the underlying lower-level dynamics.
For example, features such as determinism or indeterminism are not generally pre-
served under coarse-graining: the lower-level system may be deterministic, while the
higher-level system is not (or vice versa). Thus indeterminism could be an emergent
property (see, e.g., Butterfield 2012; List 2014; List and Pivato 2015; and relatedly
Werndl 2009b).

76 This construction, under the present notational conventions, was introduced in List (2014) and List and
Pivato (2015). Relatedly, see also Yoshimi (2012).

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2598 Synthese (2021) 198:2551–2612

In a similar vein, we may study the level-specificity of other properties. For instance,
this approach can be used to argue that non-trivial objective chance can be an emergent
phenomenon, consistently with lower-level determinism (List and Pivato 2015).77

5.2 Laws and regularities in the special sciences

There is much debate on whether there are laws in the special sciences, as distinct
from fundamental physics. The existence of laws is particularly contested in fields
such as biology, ecology, geology, psychology, and the social sciences. (Chemistry,
by contrast, is often viewed as a close relative of physics and thereby similar enough
to it in its lawfulness.) Examples of special-science regularities that are sometimes
describedaslawsinclude(i)Kleiber’slawinbiology,accordingtowhichanorganism’s
metabolic rate is proportional to the ¾th power of its body mass; (ii) the laws of supply
and demand in economics, according to which (except for Giffen goods) the demand
for a good is a decreasing function of its price, and the supply is an increasing function
of price; and (iii) Duverger’s law in political science, according to which, under a first-
past-the-post electoral system, the effective number of parties in the legislature will
be lower than under a proportional-representation system, ceteris paribus. The key
question is whether any of these regularities are sufficiently robust to qualify as laws.

One common view is that, as we move further away from fundamental physics,
there are fewer and fewer regularities that are genuinely law-like. Kim (2010, ch. 14),
for instance, argues that there are no “strict” laws in the special sciences. Among the
reasons he gives for this conclusion are (i) the multiple realizability of special-science
properties, which, he claims, undermines their “inductive projectibility”, and (ii) the
alleged metaphysical anomalism of the mental realm, which, he suggests, undermines
the existence of laws in psychology and the social sciences.

Other scholars defend the existence of laws in the special sciences. For exam-
ple, focusing on the social sciences, Kincaid (1990) argues that several widely cited
arguments against laws fail. He thinks that the most serious challenge to laws in the
social sciences comes from the excessive ceteris paribus qualifications that all such
laws require, but argues that the procedures we routinely employ to deal with such
qualifications in the natural sciences carry over to the social sciences.

Our framework might be used to make some progress in this debate. Using the
framework, we can in principle describe the special-science systems in question and
identify the properties these systems would have to display in order to secure the
existence of laws. Those laws would then have the testable and generalizable character
we have discussed. As we have seen, what laws there are in a given system depends on
the system’s symmetries and the properties they preserve. This is as true for a system
in the special sciences as it is for a physical system. Moreover, our analysis implies that
whether, given only local observations, we can gain knowledge of the probabilistic
dynamics of a special-science system depends on whether the system is ergodic. The

77 For earlier work defending higher-level chance, sometimes using a strategy similar in spirit to the present
one (though not fully equivalent), see, e.g., Loewer (2001), Frigg and Hoefer (2010), Glynn (2010), Strevens
(2011) and Hemmo and Shenker (2012).

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importance of ergodicity for the special sciences is much less widely recognized than
its importance for physics.

Interestingly, if a special-science system arises as a higher-level description of a
physical system, as discussed in Sect. 5.1, then it will inherit some structure from the
physical system, and it will have at least as many temporal or spatiotemporal sym-
metries as that physical system (and possibly more), and at least as much ergodicity,
for reasons explained in “Appendix A”. Another question is whether we are prepared
to recognize weaker kinds of laws corresponding to partial or local symmetries, as
defined in “Appendix B”. This question is particularly pertinent for the special sci-
ences, insofar as the systems investigated in fields ranging from biology to the social
sciences often have special initial or boundary conditions. While all of these issues
are difficult, our framework can help us clarify what is at stake in the debate about
special-science laws and thereby render the debate more tractable. For earlier appli-
cations of dynamical-systems theory to the special sciences, see Auyang (1998) and
Yoshimi (2012).

5.3 Intentional systems

Although there has been no such thing as intentionality in our paradigmatic examples
of systems, there is no barrier, in principle, to using our framework also for describing
systems involving intentional agents. Indeed, van Gelder (1995) and Juarrero (1999)
have argued for understanding cognitive systems as special kinds of dynamical systems
(see also Spivey 2008; Hotton and Yoshimi 2010; Silberstein and Chemero 2012); and
more recently, a precursor of the present formalism has proved useful for the analysis
of free will and agency (List 2014; List and Rabinowicz 2014). We can think of an
agent, together with the relevant environment, as a temporally evolving system. This
system can be described at different levels: at a physical level, at which we would not
take an “intentional stance” towards the system, and at an agential level, at which we
would take such a stance (on the notion of an “intentional stance”, see Dennett 1987).
Physical-level descriptions capture the states of the agent’s brain and body, while
agential-level descriptions capture the agent’s higher-level mental or psychological
states, thereby focusing on the agent’s beliefs, desires, and intentions, rather than the
underlying neuronal or bodily states.

The present framework then allows us to explain, for instance, how agential-level
indeterminism and an agent’s possibility of doing otherwise can co-exist with physical-
level determinism (List 2014). The framework might also shed some light on how other
psychological properties can emerge from the underlying physical dynamics of the
system. In particular, as a factor system of the original physical system, the agential
system may exhibit additional symmetries not present at the physical level—a point
already alluded to in Sect. 5.2. This may, in turn, be used to explain why some higher-
level regularities in an intentional system (e.g., regularities involving beliefs, desires,
intentions, and norms) may qualify as “real patterns”, as Dennett (1991) has argued,
and not merely as illusions due to our ignorance of the physical-level details.

Needless to say, all of these applications are challenging and raise controversial
philosophical issues. We hope, however, that our framework will be a clarifying con-

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tribution to formal metaphysics and the philosophy of science and will inspire further
work.

Acknowledgements List’s work was supported by a Leverhulme Major Research Fellowship, and Pivato’s
work was supported by an NSERC Grant #262620-2008 and by Labex MME-DII (ANR11-LBX-0023-01).
We are very grateful to George Musser, Bryan Roberts, Timothy Williamson, and the anonymous reviewers
of this paper for helpful comments.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-
tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.

Appendix A: Factor systems

One possible objection to our framework is that it is both unrealistic and unwieldy.
It is unrealistic because the actual universe is not sufficiently regular (e.g., it might
lack a large enough monoid of symmetries or ergodicity). It is unwieldy because the
universe as a whole is far too complex a system for us to analyze within this framework
anyway.

However, there is no need to insist on applying our framework to the universe as a
whole. Instead, we can apply it to a “factor” system, as we now explain. Consider a
temporally evolving system, consisting of a set � of nomologically possible histories
(each of which is a function from a set T of times into a set X of possible states),
along with a conditional probability structure {PrE}E⊆�. Let X′ be another set, and
let σ be a function from X into X′. For every history h in �, let σ(h) be the function h′
from T into X′ defined by h′(t) � σ(h(t)) for all t in T. Let �′ � {σ(h): h ∈ �}. For
any subsets D′ and E′ of �′, let D and E be their inverse images under σ, where these
are subsets of �, and define Pr ′E ′(D

′) � Pr E (D). Then { Pr ′E ′ }E ′⊆�′ is a conditional
probability structure on �′. The system specified by �′ and { Pr ′E ′ }E ′⊆�′ is called a
factor system of the original system specified by � and {PrE}E⊆�. The function σ
from � into �′ is called a factor map. We can define factors of spatially extended and
amorphous systems analogously.

For a concrete example, suppose that (�, {PrE}E⊆�) is a classical-physics descrip-
tion of the entire solar system at an atomic level of detail. So X is an extremely
high-dimensional space, whichmust specifythepositionandmomentumof everyatom
intheentiresolarsystem,alongwithalloftheirgravitationalandelectromagneticinter-
actions.78 Meanwhile, let (�′, { Pr ′E ′ }E ′⊆�′) be the very simple celestial-mechanical
system consisting only of the Earth, Moon, and Sun, described as gravitationally inter-
acting point masses. So X′ � R18, because we must specify the three-dimensional
position and momentum vectors of each of the three objects in the system, and 3 × 6
� 18. Let σ be the function from X into X′ which translates each highly detailed atomic-
level description of the solar system into the crude 18-dimensional celestial mechanical
description. Then σ is a factor mapping from (�, {PrE}E⊆�) into (�′, { Pr ′E ′ }E ′⊆�′),
and thus (�′, { Pr ′E ′ }E ′⊆�′) is a factor of (�, {PrE}E⊆�).
78 For simplicity, we eschew a quantum–mechanical description in this example.

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Synthese (2021) 198:2551–2612 2601

As this example illustrates, a factor system can be seen as a sort of “abstraction”
or “simplification” of the original system, obtained by discarding some properties.
Now, suppose ψ is a function from T into itself (e.g., a time shift) which is a temporal
symmetry of the original system (�, { Pr E }E ⊆�). Then it is easy to verify that ψ
will also be a temporal symmetry of the factor system (�′, { Pr ′E ′ }E ′⊆�′). Thus, the
temporal symmetry monoid of the factor system (�′, { Pr ′E ′ }E ′⊆�′) is at least as large
as the temporal symmetry monoid of the original system (�, { Pr E }E ⊆�). In a spatially
extended system, the exact same statement applies to spatiotemporal symmetries.
Furthermore, if � is an amenable monoid of temporal (or spatiotemporal) symmetries,
and (�, {PrE}E⊆�) is ergodic relative to �, then (�′, { Pr ′E ′ }E ′⊆�′) will also be
ergodic relative to �. In other words, (�′, { Pr ′E ′ }E ′⊆�′) is at least as ergodic as
(�, { Pr E }E ⊆�).

This means that, even if the original system (�, { Pr E }E ⊆�) lacks certain symme-
tries or ergodic properties, the factor system (�′, { Pr ′E ′ }E ′⊆�′) may well possess these
properties. Furthermore, even if the original system (�, { Pr E }E ⊆�) is too complicated
to analyze using the formal tools we have described, the system (�′, { Pr ′E ′ }E ′⊆�′) may
well be simple enough. To illustrate this, consider our example of the solar system. The
original system (�, { Pr E }E ⊆�) describes the entire solar system at an atomic level of
detail. Whether or not the system possesses the desired symmetries or ergodic prop-
erties, it is certainly too complex to analyze. In contrast, the abstract Earth-Moon-Sun
system (�′, { Pr ′E ′ }E ′⊆�′) is very simple. In fact, it is an example of a quasiperiodic
dynamical system: it can be described as two independently rotating “wheels”, one
describing the orbit of the Moon around the Earth, and the other describing the orbit
of the Earth around the Sun. This is a prototypical example of an ergodic dynamical
system.

Appendix B: Partial symmetries and local symmetries

An important assumption of this paper has been that there is a fairly large monoid �
of symmetries acting on the set � of nomologically possible histories. We have argued
that a property of the system qualifies as a “law” only if it is invariant under all of
these symmetries. But this argument runs into a problem: many systems studied in the
sciences lack sufficient symmetries to account for all of their “law-like” features.

For example, suppose space is represented by the set of all integers, while time
is represented by the set of positive integers, i.e., S � {. . . , −1, 0, 1, 2, . . .} and
T � {1, 2, 3, . . .}, and consider the simple random-walk system described in Sect. 2.9.
Nomologically speaking, the token could begin at any spatial location at time one. But
suppose the conditional probability structure {PrE}E⊆� is such that, with probability
one, the token begins at spatial location zero at time one.79 In that case, the probability

79 The following argument does not depend on this assumption. Indeed, our argument would apply to
any initial probability distribution for the token. Note that there is no such thing as a uniform probability
distribution over the set of integers.

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2602 Synthese (2021) 198:2551–2612

distribution of its location at time t is a (t − 1, ½)-binomial distribution.80 Evidently,
this distribution is not invariant under spatial translations, since it is centred around
zero. Furthermore, it changes over time. Thus, spatiotemporal translations are not
symmetries of this system. But this contradicts our intuition that the motion of the
token is highly “law-like”: it can be described by a simple rule which is the same
everywhere in space and time.

To solve this problem, we now introduce the notion of “partial” symmetries. We use
the framework of spatially extended systems. Recall that H is the set all logically pos-
sible spatially extended histories. A partial symmetry monoid of a spatially extended
system is a collection � of transformations of H, along with a collection E of ordered
pairs of events (E, D), such that:

• ψ(�) � � for all ψ in �; and
• for any event pair (E′, D′) in E and any ψ in �, if E and D are the inverse images
(in �) of E′ and D′ under ψ, then (E, D) is also in E, and PrE(D) � PrE′(D′).81
For example, in our random-walk example (re-construed in the framework of

spatially extended systems), let E be the set of all ordered pairs of events (E, D)
such that event E exactly specifies the location of the particle at some time t, while
event D happens at some later time t′. Thus, PrE(D) is the conditional probability
that the token satisfies such-and-such property at time t′, given that it was at such-
and-such location at time t. Let � be the monoid of all spatiotemporal translations
of S × T. If ψ is any element of �, then the set E of pairs of events is invariant under
ψ, and the conditional probability PrE(D) is preserved by ψ for any (E, D) in E, in
the sense described above. Thus, the pair (�,E) is a partial symmetry monoid for the
random-walk system. Crucially, the set E does not include pairs of the form (�, D),
so we do not require unconditional probabilities of the form Pr�(D) to be preserved
by spatiotemporal translations.

Seen from this perspective, the transition probabilities of the random walk are “law-
like”, because they are preserved by all the transformations in �. In contrast, the initial
probability distribution of the system is merely a brute necessity of the present system,
since it is not preserved by any symmetries.

For another example suppose that the temporally evolving (or spatially extended)
system (�′, { Pr ′E ′ }E ′⊆�′) is a factor of the system (�, { Pr E }E ⊆�), via some factor
map σ, as described in “Appendix A”. For any event E′ ⊆ �′, let σ−1(E′) denote its
inverse image under σ (here defined as a subset of �). Then define E � {(σ−1(E′),
σ−1(D′)): E′, D′ ⊆ �′}. Let � be a monoid of spatiotemporal symmetries of the factor
system (�′, { Pr ′E ′ }E ′⊆�′). The elements of � might not be symmetries of the original
system (�, { Pr E }E ⊆�). However, they will be partial symmetries, with respect to
the set E. So (�, E) is a partial symmetry monoid for (�, { Pr E }E ⊆�). As already
explained in “Appendix A”, one can greatly extend the scope of our framework by
focusing attention on a factor system rather than the original system. We now see that
this is a special case of the broader concept of a partial symmetry monoid.

80 To be precise: if t is odd, and t′ � t − 1, then for any even s between − t′/2 and t′/2, the probability that
the token will be at spatial location s at time t is 2−t′ B((t′ + s)/2, t′), where B is the binomial coefficient
function. The formula for even times is similar, but more complicated.
81 Formally, E � ψ−1(E′) ∩ � and D � ψ−1(D′) ∩ �.

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Synthese (2021) 198:2551–2612 2603

However, partial symmetry monoids cannot accommodate another feature of many
systems. To illustrate this, consider a temporally evolving system where time is a finite
sequenceofintegers,suchas T � {1,2,…,100}.Forsuchasystem,timetranslationsare
not even well-defined.82 But in most such systems, we still want to say that the system
obeys the same causal laws at all times, except perhaps at times 0 and 100. A similar
problem arises in a spatially extended system where the space S is bounded (e.g., a
partial differential equation defined on a cube, with specified boundary conditions)
or a finite set of points (e.g., a cellular automaton defined on a 100 × 100 grid, with
specified boundary conditions). In such a system, spatial translations are not well-
defined. But in most such systems, we still want to say that the system obeys the same
causal laws everywhere in the “interior” of the spatial domain.

To solve this problem, we now introduce “local” symmetries. We begin with some
preliminary definitions. Let N be a subset of S × T; call this a neighbourhood of
space–time. Extending our earlier terminology, we say that an event E ⊆ � happens
inside N if, for all histories h and h′ in �, if h N � h′N , then hN is in E if and only if h′N
is in E. Let �N � {hN : h in �}; this is the set of all nomologically possible histories
restricted to N (a set of functions from N into X). Let N′ be another neighbourhood
of S × T, and suppose ψ is a function from N′ into N. We use this to construct a
function from histories restricted to N into histories restricted to N′. Specifically, for
any hN in �N , we define ψ(hN ) to be the function h

′
N′ from N

′ into X given by
h′N ′(n

′) � h N (ψ(n′)), for all n′ in N′. Note that while h′N ′ is a logically possible
history restricted to N′, it is not necessarily an element of �N′.

We now define a local symmetry groupoid of a spatially extended system to be a
combination of three components:

• a collection N of subsets of S × T (called neighbourhoods);
• for each neighbourhood N in N , a set EN of ordered pairs of events (E, D) which
happen inside N (called local events); and

• for each pair of neighbourhoods N and N′ in N , a collection �N,N′ of bijections
from N′ into N (called local symmetries).
We refer to the collection {�N,N′: N, N

′ ∈ N} as a groupoid because it must satisfy
two algebraic closure properties:

• for all neighbourhoods M and N in N , and any local symmetries ψ in �M,N , its
inverse ψ−1 is in �N,M ; and

• for all neighbourhoods L, N, and M in N , and all local symmetries α in �L,M and
β in �M,N , the composition α o β is in �L,N .

For any neighbourhoods N and N′ in N , and any ψ in �N,N′, we call ψ a local
symmetry because it must preserve the modal and probabilistic structure of the system
in the following sense:

• ψ(�N ) � �N′; and
• for all event pairs (E′, D′) in EN ′, if E and D are the inverse images (in �N ) of E′
and D′ under ψ, then (E, D) is in EN , and PrE(D) � PrE′(D′).83

82 For example, suppose we try to define ψ(t) � t + 1 for all t in T; then ψ(100) is not well-defined, because
101 is not an element of T.
83 Formally, E � ψ−1(E′) ∩ �N and D � ψ−1(D′) ∩ �N .

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2604 Synthese (2021) 198:2551–2612

For example, suppose that S � {1,2,…,10} and T � {1,2,…,100}. For any s in
{2,…,9} and t in {2,…,99}, let N s,t be the 3 × 2 “space–time rectangle” of the form
Ns,t � {s − 1, s, s +1} × {t , t +1}. Let N be the set of all such space–time rectangles.
For any s and s′ in {2,…,9}, and any t and t′ in {2,…,99}, if N � N s,t and N ′ � Ns′,t ′,
then we define �N ,N ′ � {ψs′,t ′→s,t }, where ψs′,t ′→s,t is the function from N′ into N
which sends each space–time point (s0, t0) in N

′ to the point (s0 −s′ +s, t0 −t ′ +t) in N .
Then, with a suitable specification of the local event sets EN for all N in N , we could
construct a local symmetry groupoid for many of the spatially extended systems (such
as cellular automata) that one might define on S × T. However, a fully worked out
example would be rather technically involved and is beyond the scope of this paper;
see Golubitsky et al. (2003) and Guay and Hepburn (2009).

Most of the ideas we have developed in this paper for the monoid of “full” symme-
tries can be generalized to partial symmetries and local symmetries. However, this is
also beyond the scope of this paper.

Appendix C: Inferential modesty, informational parsimony,
and the nomological hypothesis

Our version of Occam’s Razor requires us to assume that the symmetries of our system
are given by a maximal symmetry monoid consistent with our total nomological evi-
dence E (a superset of �). Under natural assumptions, at least one maximal consistent
monoid will indeed exist.84 However, there may be more than one. In this case, we
need a criterion to choose one maximal symmetry monoid rather than another. We
now develop such a criterion.

Let us begin with an example. Consider a very simple temporally evolving system,
where the set T of times contains only a single element. So, histories can be identified
with states at that single time; this expositional simplification has no substantive con-
sequences. Suppose that the state of the system is described by a two-dimensional grid
of zeros and ones, which is infinite in every direction. Let X be the set of all logically
possible grids of this kind. Then the set H of all logically possible histories can be
identified with X. In this system, one elementary kind of nomological constraint is
one that constrains the values of one or more cells, for example the constraint “in any
possible history, the cell (2, 3) must have the value zero”.85 Suppose we have obtained
evidence that any possible history must satisfy the constraints shown in Fig. 1. This
evidence would be represented by the subset E of H consisting of all single-period
histories in which the grid coincides with Fig. 1 in all non-empty cells. For the sake
of argument, let us treat E as our total nomological evidence about the system.

Now, for any integer n, let ψ→n be the transformation that shifts the entire grid to
the right by n spaces.86 Let �→ :� {. . . , ψ→−1, ψ→0 , ψ→1 , ψ→2 , ...} denote the monoid
84 For example, if (i) H has a topology, (ii) E is a compact subset of H, and (iii) all the transformations in
question are continuous, then Zorn’s lemma implies the existence of a maximal consistent monoid.
85 Formally, this constraint corresponds to the set E � {h ∈ H: h(2, 3) � 0}. Of course, we have chosen this
rather artificial example only for expositional simplicity. Typically, we would be interested not so much in
nomological constraints on single coordinates, but in constraints on the relationships between two or more
coordinates, such as the constraint “No two adjacent cells can both contain a zero”.
86 Of course, if n is negative, then ψ→n is actually a shift to the left.

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Synthese (2021) 198:2551–2612 2605

Fig. 1 The empirical evidence : : : : : :

... 1 ...

... ...

... 1 ...

... 1 ...

... 0 0 ...

... 1 ...

: : : : : :

Fig. 2 The consequences of
Hypothesis 1

: : : : : :

... 1 1 1 1 1 1 ...

... ...

... 1 1 1 1 1 1 ...

... 1 1 1 1 1 1 ...

... 0 0 0 0 0 0 ...

... 1 1 1 1 1 1 ...

: : : : : :

of all such horizontal shifts. Meanwhile, let ψ
↑
n be the transformation that shifts the

entire grid upwards by n spaces, and let �↑ :� {. . . , ψ↑−1, ψ↑0 , ψ↑1 , ψ↑2 , ...} denote
the monoid of all such vertical shifts. Consider two hypotheses:

Hypothesis 1: All transformations in �→ are symmetries of the system.
Hypothesis 2: All transformations in �↑ are symmetries of the system.

The evidence represented in Fig. 1 is consistent with either of these hypotheses.
However, it cannot accommodate both simultaneously. If Hypothesis 1 were true, then
the constraints in Fig. 1 would entail the constraints shown in Fig. 2. If Hypothesis
2 were true, then they would entail the constraints shown in Fig. 3. In each figure,
the constraints that were part of the initial nomological evidence are highlighted in
boldface; extrapolated constraints (based on the postulated symmetries) appear in non-
bold font. Clearly, Hypotheses 1 and 2 cannot both be true, since they yield mutually
contradictory constraints on the values of the grey cells.

Let � be some maximal consistent monoid of transformations that we postulate as
the symmetry monoid, in accordance with Occam’s Razor. Hypothesis 1 then asserts

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2606 Synthese (2021) 198:2551–2612

Fig. 3 The consequences of
Hypothesis 2

: : : : : :

... 1 0 0 ...

... 1 0 0 ...

... 1 0 0 ...

... 1 0 0 ...

... 1 0 0 ...

... 1 0 0 ...

: : : : : :

Fig. 4 Weaker evidence to obtain
Fig. 2 from Hypothesis 1

: : : : : :

... 1 ...

... ...

... 1 ...

... 1 ...

... 0 ...

... 1 ...

: : : : : :

that �→ ⊆ �, while Hypothesis 2 asserts that �↑ ⊆ �. Since both hypotheses cannot
simultaneously be true, it follows that there are at least two distinct ways in which we
could specify �: one including �→ and another including �↑. So even in this very
simple example, there is no unique maximal consistent monoid.

At first sight, the choice between these two maximal symmetry monoids seems
arbitrary. But it is not. To see this, note that both hypotheses could have entailed the
same constraints they did, using less initial evidence. For example, Hypothesis 1 would
have entailed the same constraints from the weaker evidence represented in Fig. 4.

The original evidence in Fig. 1 constrained six cell values (i.e., six “bits” of infor-
mation). But Hypothesis 1 can make do with only five of them (in particular, the second
zero is redundant). Meanwhile, Hypothesis 2 would have entailed the same constraints
from only three bits of information, as represented in Fig. 5.

In other words, Hypothesis 2 could have entailed all of its original constraints,
using less information than Hypotheses 1 needed to obtain its original constraints. Thus
Hypothesis 2 can be viewed as more informationally parsimonious than Hypothesis 1.
Hypothesis 2 stands out in another way too: from the same initial evidence, it constrains

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Synthese (2021) 198:2551–2612 2607

Fig. 5 Weaker evidence to obtain
Fig. 3 from Hypothesis 2

: : : : : :

... ...

... ...

... ...

... ...

... 0 0 ...

... 1 ...

: : : : : :

fewer cell values than Hypothesis 1. So, Hypothesis 2 is also more inferentially modest
than Hypothesis 1.

This simple example illustrates two general points. First, different symmetry
monoids may lead to different nomological hypotheses—hypotheses about what the
nomologically possible histories are—even starting from the same body of nomologi-
cal evidence. Formally, we may have �(�1, E) �� �(�2, E) where �1 and �2 are two
distinct symmetry monoids that are each consistent with E. Second, one symmetry
monoid could generate the same nomological hypothesis from two different bodies
of nomological evidence. Formally, we may have �(�, E1) � �(�, E2) for the same
symmetry monoid � and two distinct sets E1 and E2.

Thus, given two symmetry monoids �1 and �2, which are each compatible with
the same total nomological evidence E, we can compare them along two dimensions:

Inferential modesty: If �(�2, E) ⊆ �(�1, E), then we say that �1 is (at least
weakly) more inferentially modest than �2.
Informational parsimony: Let E1 be the largest superset87 of E such that
�(�1, E1) � �(�1, E). Let E2 be the largest superset of E such that �(�2, E2)
� �(�2, E). If E2 ⊆ E1, then we say that �1 is (at least weakly) more informa-
tionally parsimonious than �2.

Returning to our earlier example with the infinite grid, let E be the nomological
evidence described by Fig. 1. Then �(�→, E) is the set of single-period histories
satisfying the constraints described by Fig. 2, and �(�↑, E) is the corresponding set
for Fig. 3. Meanwhile, if E→ is the nomological evidence described by Fig. 4, then
we have �(�→, E →) � �(�→, E). Likewise, if E↑ is the nomological evidence
described by Fig. 5, then we have �(�↑, E ↑) � �(�↑, E).

In this example, neither �(�→, E) nor �(�↑, E) includes the other, so neither
monoid is more inferentially modest than the other, according to our definition. Like-
wise, neither E→ nor E↑ includes the other, so neither monoid is more informationally

87 Recall that larger subsets of H encode less information. In particular, if E1 is a superset of E, then E1
encodes less information than E.

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2608 Synthese (2021) 198:2551–2612

parsimonious. So our formal definitions up to this point are not sensitive enough to
capture the plausible intuition that �↑ is both more inferentially modest and more
informationally parsimonious than �→.

One possible way of capturing this intuition is to use concepts from information
theory, such as entropy. To do this, we must introduce a prior probability distribution
Pr0 on the set H of all logically possible histories. In the example with the infinite
grid, this could be the uniform Bernoulli distribution, which treats all the cells in the
grid as independent, identically distributed random variables, where zero and one each
appear with probability ½. Given two different symmetry monoids �1 and �2 that are
consistent with the same nomological evidence E, we can use Pr0 to compare them:

Inferential modesty (relative to Pr0): If Pr0(�(�2, E)) ≤ Pr0(�(�1, E)), then
we say that �1 is (at least weakly) more inferentially modest than �2, relative
to Pr0.
Informational parsimony (relative to Pr0): Let E1 be the largest superset of
E such that �(�1, E1) � �(�1, E). Let E2 be the largest superset of E such that
�(�2, E2) � �(�2, E). If Pr0(E2) ≤ Pr0(E1), then we say that �1 is (at least
weakly) more informationally parsimonious than �2, relative to Pr0.

Do these criteria enable us to prefer �↑ to �→, as intuition suggests? Let us begin
with the second criterion. Comparing Figs. 4 and 5, we see that Pr0(E

→) � 2−5,
whereas Pr0(E

↑) � 2−3, and so �↑ is indeed more informationally parsimonious than
�→, relative to the uniform Bernoulli distribution. The first criterion, by contrast, does
not help. Comparing Figs. 2 and 3, we see that Pr0(�(�

→, E)) and Pr0(�(�↑, E)) are
each zero, because they constrain an infinite number of cells. So, they do not differ
in inferential modesty relative to Pr0. They do differ in more sensitive measures of
inferential modesty, computed using more advanced notions from information theory,
such as “entropy density”. But the details are beyond the scope of this paper.

Note that, if �1 is more inferentially modest than �2 in the original sense, which
did not refer to any prior probability, then �1 is more inferentially modest than
�2 in the information-theoretic sense, relative to any prior Pr0. This is because if
�(�2, E) ⊆ �(�1, E), then Pr0(�(�2, E)) ≤ Pr0(�(�1, E)). Likewise, if �1 is more
informationally parsimonious than �2 in the original sense, then �1 is more informa-
tionally parsimonious than �2 in the new sense, relative to any prior Pr0. The reason
is that if E2 ⊆ E1, then Pr0(E2) ≤ Pr0(E1).

Appendix D: Spatial distance in quantum mechanical systems

In Sect. 3.9, we proposed a definition of the distance between two regions R and R′
in space, based on the minimum time required for a “signal” to travel from R to R′.
As we have already observed, this definition is not entirely satisfactory in systems
where signals can travel at arbitrarily high speeds (such as classical mechanics). This
is particularly problematic in quantum mechanics, for two reasons.

First, there is the well-known phenomenon of entanglement, where two particles,
perhaps separated by a large spatial distance, can apparently correlate their behaviour.
But, in fact, this is less of a problem than it first appears. Rather than interpreting

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Synthese (2021) 198:2551–2612 2609

entanglement as “spooky action at a distance”, we can interpret it as a sign that we
have not correctly specified the space S for this spatially extended system. A three-
dimensional quantum system with n particles is not a collection of n wave functions
on a three-dimensional space; rather, it should be viewed as a single wave function on
a 3n-dimensional space. So, we should define S � R3n. Even if two particles appear
widely “separated” from our three-dimensional perspective, their joint location is
described by a single “hump” of the wave function in a six-dimensional space.88

From this perspective, the entangled behaviour of the two particles does not appear as
a non-local phenomenon.

However, there is a more fundamental problem, which affects even a single-particle
quantum system. Solutions to the Schrödinger equation on unbounded domains gen-
erally have full support: they give non-zero probability to every part of the space. This
means, in effect, that the particle has a non-zero probability (albeit tiny) of “jumping”
arbitrarily large distances through space.89 Thus, no two regions of space are ever
unreachable from one another in any time duration, no matter how short, and so the
distance between any two regions will be zero, according to the definition given in
Sect. 3.9.

To address this problem, we must introduce a slightly more nuanced version of
“unreachability”. Let ε >0 be some small “error tolerance”. Given three events E,
F, and G in �, we say that E and G are ε-conditionally independent given F, if
1 − ε < PrF (E ∩ G)/ PrF (E) · PrF (G) < 1 + ε. In other words, the conditional
probability PrF(E ∩ G) is “almost” the same as the product PrF(E)·PrF(G), which
means that E and G are “almost” conditionally independent, given F. If R and R′ are
two regions of S, and t > t0, then we say that R

′ is ε-unreachable from R in time t if,
for any h in �, any event which happens in R′ at time t is ε-conditionally independent
of any event which happens in R at time t0, given the event [h RC ×{t0}]. If ε is small,
this means that, with very high probability, a signal which originates in R at time t0
cannot reach R′ before time t. We then define the ε-distance between R and R′ to be
the supremum of the set of all t such that R′ is ε-unreachable from R in time t (if this
supremum exists).

By using a small but non-zero ε, we can thus define a non-trivial notion of ε-
distance between different regions of space, even in a quantum–mechanical system.
This measure of distance will obviously depend on the value of ε, but it will roughly
approximate the “classical” notion of distance. However, a detailed development of
this approach is beyond the scope of this paper.

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	Dynamic and stochastic systems as a framework for metaphysics and the philosophy of science
	Abstract
	1 Introduction
	2 Temporally evolving systems
	2.1 Basic definitions
	2.2 Determinism and indeterminism
	2.3 Nomological possibility and necessity
	2.4 Modal and probabilistic properties
	2.5 Symmetries
	2.6 Laws and their significance
	2.7 Ergodicity and its significance
	2.8 Occam’s Razor
	2.9 The role of time

	3 Spatially extended systems
	3.1 Basic definitions
	3.2 Determinism and indeterminism
	3.3 Nomological possibility and necessity
	3.4 Modal and probabilistic properties
	3.5 Symmetries
	3.6 Laws and their significance
	3.7 Spatiotemporal ergodicity and its significance
	3.8 The role of space
	3.9 Duration and distance

	4 Amorphous systems: space–time as an emergent property
	4.1 Basic definitions
	4.2 Adjacency structure and the Markov property
	4.3 Time and predictability
	4.4 Which features of a system are real?

	5 Concluding discussion
	5.1 Higher-level versus lower-level properties
	5.2 Laws and regularities in the special sciences
	5.3 Intentional systems

	Acknowledgements
	Appendix A: Factor systems
	Appendix B: Partial symmetries and local symmetries
	Appendix C: Inferential modesty, informational parsimony, and the nomological hypothesis
	Appendix D: Spatial distance in quantum mechanical systems
	References