# A Proposal for a Metaphysics of Self-Subsisting Structures.
I. Classical Physics

Antonio Vassallo<EMAIL_ADDRESS>Faculty of Administration and
Social Sciences, Warsaw University of Technology, Plac Politechniki 1, 00-661
Warsaw, Poland  Pedro Naranjo<EMAIL_ADDRESS>Faculty of Administration
and Social Sciences, Warsaw University of Technology, Plac Politechniki 1,
00-661 Warsaw, Poland  Faculty of Philosophy, University of Warsaw, Krakowskie
Przedmieście 3, 00-047 Warsaw, Poland  Tim Koslowski<EMAIL_ADDRESS>University of Würzburg, Theoretical Physics II, Campus Hubland Nord, Emil-
Hilb-Weg 22, 97074 Würzburg, Germany

###### Abstract

We present a new metaphysical framework for physics that is conceptually
clear, ontologically parsimonious, and empirically adequate. This framework
relies on the notion of _self-subsisting structure_ , that is, a set of
fundamental physical elements whose individuation and behavior are described
in purely relational terms, without any need for a background spacetime.
Although the specification of the fundamental elements of the ontology depends
on the particular physical domain considered—and is thus susceptible to
scientific progress—, the empirically successful structural features of the
framework are preserved through theory change. The kinematics and dynamics of
these self-subsisting structures are technically implemented using the
theoretical framework of _Pure Shape Dynamics_ , which provides a completely
relational physical description of a system in terms of the intrinsic geometry
of a suitably defined space called _shape space_.

Keywords: Self-subsisting structure; relationalism; ontic structural realism;
Humeanism; pure shape dynamics; parsimony

Published in _Synthese_ 200, 374 (2022).
https://doi.org/10.1007/s11229-022-03865-x

###### Contents

  1. 1 Introduction
  2. 2 Pure Shape Dynamics
    1. 2.1 Motivation
    2. 2.2 Comparison with Shape Dynamics
    3. 2.3 Dynamics
    4. 2.4 Examples
      1. 2.4.1 Geodesic System
      2. 2.4.2 Newtonian $E=0\,\,\,N$-body System
      3. 2.4.3 Newtonian $E>0\,\,\,N$-body System
      4. 2.4.4 $E=0$ Bianchi IX cosmological model
  3. 3 From Shapes to Self-Subsisting Structures
  4. 4 From Self-Subsisting Structures to Spacetime
    1. 4.1 A Supervenience Basis for Laws and Dynamical Structure
    2. 4.2 A Supervenience Basis for Space and Time
  5. 5 Conclusion: Pure Shape Dynamics in Perspective

##  1 Introduction

In this paper, we take the word “physics” very seriously. In fact, we take it
literally as “(the study of) natural things.” This is because we believe that
physics goes hand in hand with a simple thesis, namely, that there are mind-
independent “things” making up the natural world, which we get to know by
taking scientific claims at face value.111It may be argued that such a simple
thesis is tripartite since it involves metaphysical, epistemic, and semantic
connotations. See for example Chakravartty (2017, §1.2), for a discussion of
this threefold nature of scientific realism.

This realist understanding of physics—and science in general—is often
criticized on historical grounds: Past theories were considered successful
until eventually proven wrong, and we have no reason to exclude that the same
fate awaits currently well-established theories. Hence, how can we take _any_
theory at face value? Here we do not want to enter this debate (see Psillos,
2018, for an excellent overview of the subject), and we will endorse the
standard structuralist response to this challenge firstly put forward in its
modern form by Worrall (1989). In a nutshell, structuralists accept that there
is a change in the theoretical entities postulated by physical theories as
physics develops (e.g., string theory is quite obviously not fundamentally
about Newtonian material particles). However, they maintain that the way such
entities are related is preserved through theory change, so past theories were
successful insofar as they captured these structural aspects of the world.

At first sight, structuralism might be taken as _just_ accepting that true
scientific claims provide us with knowledge about the structural aspects of
the world. In this sense, structuralism would represent a merely
epistemological refinement of scientific realism. A historically remarkable
example of this epistemic attitude can be found in Henri Poincaré, who
famously claimed that the empirical success of a physical theory relies on its
capability to provide an accurate description of the relational aspects of
reality. For him, it is these relations that we can epistemically access,
whereas the true nature of the entities bearing these relations to each other
we will never get to know (Poincaré, 1900, p. 15).

However, structural realism may also be taken as a metaphysical thesis: The
ontology of the natural world _consists_ of physical structures (see Ladyman,
1998, for one of the first papers acknowledging the difference between
_epistemic_ and _ontic_ structural realism). There are many ways to shape this
ontic structural realist thesis. One of them may, for example, be to take
Poincaré’s intuition to its extreme ontological consequences. This is done by
denying that theoretical entities directly refer to anything like individual
entities: There are only relations “all the way down” (see French, 2010, for a
presentation of this radical ontic thesis). There is also a more moderate
ontic take on physical structures. According to moderate ontic structural
realism, individual physical objects are to be taken ontologically on a par
with the relations they instantiate, which means that neither of the two is
reducible to the other (see Esfeld and Lam, 2011, for an articulation of this
position). In the following, we will focus on structural realism as a
metaphysical position instead of a purely epistemic one.

The discussion regarding structural realism in physics is usually carried out
in the context of “standard” physical theories (classical mechanics, general
relativity, and even quantum theory). However, to our knowledge, not much
emphasis has been placed so far on putting forward a theoretical framework
that patently encodes the core tenet of structuralism, i.e., the idea that a
physical theory has to be taken seriously only insofar as its relational
aspects are concerned. The aim of this paper is precisely to discuss a general
theoretical framework that can encompass the empirically adequate structural
aspects of classical and quantum physics and which may serve as a template to
implement a quantum gravity theory. However, in order to keep the discussion
at a reasonable length, the analysis will focus on classical (i.e., non-
quantum) physics, leaving the discussion of the quantum case to a companion
paper.

The first step in characterizing this theoretical framework is to recognize
that this structuralist reading naturally fits in with a relationalist take on
dynamics in the context of theories of motion: Simply speaking, theories like
classical mechanics are not about individual objects located in a background
spacetime but, instead, about a web of spatiotemporal relations instantiated
by said objects. Recently, some extensive attempts have been made to recast
physics in Leibnizian/Machian relational terms according to the program set
out in Barbour and Bertotti (1982). This brand of relationalism accords a
privileged status to spatial relations and seeks to reduce time to a mere
change in these spatial relations. This approach has the advantage of being
ontologically parsimonious while preserving empirical adequacy (with this
respect, see the discussion in Vassallo et al., 2017; Vassallo and Esfeld,
2016). The present paper will show how these Leibnizian/Machian ideas are
essential in setting up a unified framework for physics, which fully captures
a structuralist understanding of theory change.

The plan of the paper is the following. In §2, we will introduce the technical
backbone of the theoretical framework, including its main principles and some
concrete applications. In §3, we will elaborate on the kind of metaphysics
that goes along with the formalism. We will show how the best understanding of
the framework is given in terms of _self-subsisting_ structures, which are
configurations of fundamental objects individuated by a set of fundamental,
world-building relations. We will also point out that, even if the concrete
characterization of these fundamental objects and relations depends on the
particular physical domain considered, the “gist” of these structures is
preserved throughout these domains (and, hence, through standard theory
change), in line with scientific structuralism. §4 will be devoted to
providing a metaphysical story about how space, time, and other salient
dynamical features of the standard physical description of the world can be
recovered from an ontology of self-subsisting structures. Finally, in §5, we
will put forward some preliminary reflections on how the framework presented
may play out in the context of the quantum gravity program.

##  2 Pure Shape Dynamics

###  2.1 Motivation

One of the primary motivations for pursuing a relational strategy in
constructing physical theories is epistemic and amounts to the realization
that every measurement of a physical magnitude boils down to the comparison of
said magnitude with a chosen standard unit. Hence, only ratios of physical
quantities carry objective information.222This comparativist attitude with
respect to physical magnitudes is not immune to philosophical controversy (see
Dasgupta, 2013, 2020; Baker, 2020, for a recent example of the debate). Here
we will take comparativism for granted, leaving a philosophical defense of our
choice to future work. This means that it makes no sense to speak of values—or
change thereof—of a physical magnitude _simpliciter_ : This information
becomes physically meaningful—indeed, empirically _observable_ —only in
relation to something else. It does not take a giant conceptual leap to
elevate such an epistemic consideration to a metaphysical guideline: There are
no “absolute” physical magnitudes and, therefore, any empirically adequate
ontology of the physical world should avoid elements whose variation makes no
observable physical difference. Historically, the main targets of the
relationalist despise for unobservable entities are absolute spatial and
temporal structures: Think for example of Leibniz’ famous arguments against
Newtonian space and time (see, e.g., McDonough, 2019, §5.1).

The modern relationalist way endorsed in this paper to formally implement the
rejection of an external space in which material bodies are placed can be
roughly summarized as follows (cf. Barbour, 2012, §2.1): Take a spatial
configuration of material bodies and “quotient out” all the degrees of freedom
associated with its being embedded in such an external space. In the case of
an external $3$-dimensional Euclidean space $\mathbb{R}^{3}$, this amounts to
identifying all the configurations related by transformations that belong to
the groups of rigid translations $\mathsf{T}$, rotations $\mathsf{R}$ and
dilatations $\mathsf{S}$, which jointly define the so-called _similarity
group_. Thus, if $Q^{N}$ is the configuration space associated with the
geometrical configuration of a $N$-body system in $\mathbb{R}^{3}$, with each
of the bodies placed at one of the vertices of the associated $N$-gon, the
only objective spatial information encoded in the configuration is captured by
the _shape_ of the $N$-gon or, more precisely, by its conformal structure.

Mathematically, this means that the relevant configuration space is
$Q^{N}_{\mathrm{ss}}:=Q^{N}/\mathsf{T}\mathsf{R}\mathsf{S}$, which is known as
_shape space_. This quotienting out procedure can be extended to more general
cases—including dynamical geometry—provided that there be a suitable sense in
which the redundant structure to be washed away has an associated symmetry
group.333This may not be the case for structures that represent conditions
fixed a priori in the theory such as, for example, having the structure of a
Hausdorff space. We will gloss over these cases for simplicity’s sake, given
that they have no major impact on our analysis. All shape spaces are instances
of stratified manifolds (on this score see, e.g., Anderson, 2016; see also
Kendall et al., 1999, chapters 6 and 7, for a thorough technical discussion of
shape spaces). Throughout the discussion, we will denote shape spaces in
general by $\mathfrak{S}$. It is important to reiterate that the conformal
structure left after the quotienting out is performed represents _all_ the
information needed to characterize a relational configuration, which means
that facts about the identity of the bodies making up an $N$-gon are
unimportant with this respect. In order to implement this fact formally, we
should perform the quotienting out procedure also with respect to the
permutation group.444To make a concrete case, the permutation invariant
version of the configuration space of $N$ point-particles in Euclidean
$3$-space—$\mathbb{R}^{3N}$—is the space of all $N$-element subsets of
$\mathbb{R}^{3}$—which is symbolized by ${}^{N}\mathbb{R}^{3}$. It is easy to
see that all possible permutations of particles in a fixed configuration
represent the same $N$-element subsets of $\mathbb{R}^{3}$. Hence, reducing
$\mathbb{R}^{3N}$ to ${}^{N}\mathbb{R}^{3}$ accounts for the fact that there
is no label attached to the particles in a configuration so that they can be
swapped without changing the physical information encoded in the
configuration. Historically, the search for a full implementation of this
Leibnizian take on space led to _Shape Dynamics_ (SD; see Barbour, 2012 for a
nice introduction to the subject, with an emphasis on conceptual matters, and
Mercati, 2018, for a pedagogical, yet comprehensive, account). In the context
of SD, only the ratios of distances between bodies carry physical meaning,
which is achieved exactly by demanding that angles be preserved under the
quotienting out procedure.555We acknowledge the debate, especially in the
theoretical physics community, over the physical significance that _absolute_
scale or size possess. In particular, a different implementation than ours of
relational ideas in physics is exhibited in general relativity, which does
_not_ perform the quotient by scaling transformations (see Barbour, 1994a, b,
for thorough discussions, and Barbour, 2003; Anderson et al., 2003, for the
first models exhibiting scale-invariance in particle dynamics and dynamical
geometry, respectively).

In order to fully specify a dynamical description of the system, it is
important to implement the temporal side of relationalism. A simple
articulation of this approach can be found in the writings of Ernst Mach
(Mach, 1883). According to Mach, time is not a physical structure whose
existence is independent of the material happenings that make up the physical
world (think of the Newtonian universal clock that ticks without relation to
anything else in the universe); rather, it is the result of an act of
abstraction over the ordered change inherent into said material happenings.
Hence, any adequate relational theory of dynamics must dispense with the
absolute and external metric structure that usually models Newtonian time.
Consequently, it is not the case that relational dynamics unfolds in a space
which is the Cartesian product of $\mathfrak{S}$ with the real line: In that
case, the ordered change in the shape of a system as encoded in its associated
curve $\gamma$ in shape space would be labeled by a unique and global
parameter $t\in\mathbb{R}$. The standard formulation of SD put forward in
Barbour et al. (2013) replaces such privileged global parameter
$t\in\mathbb{R}$ with an arbitrary dimensionless local parameter
$t/t_{0}$—$t_{0}$ being some arbitrary initial time value. In this way, SD
exhibits an evolution of the initial data that does not depend on the
particular choice of time units (in §2.2 we shall give a detailed analysis of
how to get rid of external temporal structures in dynamical geometry as well).
However, once such a choice is made, the units of time remain fixed throughout
the evolution, thus introducing a _preferred_ parametrization of the dynamical
system in terms of the dimensionless parameter $t/t_{0}$. Hence, it seems that
a reference structure is introduced in the dynamical description of the
system, which is independent of its relational details. This goes against
Mach’s intuition that temporal facts should completely depend on facts about
intrinsic changes in the system. Indeed, a truly relational theory requires
that _nothing_ over and above the structure of shape space be needed to
account for the evolution of physical systems.

Therefore, in order to make a step forward in the construction of a fully
relational framework, it is crucial to get rid of the reference structures
that appear in the definition of units of (local) increments of time. This can
be done by banishing any concept whatsoever of parametrization of the
dynamical curve $\gamma$, hence taking it as an _unparametrized_ curve
$\gamma_{0}$.666An unparametrized curve $\gamma_{0}$ is an equivalence class
$[\gamma]$ of paths $\gamma_{i}$ in the following sense: two paths
$\gamma_{1}:\ell_{1}\rightarrow X,\,\gamma_{2}:\ell_{2}\rightarrow X$ are
equivalent if there is a non-decreasing, continuous map
$\phi:\ell_{1}\rightarrow\ell_{2}$ such that $\gamma_{1}=\gamma_{2}\circ\phi$.
Here, $\ell_{1},\ell_{2}$ are the domains of the paths (usually,
$\ell_{i}\in\mathbb{R}$), $X$ is a set or some structure thereof, as a
manifold (again, usually, $X\simeq\mathbb{R}^{n}$). This is the essential idea
underlying _Pure Shape Dynamics_ (PSD), which can be considered a natural
evolution of SD (see Koslowski et al., 2021, for a thorough technical
introduction to this new relational framework).

The major novelty of the relationalist picture advocated by PSD is exactly its
insistence on using only _intrinsic_ geometric properties of $\gamma_{0}$ in
$\mathfrak{S}$ in the description of the evolution of a given physical system,
which is expressed in terms of the _equation of state_ of $\gamma_{0}$: A
point $q^{a}\in\gamma_{0}$ corresponds to the full configuration of the
system, which by construction is its shape _qua_ objective data, to which it
is added the set {$\alpha_{I}^{a}$} of any intrinsic geometric properties of
$\gamma_{0}$ needed to fully specify the evolution. The mathematical structure
underlying this manifestly intrinsic nature of the description is the
directional action of a local section $A(q^{a},\alpha_{I}^{a})$ in a suitable
unit tangent bundle over $\mathfrak{S}$:
$A(q^{a},\alpha_{I}^{a})=\frac{dq/ds}{d\alpha_{I}^{a}/ds}=\frac{dq}{d\alpha_{I}^{a}}$,
i.e., the equation of state of $\gamma_{0}$ expresses the ratio of change of
its intrinsic geometric degrees of freedom (see, again, Koslowski et al., 2021
for the technical details).

Thus, the fundamental structure of $\mathfrak{S}$ describing the equation of
state of $\gamma_{0}$—and, hence, the evolution of a physical system—is
largely topological, which in turn guarantees the parametrization-
irrelevant777The notion of parametrization irrelevance (i.e., no reliance on
any parametrization whatsoever) was first introduced in Anderson (2013,
§1.4.1) to make a distinction with parametrization invariance (i.e., no
reliance on a fixed parameter). nature of PSD even if some parametrization $s$
is used in computations. However, some minimal geometrical structure is
certainly needed to describe the curve, namely a metric on shape space
$g_{ab}$ (cf. the definitions of the unit tangent vector, (2), and of the
curvature degree of freedom $\kappa$ introduced after (14)). Two facts are
worth stressing, at this point: (i) A metric on shape space does _not_ measure
size, but the amount of similarity in the configurations, and (ii) there
exists a naturally induced metric on shape space, the _kinematic metric_ ,
which is the metric entering the definition of the kinetic energy associated
with a given physical system. We shall consider the explicit case of Euclidean
geometry in §2.4.2; in the case of dynamical geometry, the kinematic metric is
the DeWitt supermetric (cf. Barbour, 2012, in particular, footnote 11).

Let us formally state the ideas above as the fundamental principle underlying
PSD:

######  Definition 2.1.

The evolution of any physical system is uniquely determined by the equation of
state of the intrinsic geometric properties {$q^{a},\alpha_{I}^{a}$} of the
_pure_ , i.e., unparametrized curve $\gamma_{0}$ in shape space
$\mathfrak{S}$. This equation of state determines the ratios of change of
{$q^{a},\alpha_{I}^{a}$}.

Thus, given a physical system, we shall express the equation of state of the
unparametrized curve in its associated shape space as888(1) is the
“components” version of the equation of state, which is more convenient to
work with than its directional action counterpart.

$\begin{array}[]{rcl}dq^{a}&=&u^{a}(q^{a},\alpha_{I}^{a}),\\\
d\alpha_{I}^{a}&=&\Omega_{I}^{a}(q^{a},\alpha_{I}^{a})\,,\end{array}$ (1)

and demand that the right-hand side be described in terms of dimensionless and
scale-invariant quantities, whose intrinsic change is obtained employing
Hamilton’s equations of motion. For consistency, the elements in
$\alpha_{I}^{a}$ in (1) must exhaust the set of all possible dimensionless and
scale-invariant quantities that can be formed out of the different parameters
entering a given theory. Note also that dynamical parameters, such as mass,
charge, and spin, enter the equation of state through _scale-invariant_ and
_dimensionless_ quantities, as shall be illustrated in §2.3 and §2.4. Hence,
the conformal structure alone does not individuate any dimensionful quantity.
Thus, it may be argued that dimensionful dynamical parameters and physical
units are not needed by the theory and serve only as extra descriptive
information about the physical system.

In (1), $u^{a}$ is the unit tangent vector defined by the shape momenta
$p_{a}$:

$u^{a}\equiv g^{ab}(q)\frac{p_{b}}{\sqrt{g^{cd}p_{c}p_{d}}}\,,$ (2)

which allows us to define the direction $\phi^{A}$999A direction is defined
through $n-1$ components in an $n$-dimensional manifold, hence the difference
in the label. at $q^{a}$. It is through the unit tangent vector and the
associated direction that the shape momenta enter Hamilton’s equations, which
are in turn used in the intermediary steps leading to the equation of state
(1).

###  2.2 Comparison with Shape Dynamics

At this point, it is important to spell out in more detail the difference
between PSD and SD. As we have already hinted, such a difference is crucial to
achieving a better relational dynamics.

In a nutshell, the essential difference amounts to the fact that SD relies on
the notion of parametrization of a curve to make sense of the dynamics,
whereas PSD’s fundamental physical description is given in unparametrized
terms. This, of course, does not mean that dynamical curves in PSD cannot be
described using a suitable parametrization, but bringing in such a piece of
formalism is by no means necessary and just amounts to adopting a notational
shortcut to simplify the calculations. In order to understand why the same
cannot be said of SD, we should take a closer look at how the notion of
parametrization has developed in the literature on SD.

The formulation of SD laid down in Barbour et al. (2014b) is motivated by the
desire to avoid the “many-fingered” time featuring General Relativity (GR),
i.e., to identify a _global_ variable that could serve as _physical_ time and,
hence, give rise to physical evolution. This is achieved by considering a
time-dependent Hamiltonian, which naturally induces a curve parametrization in
terms of a genuinely _physical_ temporal parameter related to York time.
Although the “many-fingered” time does not arise in particle models, it is
nonetheless illuminating to run the argument keeping both particles and
dynamical geometry, in order to stress their remarkable structural properties.

First, consider the following pairs of conjugate variables: (i) {$\log R,D$}
in the particle model, with $R=\sqrt{I_{\mathrm{cm}}}$ (where
$I_{\mathrm{cm}}$ is the center-of-mass moment of inertia) and
$D=\sum_{1}^{N}\mathbf{r}_{a}^{\mathrm{cm}}\,\cdot\mathbf{p}^{a}_{\mathrm{cm}}$
is the dilatational momentum (where $\mathbf{r}_{a}^{\mathrm{cm}}$ and
$\mathbf{p}^{a}_{\mathrm{cm}}$ are, respectively, the position and momentum of
particle $a$ relative to the center of mass), and (ii) {$V,Y$} in dynamical
geometry, where $V$ is the spatial volume and $Y$ is related to York time. As
detailed in Barbour et al. (2014b), a deparametrization procedure with respect
to $D$ (respectively, $Y$) allows us to transform $\log R$ (resp., $V$) into a
physical Hamiltonian and $D$ (resp., $Y$) into a monotonic _physical_ time
variable. Importantly, $D$ (resp., $Y$) is an _independent_ variable, on top
of the shape degrees of freedom.

Next, Barbour et al. (2013, 2014a) aim to eliminate these independent physical
time variables from the theory: This is obtained through the introduction of a
logarithmic time $\tau=\ln(|D/D_{0}|)$ (resp., $\lambda=\ln(|Y/Y_{0}|)$),
which renders the Hamiltonian dimensionless and time-independent.

Crucially, although the time dependence has been eliminated through the
logarithmic time, the fact remains that _the parameter $\tau$ (resp.,
$\lambda$) is a necessary initial datum in the standard formulation of SD._
Hence, although there remains a freedom in choosing the origin of $\tau$
(resp., $\lambda$) by selecting a different value of $D_{0}$ (resp., $Y_{0}$),
for two points $x,y$ in shape space the curves that pass through them with
different values of $\Delta\tau:=\tau(y)-\tau(x)$ (resp., $\Delta\lambda$) do
not coincide away from these two points. Therefore, the difference
$\Delta\tau$ (resp., $\Delta\lambda$) is not just “descriptive fluff”: It
refers to a salient physical feature of the dynamics and, as such, should be
taken ontologically seriously. In the end, it seems that SD is committed to
the existence of external (i.e., non-shape) features of the physical world.
This is what motivates the PSD program. As already emphasized, PSD’s goal is
to eschew _all_ external structures from the curve in shape space, whereby the
dynamics is expressed in terms of the intrinsic, geometric properties of said
curve—that is, solely in terms of shape degrees of freedom
{$q^{a},\alpha_{I}^{a}$} (as per principle 2.1, formalized by (1)). An
_independent_ variable, however physical, is no exception. Thus, we demand
that the curve in shape space be unparametrized.

The core difference between SD and PSD is best conveyed by considering how
these two theories mathematically implement the main relational tenets.
Standard SD satisfies the so-called _Mach-Poincaré principle_ , whereby a
point in shape space and a tangent vector to it suffice to uniquely generate a
$\tau$\- (resp., $\lambda$-) parametrized curve in shape space. PSD, on the
other hand, satisfies a modified version of this principle (enunciated in
definition 2.1): A point and whatever number of _higher_ -order derivatives of
the curve are needed to uniquely determine the curve in shape space. These
higher-order derivatives are the elements of the set {$\alpha_{I}^{a}$}.

To make apparent how PSD improves upon SD, it is perhaps useful to distinguish
two forms of relationalism: On the one hand, one can define that any ratio of
sizes, even if compared at two distinct times, are relational quantities. This
notion of relationalism does not introduce an absolute scale, but does require
that an omniscient describer, as it were, of the universe be able to
“remember” a previous size for comparison at a later time. On the other hand,
one can define that only instantaneous ratios are relational, thereby not
requiring that the omniscient describer of the universe possess the ability to
“remember” a previous size for later comparison.

The relational meaning of $\Delta\tau$ (resp., $\Delta\lambda$) depends on
which interpretation of relationalism is used: If one adopts the notion of
relationalism that compares sizes at different instants of time, then
$\Delta\tau$ (resp., $\Delta\lambda$) is the logarithmic ratio of expansion
rates at the two instants. If one adopts the notion of relationalism that only
shapes can be “remembered,” then $\Delta\tau$ (resp., $\Delta\lambda$) can be
expressed in terms of the geometric data of the curve in shape space, so it is
not an independent quantity in this case.

Thus, if one accepts that “remembering” sizes for later comparison is less
relational than not being required to “remember” sizes at all—as we
submit—then PSD is definitely more relational than the formulation of SD that
requires the initial datum $\tau$ (resp., $\lambda$). In other words, the
version of the Mach-Poincaré principle that PSD satisfies is more relational
than the original version of it satisfied by standard SD, which means that PSD
is an improvement on SD.

###  2.3 Dynamics

One question of the utmost importance is how an unparametrized curve is to
account for evolution. In standard relational accounts, a primitive notion of
labeling is employed to describe the physical change of a system. The only
property this label must meet is that it be monotonically increasing if it is
to serve the role of a temporal bookkeeping device. Thus, by letting this
label take on an ordered set of values, one gets the corresponding set of
configurations of the physical system, the succession of which generates the
associated curve in the relevant configuration space.

However, PSD features unparametrized curves, which means the above account of
dynamics is not available. The challenge is hence to find an intrinsic feature
of the system that serves the purpose of a physically meaningful labeling of
change. Fortunately, PSD does in fact possess the resources to handle the
dynamical evolution of a physical system. This is due to the
insight—originally put forward in Barbour et al. (2014a)—that a shape contains
structure encoded in stable records, whereby the evolution is towards
configurations (i.e., shapes) that maximize the _complexity_ of the system.
The result provides the desired ground for introducing a direction of change,
which boils down to the direction of accumulation of the above-mentioned
stable records.101010Note that this idea is far from being uncontroversial.
See, for example, the exchange between Zeh (2016) and Barbour et al. (2016).

Given that the only fully worked-out physical system exhibiting generic
formation of stable records is the $N$-body system, we shall use it as an
example to motivate our approach to dynamics. It is worth pointing out that
promising results come from the vacuum Bianchi IX cosmological model, where a
natural candidate exists for a measure of shape complexity in this simplified
model of dynamical geometry (see Barbour et al., 2013, section 3.5). However,
the extension of our arguments to full dynamical geometry and quantum
mechanics is still a work in progress.

For current purposes, complexity is essentially the amount of clustering of a
system, with a cluster being a set of particles that stay close relative to
the extension of the total system. Next, we demand that the complexity
function grow when (i) the number of clusters do, and (ii) the clusters become
ever more pronounced, namely when the ratio between the extension of the
clusters to the total extension of the system grows.

In the case of the $N$-body problem, a natural measure of the complexity of a
system is:

$\mathsf{Com}(q)=-\frac{1}{m_{\mathrm{tot}}^{5/2}}\sqrt{I_{\mathrm{cm}}}\,V_{N}=\frac{\ell_{\mathrm{rms}}}{\ell_{\mathrm{mhl}}}\,.$
(3)

In the above expression, $I_{\mathrm{cm}}$ is the center-of-mass moment of
inertia:

$I_{\mathrm{cm}}=\sum_{a=1}^{N}m_{a}\,\mathbf{r}_{a}^{\mathrm{cm}}\cdot\mathbf{r}_{a}^{\mathrm{cm}}\equiv\sum_{a<b}\frac{m_{a}\,m_{b}}{m_{\mathrm{tot}}}r_{ab}^{2}:=m_{\mathrm{tot}}\,\ell^{2}_{\mathrm{rms}}\,,$
(4)

where $\mathbf{r}_{a}^{\mathrm{cm}}$ is the position of particle $a$ relative
to the center of mass, $m_{\mathrm{tot}}=\sum_{a}m_{a}$,
$r_{ab}=|\mathbf{r}_{a}-\mathbf{r}_{b}|$, and $V_{N}$ is the Newton potential:

$-V_{\mathrm{N}}=\sum_{a<b}\frac{m_{a}\,m_{b}}{r_{ab}}:=m_{\mathrm{tot}}^{2}\,\ell_{\mathrm{mhl}}^{-1}\,.$
(5)

In (3), $\ell_{\mathrm{rms}}$ and $\ell_{\mathrm{mhl}}$ account for the
greatest and least inter-particle separations, respectively. Thus, their
ratio, (3), measures the extent to which particles are clustered. In this
simple case, it so happens that the opposite of (3) is none other than the
shape space version, $C(q)$, of (5). Hence, $C(q)=-\mathsf{Com}(q)$ is,
accordingly, referred to as the _shape potential_ , modulo the mass
factor.111111To be precise, all PSD models have an associated shape potential,
which is related to the measure of complexity defined in each particular
context.

In Barbour et al. (2014a), it is shown that, for $E_{\mathrm{cm}}\geq 0$,
$I_{\mathrm{cm}}$ is concave upwards as a function of Newtonian time, and its
time derivative (essentially, the dilatational momentum, $D$) is monotonic.
This implies that $I_{\mathrm{cm}}$ is U-shaped, with a unique minimum at
$D=0$ that divides all solutions in half. This is the so-called _Janus
Point_.121212This designation was first put forward in Barbour et al. (2015).
Interestingly enough, the complexity function (3) has a minimum near this
point and grows in either direction away from it, whereby automatically
defining a time-_asymmetric_ dynamics for internal observers, i.e., ones
within one of the two branches at either side of the Janus Point.

Moreover, it is a well-known result that the $N$-body system features generic
solutions in which the original system splits into subsystems consisting of
individual particles and clusters. Such almost isolated subsystems become
increasingly isolated in the asymptotic regime (Marchal and Saari, 1976), and
will develop approximately conserved charges, namely the energy $E$, linear
momentum $\bf P$, and angular momentum $\bf J$. These charges, in turn, enable
us to define units $X$ of spatial scale and $T$ of time duration as:

$X^{2}=\frac{\bf J^{2}}{\bf P^{2}}\,,\,\,\textrm{ and }\,\,T^{2}=\frac{\bf
J^{2}}{E^{2}}\,.$ (6)

This discussion naturally leads to two crucial remarks. First, the ever better
isolated subsystems serve as local and _stable_ substructures, whose ever
better conserved charges give rise to stable _records_. This dynamically
defines a direction of increasing complexity, measured by the complexity
function (3), which tends to grow secularly. We argue that this direction of
increasing complexity should be identified with the so-called _arrow of time_
for internal observers. Thus, we arrive at a description of such an arrow of
time in terms of purely intrinsic properties of the unparametrized curve in
shape space (see §4.2 for a more in-depth discussion of this point). Second,
within the dynamically formed subsystems, there are pairs of particles that
may function as physical rods and clocks, referred to as _Kepler pairs_ , due
to their asymptotic dynamics tending to elliptical Keplerian motion.

A final remark is in place at this point. This treatment of temporal
structures in terms of the complexity function is a common trait of PSD and
SD—indeed, the above discussion was originally introduced in the context of
SD. However, SD exhibits a patent redundancy in its description of temporal
structures since it introduces the complexity function while retaining the
_physical_ parameter $\Delta\tau$ (resp., $\Delta\lambda$). PSD, on the other
hand, codes all reference to temporal structure in the complexity function
alone, thus resulting conceptually cleaner.

###  2.4 Examples

In order to illustrate how PSD describes dynamical systems, in this subsection
we will consider some physically interesting models and their associated
equations of state of the respective unparametrized curves in shape space.
Furthermore, this will explicitly show one of the central tenets of the
program, namely, its use of only intrinsic geometric properties
$\alpha_{I}^{a}$ of the curve. The discussion will also make manifest how the
size of the set $\alpha_{I}^{a}$ depends on the complexity of the system to be
described: The more complex the system, the more geometric properties are
needed in its physical description.

The examples below shall show the equations of state of a number of physical
systems, but, in order not to obscure the presentation, we will not exhibit
the mathematical structure underlying the manifest unparametrized nature of
said equations (see Koslowski et al., 2021, for this construction for the
$E=0\,\,\,N$-body system). Finally, a word is in order: As already mentioned
above, the use of a parametrization is legitimate for the sake of
computational ease, but we stress again that it does not belong to the
fundamental structure of PSD itself.

####  2.4.1 Geodesic System

The dynamical system described by equation (1), with the tangential direction
$\phi^{A}$ at the point $q^{a}$ as the only element in the set
$\alpha_{I}^{a}$, corresponds to the simplest system in shape space, namely
one whose curve is a geodesic, given by the Hamiltonian:

$H=\frac{1}{2}g^{ab}(q)p_{a}p_{b}\,,$ (7)

where $g^{ab}$ is the kinematic metric. Absorbing the dimensionless mass
ratios $\mu_{i}:=\frac{m_{i}}{M}$ into the configuration space metric (see
(13) below) allows us to set the overall mean mass
$M:=\frac{1}{N}\sum_{i=1}^{N}m_{i}$ to unity. In order to work out the
equation of state, let us conveniently use the arc-length parametrization of
the curve with respect to $g_{ab}(q)$:

$\left(\frac{ds}{dt}\right)^{2}=g_{ab}(q)\frac{dq^{a}}{dt}\frac{dq^{b}}{dt}=g^{ab}(q)p_{a}p_{b}\,,$
(8)

where $s$ is the arc-length parameter, which readily yields

$\frac{dq^{a}}{ds}=g^{ab}(q)\frac{p_{b}}{\sqrt{g^{cd}(q)p_{c}p_{d}}}\,,$ (9)

where the right-hand side is the unit tangent vector $u^{a}$, (2), which,
recall, defines the direction $\phi^{A}$ at $q^{a}$. Taking any explicit
functional expression $\Phi(q,p)$ for the direction, one gets:

$\frac{d\phi^{A}}{ds}=\frac{\partial\Phi}{\partial
q^{a}}\,\frac{dq^{a}}{ds}+\frac{\partial\Phi}{\partial
p_{a}}\,\frac{dp_{a}}{ds}\,,$

which, by means of (2), (8) and (9), enables us to write the equation of state
of the geodesic curve by means of the canonical equations of motion generated
by the Hamiltonian (7) as:

$\begin{array}[]{rcl}d\,q^{a}&=&u^{a}(q,\phi)\,,\\\
d\,\phi^{A}&=&\frac{\partial\Phi}{\partial
q^{a}}\,u^{a}(q,\phi)-\frac{1}{2}\frac{\partial\Phi}{\partial
u^{a}}g^{bc}_{\phantom{bc},a}(q)u_{b}(q,\phi)u_{c}(q,\phi)\,,\end{array}$ (10)

where we have dropped the arc-length parameter $s$ to emphasize the underlying
unparametrized nature of the curve.

An example of geodesic system is a universal configuration of classical non-
interacting particles that indefinitely expands. Note how such a system, by
construction, does not exhibit the formation of stable records.

####  2.4.2 Newtonian $E=0\,\,\,N$-body System

Let us consider the Hamiltonian with a generic potential $V$:

$H=\frac{1}{2R^{2}}(D^{2}+g^{ab}(q)p_{a}p_{b})+V(R,q)\,,$ (11)

where the overall mean mass $M:=\frac{1}{N}\sum_{i=1}^{N}m_{i}$ has been
absorbed into the coupling constant of the potential, {$q^{a},p_{a}$} are
coordinates and momenta in shape space, and $D$ is the dilatational momentum.
Moreover, $g^{ab}$ is the kinematic metric, which, in the case of Euclidean
space, is given by the scale-free Euclidean metric on configuration space:

$ds^{2}=\frac{\sum_{I=1}^{N}\,d\bf r_{I}^{2}}{R^{2}}\,,$ (12)

where $R^{2}:=\sum_{I=1}^{N}\bf r_{I}^{2}$ denotes the square of the total
scale in the center-of-mass frame. In “global” coordinates, the induced
kinematic metric on shape space takes the explicit form:

$\frac{1}{R^{2}}\,g^{ab}(q):=g_{IJ}\frac{\partial q^{a}}{\partial
r^{I}}\frac{\partial q^{b}}{\partial r^{J}}\,,$ (13)

where $r^{I}$ and $g_{IJ}$ —which contains the dimensionless mass ratios
$\mu_{i}:=\frac{m_{i}}{M}$ — are some coordinates and metric, respectively, in
configuration space. We shall focus on the case of a homogeneous potential,
$V(R,q)=\beta R^{k}C(q)$, with $\beta$ an arbitrary coupling constant into
which the overall mass $M$ above has been absorbed, which enables us to write
(11) as an energy conservation constraint:

$H=\frac{1}{2}(D^{2}+p^{2})+\beta R^{k+2}C(q)=0\,,$ (14)

where $p\equiv\sqrt{g^{ab}p_{a}p_{b}}$ is the length of the shape momenta.
Unlike the geodesic system (10), now one can build a further degree of freedom
{$\alpha_{I}^{a}$}, namely $\kappa\equiv\frac{p^{2}}{\beta R^{k+2}}$, which is
related to $K^{-1}$, with $K$ the curvature of the curve, which, in turn, is
given in terms of the acceleration vectors. Thus, $\kappa$ can be thought of
as an _intrinsic acceleration_ , i.e., a measure of how much the curve traced
out by a given physical system deviates from geodesic dynamics, (10). The
intrinsic change of $\kappa$ is obtained, once again, by Hamilton equations of
motion, yielding the following equation of state:

$\begin{array}[]{rcl}dq^{a}&=&g^{ab}(q)u_{b}(q,\phi)\\\
d\phi^{A}&=&\frac{\partial\Phi_{A}}{\partial
q^{a}}\,g^{ab}(q)u_{b}(q,\phi)-\frac{\partial\Phi_{A}}{\partial
u_{a}}\left(\frac{1}{\kappa}\frac{\partial C(q)}{\partial
q^{a}}+\frac{1}{2}{g^{bc}}_{,a}(q)\,u_{b}(q,\phi)u_{c}(q,\phi)\right)\\\
d\kappa&=&-(k+2)\kappa\varepsilon(q,\phi,\kappa)-2u^{a}(q,\phi)C_{,a}(q)\,,\end{array}$
(15)

where
$\varepsilon\equiv\frac{D}{p}=\pm\sqrt{-\left(1+2\frac{C(q)}{\kappa}\right)}$
can be solved for in terms of {$q^{a},\phi^{A},\kappa$} by means of the energy
conservation constraint (14).

The immediate question that arises at this point is: How do we recover the
standard Newtonian notions of scale and duration from (15)? Although it is
clear that it will be impossible to get absolute units of scale and duration,
since the curve is void of any such structures, one can nonetheless
meaningfully ask how definitions of absolute scales evolve. To explain the
question, let us consider a curve $\gamma$ in shape space and two points
$q^{a}$ and $q^{b}$ on it. Next, we will define the total scale $R$ of the
system at point $q^{b}$ to be the unit of size $R_{0}$ and the total duration
between $q^{a}$ and $q^{b}$ to be the unit of time $T$. Then, given a third
point $q^{c}$ on $\gamma$, one may ask: what is the total scale $R$ measured
in units of $R_{0}$? Likewise, what is the duration between $q^{b}$ and
$q^{c}$ in units of $T$?. To answer these questions, we will give explicit
equations for standard scale and duration in the case of the homogeneous
system (14). Following the suggestion in Barbour (1994a, §4), we will call
these standards _ephemeris scale_ and _ephemeris duration_. Historically, the
ephemeris time was a duration standard used by astronomers and based on the
intrinsic properties of the solar system (considered as a closed system). More
precisely, the ephemeris time was the duration standard which made all the
motions of the dynamically relevant bodies compatible with Newtonian dynamics
(see also Mercati, 2018, §13.2.4, for a discussion of ephemeris scale and time
in the context of SD).

Using the arc-length parametrization condition (8), we obtain the ephemeris
scale equation (cf. Koslowski et al., 2021, §3.2, for the technical details of
the derivation):

$\frac{d}{d\,s}\log
R=\frac{D}{p}=\pm\sqrt{-\left(1+2\frac{C(q)}{\kappa}\right)}\,.$ (16)

Notice that the shape potential $C(q)$ in Newtonian gravity is negative
definite, which must be taken into account in (16).

Likewise, the ephemeris duration equation is derived from

$\frac{d}{d\,s}\log\left(\frac{ds}{dt}\right)=\frac{d}{d\,s}\log
p=-\frac{2}{\kappa}u^{a}(q,\phi)\,C_{,a}(q)\,.$ (17)

The unit of time $T$ is obtained by integrating (17) between configurations
$q^{a}$ and $q^{b}$.

As expected, the right-hand sides of both (16) and (17) refer only to
intrinsic properties of the unparametrized curve in shape space. In
particular, the (rate of change of the) complexity function (3) gives rise to
the standard, global notion of duration, exhibiting the general result of the
emergence of an arrow of time in terms of the increase in complexity of the
$N$-body system. It is important to point out that (i) the ephemeris equations
are model-dependent and (ii) there exists a many-to-one correspondence between
Newtonian models and equations of state of curves in shape space, so the
ephemeris equations are in general not uniquely associated with an equation of
state of the curve in shape space.

We should also emphasize one key fact: Although the complexity function, (3),
is, by construction, genuinely a shape quantity, it does _not_ belong to the
fundamental geometric properties of the curve in shape space, for the simple
reason that it functionally depends on—and hence is reducible to—more basic
data, namely, inter-particle separations.

####  2.4.3 Newtonian $E>0\,\,\,N$-body System

In this case, the potential reads $V(R,q)=\beta\,R^{k}C(q)-E$, with the non-
vanishing energy breaking homogeneity, which implies that one can no longer
solve the energy conservation constraint,
$0=\frac{1}{2}(p^{2}+D^{2})+\beta\,R^{k+2}\,C(q)-R^{2}\,E$, for $\varepsilon$
in terms of {$q^{a},\phi^{A},\kappa$}. Thus, we have to augment the dynamical
system (15) with an equation of motion for $\varepsilon$ (which, again, is
obtained by means of Hamilton equations). We obtain:

$\begin{array}[]{rcl}d\,q^{a}&=&u^{a}(q,\phi)\\\
d\,\phi^{A}&=&\frac{\partial\Phi_{A}}{\partial
q^{a}}\,u^{a}(q,\phi)-\frac{\partial\Phi_{A}}{\partial
u^{a}}\left(\frac{C_{,a}(q)}{\kappa}+\frac{1}{2}g^{bc}_{,a}(q)u_{b}(q,\phi)u_{c}(q,\phi)\right)\\\
d\,\kappa&=&-(k+2)\kappa\,\varepsilon-2\,C_{,a}(q)u^{a}(q,\phi)\\\
d\,\varepsilon&=&\varepsilon\frac{u^{a}\,C_{,a}(q)}{\kappa}-(\frac{C(q)}{\kappa}+2\sigma)\,,\end{array}$
(18)

where a further dimensionless ratio $\sigma\equiv\frac{R^{2}E}{p^{2}}$ can be
formed out of the parameters of the model, and can be solved for by use of the
energy conservation constraint as:

$\sigma:=\frac{R^{2}\,E}{p^{2}}=\tfrac{1}{2}(1+\varepsilon^{2})+\frac{C(q)}{\kappa}\,.$
(19)

####  2.4.4 $E=0$ Bianchi IX cosmological model

Finally, we shall consider the simplest non-trivial case of a relativistic
cosmological model, the so-called _Bianchi IX_ or _Mixmaster_ model (Misner,
1969), whose relational version was first analyzed in Koslowski et al. (2018).
Consider the following Hamiltonian:

$H_{\rm
eff}=p^{2}+\left(2\Lambda-\tfrac{3}{8}\tau^{2}\right)v^{2}-v^{\frac{4}{3}}C(q)\,,$
(20)

where $p^{a}$ are the shape momenta canonically conjugate to $q^{a}$, $v$ is
the spatial volume, $\tau$ its canonically conjugate variable, and $C(q)$ is
the associated shape potential.131313We emphasize again that there is a shape
potential for each physical system, which is related to the measure of its
complexity, with the relation between the two depending on each case. The
shape potential in (20) is _not_ (minus) the simple ratio (3) (Barbour et al.,
2013, §3.5). The curve on shape space traced out by the Bianchi IX Universe is
described by the following equation of state:

$\displaystyle dq^{a}$ $\displaystyle=u^{a}(q,\phi)\,,$ $\displaystyle
d\phi^{A}$ $\displaystyle=\tfrac{1}{2\kappa}\tfrac{\partial\Phi}{\partial
u_{a}}\,C_{,a}(q)\,,$ $\displaystyle d\kappa$
$\displaystyle=u^{a}(q,\phi)C_{,a}(q)+\tfrac{1}{2}\sigma\sqrt{\kappa}\,,$ (21)
$\displaystyle d\sigma$
$\displaystyle=\tfrac{1}{\sqrt{\kappa}}\left(\tfrac{1}{4}\sigma^{2}+\tfrac{2}{3}C(q)\right)-2\,\varepsilon\sqrt{\kappa}\,,$

where $\kappa\equiv p^{2}\,v^{-\frac{4}{3}}$, $\sigma\equiv\tau\,v^{1/3}$ and
$\varepsilon\equiv\frac{\Lambda\,v^{2}}{p^{2}}$.
Finally, because of the constraint on the effective Hamiltonian (20), $H_{\rm
eff}\approx 0$, we can solve for the parameter $\varepsilon$, yielding

$\varepsilon=\tfrac{1}{2}\left[\tfrac{1}{\kappa}\left(C(q)+\tfrac{3}{8}\sigma^{2}\right)-1\right]\,.$
(22)

The homogeneous nature of this cosmological model simplifies the equations for
the direction and curvature degrees of freedom (vanishing derivatives with
respect to $q^{a}$), but it requires nevertheless a further degree of freedom
as compared to the Newtonian $E=0$ $N$-body system: $\sigma$ comes about as a
consequence of the expansion of the Universe. By considering the transition
from (10), through (15) and (18), to (21), it is now manifest how the increase
in physical complexity of the system analyzed corresponds to an increase in
the number of geometric degrees of freedom required to describe its dynamics.

##  3 From Shapes to Self-Subsisting Structures

The question that we are interested in investigating at this point is: What is
the best metaphysical picture that goes along with the PSD framework? The
philosophically-minded reader must have noticed that PSD comes with a
conspicuous amount of metaphysics already built-in. Suffice it to recall the
main tenet around which the framework is centered, that is, the denial of the
existence of anything whose change makes no physical difference. This is a
strong metaphysical assumption, which justifies definition 2.1 and its
technical implementation, as seen in the previous section. However, the
dynamics encoded in (1) is not just informed by the metaphysical tenet
mentioned before; indeed, it in turn strongly suggests some metaphysical
morals, as we are going to argue in a moment. This, for us, is a clear signal
that a fruitful conceptual analysis of the PSD framework should treat physical
and metaphysical aspects on a par, with neither of the two being merely
entailed in a strictly logical sense by the other. Such an approach is close
to what is usually known as _natural philosophy_. According to this doctrine,
the physical and metaphysical pictures of a theoretical framework are
conceptually interwoven (see, e.g., Esfeld, 2018, for a recent discussion of
this philosophical approach). In this sense, answering the above question is
not a matter of just “reading off” an ontology from the framework’s formalism
in a neo-positivist fashion (see Ney, 2012, for a presentation and defense of
this latter stance). Therefore, in what follows, our metaphysical elaboration
on the conceptual foundations of PSD should be considered as taking place in
parallel with the technical development of the framework.

With this remark about philosophical methodology in place, let us now turn to
what the most satisfactory answer is for us to the question asked above. As
mentioned in §1, the PSD framework is in principle able to encompass not only
the Newtonian and general relativistic domains of physics—as sketched in the
previous section—but also the quantum one (which, as said, is discussed in a
companion paper). This may sound puzzling from a metaphysical standpoint,
given that such a comprehensive class of physical models is usually cast in
terms of wildly different types of theoretical entities, such as material
particles, fields, wave functions, strings, _et cetera_. Does this mean that,
according to PSD, we should take these theoretical entities at face value all
at once? To answer this question, we should recognize that “taking at face
value” can be intended in two related but slightly different senses. First of
all, we can take a theoretical entity at face value by accepting that it
directly refers to something physical, i.e., inhabiting the physical world,
rather than being just a descriptively useful piece of formalism. Thus, we may
take classical particles at face value because we can reliably assign them an
active causal role in determining some observable phenomena. However, we
cannot do the same with the Hamiltonian function, which is, in fact, just a
compact mathematical way to summarize the physical state of said particles
(the standard source for a defense of this entity-based realism is Hacking,
1982). A second, stronger sense in which we may take a theoretical entity at
face value is by claiming that it is part of the fundamental furniture of the
world, i.e., an element of the ontology of the theory, in case it is a
fundamental theory. If something is fundamental, it is “out there” in the
physical world, but the converse does not necessarily hold, whence the need
for distinguishing between these two senses of “taking at face value”.

Under the light of this distinction, it does not seem reasonable to take _all_
the theoretical entities associated with the various physical models
encompassed by PSD as the fundamental furniture of the actual world. This is
because, in this way, the ontology of PSD would become bloated and
uninformative. Indeed, adopting such an all-encompassing horizontal
fundamental ontology would _a priori_ bar the possibility to construct a tree
of ontological dependencies among entities—with its most welcome baggage of
explanatory power. Otherwise said, it seems that a metaphysics where—for
example—the existence of particles depends on fields, and the existence of
fields, in turn, depends on that of strings has much more explanatory power
than a horizontal one where all those entities are given all at once as a
brute fact of the matter.

However, there are serious reservations also in taking all of these
theoretical entities at face value in the milder sense of “being out there.”
Most importantly, this choice would obscure the fact that there has been a
conceptual shift in the use of these theoretical terms that corresponds to the
development of more and more empirically adequate theories in the history of
physics. For example, the talk of gravitational action-at-a-distance between
material particles in Newtonian mechanics has long been replaced by the
general relativistic talk of local interactions mediated by a gravitational
field. There is no doubt that general relativity does a better job than
Newtonian gravity in capturing some crucial features of gravitational
phenomena, and this justifies the conclusion that the picture of reality
provided by general relativity is more accurate than that provided by Newton’s
theory. It would then be awkward to claim that the pictures provided by the
two theories coexist in the actual world.

A moment of reflection shows, however, that the PSD framework is not about
theoretical entities _per se_ , but about the way such entities are
interrelated. This is perhaps the most straightforward metaphysical moral
associated with the quotienting out procedure introduced in the previous
section. In other words, such a procedure does not determine the fundamental
elements of reality _as individual entities_ , but just the _relational
properties_ of configurations of them, i.e., the shapes properly said. Indeed
the primary result of the quotienting out procedure is the “translation” of
the physical system under scrutiny from standard configuration space to shape
space. Nevertheless, shape space itself is just a (stratified) manifold whose
points do not carry any information regarding how the standard configuration
space was constructed in the first place. Instead, what shape space is
sensitive to is how the quotienting out procedure singles out the physical
relations making up a shape. This information is encoded in the very geometric
structure of shape space itself. Thus, the shape space corresponding to a
Newtonian $N$-body system is geometrically different from one corresponding
to, say, Bianchi IX cosmology. This implies that, whatever the physical
domain, PSD remains invariably a theory of _shapes_ —not just particles, or
fields, or any other individual theoretical entity. This is indeed the
rationale behind the claim that the dynamical law (1) represents a unified
description of all of these physical domains. What (1) describes is the common
behavior of the structural features of such domains. It is then easy to argue
that the PSD framework naturally calls for a metaphysical interpretation in
ontic structural realist terms.

We should be careful, however, in clarifying what type of ontic structuralism
we have in mind. From what we have said so far, it may be inferred that we
favor some sort of eliminativist type of structuralism, which does away with
objects, and frames structures as clusters of relations _simpliciter_. While
this is a legitimate philosophical choice (but see Lam and Wüthrich, 2015, for
a critical voice), we are not sympathetic to it in this context. The reason
for this skepticism is simple: Although the quotienting out procedure does not
_determine_ the objects in a structure, it does not _eliminate_ them either.
For any physical domain, the objects are “already there” before the procedure
occurs, in the sense that their postulation is more or less implicit in the
individuation of the degrees of freedom described by the standard
configuration space. What the procedure does is to _reconceptualize_ the
metaphysical role of these objects in the physical picture. Hence, for
example, there is a clear sense in which PSD translates a Newtonian $N$-body
system into a structure made up of material particles related through a web of
Euclidean spatial relations. In other words, there is a clear sense in which
shapes in PSD are implemented as concrete physical structures, as opposed to
merely mathematical constructs. From this point of view, it is doubtful
whether a merely eliminativist take on structures in PSD would be able to
account for such an implementation, at least without adding some additional
interpretive baggage on top of this radical structural metaphysics (see
Esfeld, 2013, §2, for an argument along these lines, and Ladyman and Ross,
2007, for a decently worked out eliminativist framework that seeks to overcome
such an objection).

The version of ontic structuralism that we favor is a more moderate one that
treats objects and relations as ontologically on a par. This choice is
motivated by the fact that, as already pointed out, the way PSD shifts the
focus from ordinary configuration space to shape space retains a robust notion
of shape as a configuration of individual elements of reality, with such
elements “adding up”, once piece after the other, to make up a whole
configuration. This is precisely in the spirit of the Leibnizian account of
space as the ordering of coexisting things, i.e., space as a whole resulting
from many local material facts put together.

The discussion so far seems to beg the question: At this point, why not going
for a metaphysics of objects _tout-court_? The answer to this question is
relatively straightforward: Because the PSD framework dispenses altogether
with any spatiotemporal background in which such objects could be placed and
hence individuated. The only way objects can be individuated in such a
framework is through the relations they instantiate. There is no other
fundamental defining feature of such objects beyond their relational aspects,
in particular no primitive intrinsic identity or distinctness (recall from §2
that the characterization of a shape does not hinge on facts regarding the
identity of the _relata_). Because of this, the shapes portrayed by PSD are
very peculiar structures, which we dub _self-subsisting_ to highlight the fact
that no aspect of their existence depends on something external to them, such
as a background spacetime in which they are embedded.

In conclusion, we submit that the best metaphysics that goes along with the
physical description encoded in (1) is moderately structuralist and is based
upon the notion of self-subsisting structure (which is a metaphysically
refined version of the concept of shape). However, still, it is not clear
whether, for us, _all_ possible self-subsisting structures are fundamental or
just “out there”. Indeed, when asked what it is that, say, both Newtonian
gravitation and dynamical geometry get right under the light of the PSD
framework, our answer would be: Some purely relational aspect of the
gravitational interaction that is hence preserved in the passage from
Newtonian shapes to dynamical geometry ones. However, are Newtonian self-
subsisting structures—i.e., point-like particles related by Euclidean spatial
relations—to be considered ontologically on a par with their dynamical
geometry counterparts—i.e., some sort of field magnitudes related by pseudo-
Riemannian spatial relations? Answering this in the positive would obviously
reintroduce the problem of mixing up radically different and perhaps
incompatible pictures of the physical world.

To this challenge, we reply that there are two ways to address metaphysical
questions in the PSD framework. The first is to do that in the context of a
specific model. For example, we can consider the $N$-body Newtonian model as a
faithful depiction of a world where fundamentally there are point-particles
instantiating a web of Euclidean spatial relations. However, we perfectly know
that such a world is not the actual one for the simple reason that such a
fundamental ontology is not (entirely) empirically adequate. With this
respect, the more empirically adequate the model, the more accurate the
ontological picture will be (so, perhaps, by developing the PSD framework to
include quantum-gravitational motions, we might get closer to the actual
fundamental ontology of the physical world; see §5). There is no way in which
we could take all the models as (fully) empirically adequate, so there is no
question whether we can mix up radically different pictures of the actual
world by going for this intra-model discourse. However, this does not mean
that our metaphysics is strictly speaking model-dependent in a neo-positivist
fashion. Indeed, there is a second sense in which we are straightforwardly
realist towards all PSD models. As we have already pointed out, there are some
purely structural inter-model aspects of PSD that correctly capture observable
phenomena, and it is precisely these shared structural features that we
consider as being “out there”. Note that this is not just epistemic structural
realism in disguise since there is in principle a class of models of PSD that
are completely empirically adequate, and we take these (yet-to-be constructed)
models to be the ones giving us the most accurate fundamental picture of
reality.

##  4 From Self-Subsisting Structures to Spacetime

###  4.1 A Supervenience Basis for Laws and Dynamical Structure

In order for the picture of the world in terms of self-subsisting structures
to be viable, a story is required about how this relational metaphysics (i)
accommodates the dynamical law (1), and (ii) recovers the familiar
understanding of material objects located in spacetime.

Section 2 made it clear that (1) is a sort of “law schema” that encompasses
the common structural features of different physical domains. The way (1) is
then actually implemented depends on the details of the quotienting out
procedure carried out on the starting non-relational theory. Hence, for
example, from $N$-particle Newtonian mechanics equation (15) is derived;
likewise, equation (21) follows from dynamical geometry, and so on. From a
metaphysical point of view, the challenge is to provide a unified “mechanism”
that links relational models to laws, thus making sense to consider the latter
domain-related laws as occurrences of (1). In light of this challenge, it
seems that primitivist and governing accounts of laws are not well-suited for
the framework. According to the former account (defended, e.g., in Maudlin,
2007, chapter 1), laws cannot be analyzed in more fundamental terms—they are
brute facts of the matter. In this case, however, the structural resemblance
of laws like (15) and (21) would remain unexplained and may very well be just
a fortuitous fact. This would, in turn, demote PSD from an all-encompassing
physical framework to a mere collection of models with no apparent
(meta)physical connection. According to the governing view of laws (see, e.g.,
Mumford, 2004, §§9.3-9.6, for a critical discussion of this stance), instead,
laws determine physical happenings (and not the other way round). In the PSD
context, this view would imply that laws such as (15) and (21) would be
external to self-subsisting structures, but that would bar such structures
from grounding the structural resemblance between these laws. This weakening
of the capability of self-subsisting structures to account for the unifying
nature of (1) would go against the very spirit of PSD, which seeks to eschew
anything external to shapes from the dynamical description. At this point, it
seems clear that a better strategy to find a common metaphysical “mechanism”
that leads from all the models of PSD to (1) is to adopt a non-primitivist and
internal view of laws, i.e., one according to which it is physical happenings
that determine the laws.

The first step to implement this strategy is to go back to the relationalist
norm put forward in §2. Simply speaking, the norm states that commitment to
any structure whose variation makes no observable difference should be
avoided. The critical insight is to realize that this norm can be restated as
a supervenience principle: We should be committed only to as much structure as
that sufficient to constitute a complete, non-redundant supervenience basis
for the dynamical laws of our framework. This provides us with the common
metaphysical “mechanism” that explains the structural resemblance of laws like
(15) and (21) and, hence, justifies their being considered occurrences of (1).
In short, the structural resemblance of, e.g., (15) and (21), is inherited via
a supervenience relation141414Here we will gloss over the conceptual
subtleties and the philosophical controversies related to the notion of
supervenience (see, e.g., Kim, 1990, for a thorough discussion of this
concept). from the (dynamical) structural resemblance of self-subsisting
structures across the models of PSD. In this sense, it becomes clear why (1)
captures the structural features common to all models of PSD: It is trivially
_determined_ by such structural features.

The problem at this point is that there are many different choices of such
supervenience basis that can recover (1), each of which carries a different
metaphysical flavor with it. Most importantly, one may or may not include in
the basis some primitive modal facts in the guise of causal powers or
dispositions borne by self-subsisting structures. In the first case, the
supervening laws would inherit some weak governing connotations (see the
review symposium French, 2006, for an exchange on this subject), while in the
second case, they would play no determining role whatsoever. Hence, it is
clear that the relationalist norm discussed above has to be supplemented with
some further requirement to adjudicate between these two choices. With this
respect, it seems natural to take ontological parsimony as such extra
requirement: All things being equal, the best supervenience basis for (1) is
the one containing the least amount of structure possible. Here we will take
for granted that there is a consistent way to count the amount of structure in
the ontology; see North, 2009, for a discussion on how to recognize and count
physical structure. “All things being equal” translates into “given a PSD
model”, which clarifies that the type of ontological parsimony we have in mind
is an intra-model one. This remark is important because, otherwise, it may be
objected that the metaphysics we are proposing violates parsimony in that it
regards a wide range of entities implementing self-subsisting structures at
least as genuine physical possibilities. The fact that parsimony is used in
this specific way poses no conceptual problem since considerations about
simplicity crucially depend on the particular context considered (see Sober,
2015, for an in-depth articulation of this view). For example, one may demand
a parsimonious fundamental ontology without posing any restriction on the type
of “higher-order” entities that metaphysically depend on such an ontology
(with this respect, see Schaffer, 2015). In fact, given that we are not
interested in restricting the range of physical possibilities encompassed by
PSD (on the contrary, we have made it clear that we consider this aspect as a
virtue of the framework), we regard intra-model parsimony as the only
simplicity requirement that counts in this context. This is because intra-
model parsimony restricts the amount of fundamental structure _at a world_
(possible or actual).

This appeal to parsimony in the context of PSD is anything but arbitrary.
Indeed, the framework (i) dispenses with external spatiotemporal structures
and (ii) reduces in a precise mathematical sense the relevant dynamical
features of a physical system to the geometric properties of a curve in shape
space. Fact (i) is a clear sign that PSD favors an ontology with very thin
spatiotemporal features (more on this in §4.2), but also fact (ii) points in
the direction of ontological parsimony. To see this, consider that the
geometry of shape space is not a “stable” feature of the formalism, i.e., its
details depend on the particular system under scrutiny. This fact can be
exploited (and will be exploited below) to argue that the shape geometric
degrees of freedom that are not common to all the models of PSD are not part
of the all-encompassing structural aspects of the framework itself—as encoded
in (1)—and, hence, should not be taken as referring to features genuinely
borne by self-subsisting structures. Instead, such model-dependent geometric
features should be considered as a useful formal tool to describe in a
succinct yet informative way how the dynamics of the self-subsisting
structures unfolds in each specific case. If we buy into this line of
argument, then it is easy to accept that the only fundamental stuff equation
(1) describes is structures featuring a set of fundamental relations
instantiated by otherwise featureless _relata_. Once we embrace such a
parsimonious metaphysics, the next step is to deny any need for the presence
of genuine modal properties in the supervenience basis. These properties are
usually individuated in terms of—or even identified with—the causal role they
play (e.g., mass being the disposition to move in a certain way in a
gravitational field), but their fundamental role becomes dubious in a
metaphysics that denies any substantial ontological import to dynamical
parameters such as mass, charge, and the like.

In the literature, it has long been recognized that an adequate ontology for
physics can, in principle, do away with fundamental modal features in the
guise of intrinsic causal properties borne by objects. This is, in a nutshell,
the central tenet of regularity theories of laws of nature, commonly referred
to as _Humeanism_. One of the best worked out examples of such a stance is the
Mill-Ramsey-Lewis Best System account of laws of nature. According to this
account, the laws at a world $w$ supervene on the arrangement of local matters
of particular fact making up $w$—the _Humean mosaic_ —as the axioms of the
simplest and most informative deductive system from which true physical
statements can be derived (see Hall, 2015, for an introduction to this topic).
This stance has been recently further developed in order to accommodate the
non-locality inherent into quantum entanglement. The crucial move with this
respect is to recognize that the mosaic does not need to include any intrinsic
property _at all_ —not even the perfectly natural and categorical properties
originally postulated by David Lewis (on this score see, e.g., Miller, 2014,
and Esfeld, 2014, but see also Bigaj and Vassallo, 2020, for a critical
assessment of this move). So how exactly does such a propertyless Humean
metaphysics apply to the case of the PSD framework?

To answer this question, let us first of all turn to the formal machinery
introduced and discussed in §2, and try to pinpoint the minimal set of degrees
of freedom that are strictly needed to construct the dynamics and are common
to all PSD models. These elements can be easily individuated to be shape space
points $q^{a}$ and the direction $\phi^{A}$ at $q^{a}$. Indeed, $q^{a}$ and
$\phi^{A}$ enter the description of the simplest models of the framework—i.e.,
geodesic motions: There is no simpler, physically meaningful motion modeled
with less degrees of freedom than this. On the other hand, more complex
motions can be modeled by introducing further geometric degrees of freedom
{$\alpha_{I}^{a}$} _alongside_ $q^{a}$ and $\phi^{A}$. Under our parsimonious
reading, {$\alpha_{I}^{a}$} are just needed to _describe_ in detail such
motions but, in the end, do not amount to adding anything metaphysically
substantial on top of the fundamental degrees of freedom $\langle
q^{a},\phi^{A}\rangle$, which are the only variables common to all models of
the framework—i.e., they are _necessary_ to make sense of the dynamics in the
first place. Think, for example, of the curvature degree of freedom $\kappa$
that arises in the simplest non-geodesic case (15). This degree of freedom is
not needed in the geodesic case _and_ it is constructed out of the kinematic
metric $g_{ab}$, which measures the “intrinsic” differences between shapes.
This suggests that facts about particle accelerations can be reduced to facts
about how the spatial relations among these particles change. Another concrete
example that motivates our restrictive realist attitude is given by (3)—the
expression of the complexity function $\mathsf{Com}(q)$ in the Newtonian
context—, which is entirely given in terms of (ratios of) inter-particle
separations. This means that the notion of complexity is not primitive but
supervenes on—indeed, it is reduced to—intrinsic facts about shapes. To sum
up, we take the structure $\langle q^{a},\phi^{A}\rangle$ to refer to the
Humean mosaic on which the law (1) and its dynamical structure supervene.

This “metaphysical” focus on the minimal set of degrees of freedom is a quite
natural choice as far as our realist and parsimonious attitude is concerned:
Since the physical degrees of freedom are the subject of the dynamical
description, they are the most plausible candidates in the PSD formalism to
refer to the features of the physical world. At this point, one may wonder why
the shape space metric $g_{ab}$ shouldn’t be taken metaphysically seriously as
well, given that it is required to define unit tangent vectors and extract the
curvature from the acceleration vectors—and, thus, to generate the dynamics.
To this we reply that, although $g_{ab}$ occurs in the right-hand sides of all
equations of state given in §2.4, there is no equation for $g_{ab}$ itself.
Hence, $g_{ab}$ is not the subject of the dynamical description, but a formal
tool used to facilitate such a description. Under this view, claiming that
$g_{ab}$ represents a _sui generis_ feature of reality would be similar to
reifying the Hamiltonian function—a rather awkward ontic commitment that would
inflate the ontology without adding explanatory value.

The next step is now to characterize the entities in the mosaic. With this
respect, we look at the minimal amount of facts needed to make sense of
fundamental degrees of freedom $\langle q^{a},\phi^{A}\rangle$. The first half
of this structure is represented by the points $q^{a}$, which already have a
fairly straightforward characterization put forward in the previous section.
Each point represents a universal configuration of fundamental elements of
reality in an intra-model sense—e.g., particles, field magnitudes. These
fundamental elements of reality are connected through a web of relations that
are spatial in nature, and which are described by the conformal structure that
“survives” the quotienting out procedure on the redundant spatial degrees of
freedom of the particular system considered. We now see why intrinsic
properties are not needed at all in this context. Simply speaking, these
properties do not enter in any way the characterization of the points $q^{a}$
representing our self-subsisting structures. Compare this to the status of the
relations making up a self-subsisting structure, which instead _determine_ the
set of points $q^{a}$ (i.e., shape space itself) through the quotienting out
procedure discussed in §2.

The second half of the structure $\langle q^{a},\phi^{A}\rangle$ tells us how
such self-subsisting structures are dynamically related. From the discussion
of the dynamical mechanism underlying (1), we already know that the
fundamental ordering of points $q^{a}$ in a dynamical curve is not metrical in
the usual sense of the word. That is, there is no fact of the matter about a
configuration $q^{a}$ coming, say, $n$ seconds after or before another
configuration $q^{b}$. However, this ordering is not the weaker one encoded in
a parametrization of a curve either, which can be rendered in terms of an
“earlier than” relation. Such a relation has, in fact, a global character that
is absent in this context. Simply speaking, given any two points on the curve,
it is impossible to say which one comes “earlier” without a parameter that
assigns a monotonically increasing numerical flag to each of them. On the
other hand, the fact that $\phi^{A}$ is defined in a neighborhood of $q^{a}$
and depends on the kinematic metric $g_{ab}$ (which measures how much two
shapes differ from each other) through (2) allows us to conceive of the
fundamental dynamical ordering as just a minimal topological ordering where
the notion of nearness of configurations is rendered in terms of a
_similarity_ relation (intended as lack of distinctness; see §4.2). According
to this topological ordering, the most we can say is whether, e.g., a
configuration $q^{b}$ is “in-between” $q^{a}$ and $q^{c}$: This happens just
in case all neighborhoods of $q^{a}$ that include $q^{c}$ also include
$q^{b}$.

It is then clear that it is challenging to _globally_ characterize the
“timelike” or dynamical part of the mosaic in terms of _succession_ or
_change_ in the configuration. Our metaphysics simply does not allow for
anything like that at the fundamental level, allowing only for a “local”
counterpart of change. The way out of this impasse is, first of all, to
remember that a model of PSD represents a possible world, and then to
recognize that such a possible world features the physically realized
configurations represented by the corresponding solution of (1) given _all at
once_ in a timeless and (almost) changeless sense. It is worth reiterating
that this does not imply that the set of physically realized configurations is
entirely unordered. On the contrary, there is still a weak, local, topological
ordering of such configurations that makes it possible for the mosaic to be
described using an unparametrized curve in shape space. To sum up, our
metaphysical take on the PSD framework is the following: The framework depicts
a cluster of physically possible worlds (models), each of which consists of a
Humean mosaic of self-subsisting structures “timelike”-arranged in a weak
topological ordering that can be described in the simplest and most
informative way by an unparametrized curve in the corresponding shape space.

It is important to note at this point that, recently, a similar metaphysical
take on physics has been proposed, which is usually referred to as _Super-
Humeanism_ (see Esfeld et al., 2018, for the standard textbook on the
subject). The Super-Humean metaphysics is relationalist, structuralist, and
parsimonious in the same vein as ours, but it differs in some essential
respects. First of all, Super-Humeanism is detached from any particular
theoretical framework. Indeed, the main Super-Humean tenet is that the laws of
any physical theory can be shown to supervene on a mosaic of permanent
material points related by ever-changing distance relations. The fact that the
fundamental picture of reality proposed by Super-Humeanism does not depend on
physics is worrisome for many metaphysicians because it seems to deprive
fundamental physics of much of its explanatory power (see Wilson, 2018, for an
argument in this sense). Indeed, Super-Humeanism seems to imply that physics
provides no key insight into the nature of reality. Consequently, physics is
threatened to be demoted to some sort of formal machinery that succinctly
describes the very complicated motions of fundamental material particles.

On the other hand, our metaphysics does depend on the particular physical
domain considered for the characterization of both relations and relata making
up a self-subsisting structure. Hence, as already pointed out, a “Newtonian”
self-subsisting structure is different from a “dynamical geometry” one in many
ontological respects that also have observational consequences. There is, of
course, an important sense in which some structural aspects of the framework
are model-independent, but this sense does not deprive physics of any
explanatory power. On the contrary, it accounts for why some empirically
adequate concepts are carried over in the development of physics. In other
words, our metaphysics is indeed sensitive to the development of physics, to
the point that we expect to radically revise the notion of self-subsisting
structure at the quantum-gravitational regime (see §5). This is enough to
defuse the charge of adopting what Lazarovici (2018, p. 81) calls “Super-
Humean subterfuge”:

> Our physical description of the world exhibits the feature $X$ because the
> contingent relational distribution of matter throughout the history of the
> universe happens to be such that the best system description exhibits the
> feature $X$.

As already pointed out, there are central features of our metaphysics that are
not related to the mosaic in the way mentioned above but are instead entirely
determined by the physics—e.g., the Euclidean nature of the spatial relations
making up a Newtonian self-subsisting structure.

Another essential difference between our metaphysics and Super-Humeanism
resides in the nature of the spatial relations postulated. For the Super-
Humeans, the distance relations making up a configuration of matter points
carry an extremely weak spatial connotation, which boils down to fulfilling
the triangle inequality (see Esfeld et al., 2018, p. 22, for a list of
requirements that any numerical assignment that coordinatizes a configuration
has to obey). This, in turn, makes it highly improbable for a very large
configuration of particles—like the one making up our universe—to be most
simply embeddable in a space as low dimensional as a $3$-dimensional
Riemannian manifold. Indeed, this would require such a configuration to obey
by sheer chance an extremely high number of geometric constraints (see, again,
Lazarovici, 2018, p. 82, for a rough estimate of how improbable this might
be). On the other hand, our self-subsisting structures do not fall prey to
this kind of “un-typicality” argument. This is because they retain a much more
robust geometric structure—mathematically represented by a conformal
structure—which makes them straightforwardly embeddable in a low dimensional
space (e.g., in the case of classical and relativistic motions). All we have
to do to perform this embedding is to “reverse” the quotienting out procedure,
thus getting back the starting embedding space (recall the discussion of the
ephemeris equations in §2.4.2). Of course, our reliance on a primitive
conformal structure renders our metaphysics less parsimonious than the Super-
Humean one, but we are more than happy to pay this price if, in return, we get
a more plausible mechanism that accounts for how we get from self-subsisting
structures to ordinary space.

###  4.2 A Supervenience Basis for Space and Time

The above discussion clarifies the sense in which ordinary space supervenes on
self-subsisting structures—at least, in the non-quantum case. In this picture,
space is reduced in the strong mathematical sense encoded in the quotienting
out procedure to the spatial relations making up a self-subsisting structure.
This is how our metaphysics conforms to the Leibnizian motto that “space is
the order of co-existing things.” Likewise, it is straightforward to recover
standard space from a self-subsisting structure. This is done by reversing the
quotienting out procedure that sets the shape space of the system under
scrutiny, thus getting the standard picture where the _relata_ in the
structure are now individual objects occupying a place in ordinary space (or
field magnitudes making up a $3$-dimensional geometry in the dynamical
geometry case). We have seen a concrete example where such a reverse procedure
is implemented, i.e., the construction—which makes use of (16)—of an ephemeris
scale from the purely intrinsic properties of a dynamical curve in shape
space. In this case, the construction makes use of the shape potential, which
can be reduced to facts regarding the structure of shapes.

However, this markedly reductionist attitude seems to face troubles when the
task becomes making sense of the notions of time and change commonly
used—especially in scientific practice. In a nutshell, we can put the
challenge in the following way: How are we to recover richly structured
notions like that of an arrow of time from the very faint-structured mosaic on
which (1) supervenes? This is a particularly delicate question, given that the
weak topological ordering among configurations postulated at the fundamental
level makes it _prima facie_ implausible to recover a picture of time as an
“abstraction from change”—borrowing the famous characterization given by Ernst
Mach. As we shall see in a moment, meeting this challenge involves some
conceptual work, but it is by no means unfeasible. The strategy we will employ
is similar to the spatial case, and amounts to showing how facts about time
and change supervene on—in the strong sense of “are reducible to”—the minimal
set of fundamental relational facts inhering into the PSD mosaic. More
precisely, we will couch the construction in terms of layers of description:
The fundamental basis (i.e., the level-$0$ domain) featuring relational facts
will be used to construct progressively more complex temporal notions. In
order to keep the paper at a reasonable length, we cannot delve too deep into
the conceptual subtleties involved in the construction. However, we submit
that the sketch we are going to offer is sufficient to establish the
feasibility and consistency of the program.

Before going through this construction, a remark is in place. One may wonder
why, in our framework, recovering temporal notions should be more tricky than
recovering spatial ones. The answer is quite easy: Because the framework
accords a privileged ontological status to space as opposed to time. More
precisely, self-subsisting structures possess a fundamental—albeit
weak—spatial connotation in terms of the conformal structure inhering into
them. From this point of view, as already said, recovering ordinary
(instantaneous) space from a self-subsisting structure $\mathfrak{C}$ just
amounts to embedding $\mathfrak{C}$ in the space that conveys the physical
information encoded in $\mathfrak{C}$ in the simplest and strongest way. In
this sense, via the embedding procedure, a self-subsisting structure becomes a
“universal snapshot” of, say, a Newtonian $N$-particle world. Clearly, such a
straightforward construction is not available in the temporal case. For
starters, the embedding story should be applied to the sequence of self-
subsisting structures constituting a possible world. However, we already know
that this fundamental level—represented by an unparametrized curve in shape
space—has no temporal connotation whatsoever, so some additional consideration
has to be added on top of the geometric construction in order to show that
some sort of temporal structure “appears” by virtue of carrying out said
construction. Secondly, even granted that the timelike part of spacetime can
be recovered in this way, still this construction would not account for the
“appearance” of temporal notions that are not encoded in the spacetime picture
(case in point, the notion of temporal passage). This remark justifies why, in
our discussion, we are placing a particular emphasis on how time and change
can be recovered in the PSD framework.

Let us now return to the discussion on the supervenience of temporal concepts
in terms of levels of description. As already said, the level-$0$ domain is
the mosaic itself. At this level, all the configurations making up a possible
world—represented by a curve fulfilling (1)—are just given at once as a brute
fact of nature. This means that the distinctness of such configurations is
assumed as a primitive concept. This level-$0$ domain features fundamental
facts regarding the “nearness” or “similarity” of neighboring configurations,
making it the case that an unparametrized curve can describe these
configurations in shape space. Facts regarding the similarity of neighboring
configurations are also given as primitive. Remember from the discussion in
§4.1 that this fundamental level just encodes facts about how self-subsisting
structures are locally (i.e., topologically) ordered, which in particular
means that there is no fact about a preferred direction of such ordering.
Thus, this undirected ordering of self-subsisting structures resembles what,
in the standard debate on the metaphysics of time, is known as a “C-series”
ordering (this concept was famously introduced in McTaggart, 1908, p. 462; see
also McDaniel, 2020, §3, for a nice historical discussion of McTaggart’s
argument against the reality of time). To be fair, this resemblance should be
taken _cum grano salis_ since, in the standard debate, time series are usually
understood as orderings among events or ordinary facts, not (facts about)
physical structures in the sense we adopt.

From this set of primitive facts obtaining at a world, more structured notions
can be shown to supervene. Thus, the level-$1$ domain involves a global
ordering over the set of configurations actualized at a world. This global
ordering is reminiscent of the “B-series” ordering in terms of an “earlier
than” relation. Such an ordering is required to make sense of ordinary change,
but it is not “there” in the mosaic, i.e., it is not a fundamental feature of
reality; rather, it can be shown to supervene on the mosaic through
mathematical reduction. Indeed, this level-$1$ ordering can be constructed by
arbitrarily fixing a reference configuration and then by assigning a
“numerical flag” to the neighboring configurations based on facts regarding
their degree of similarity to the starting one. In this way, the intrinsic
structure of shapes is exploited to establish an ordering that can be
described in terms of an arc-length parametrization of the curve (with respect
to, e.g., the kinematic metric on shape space). Thanks to this further
structure, we can traverse the whole curve as the parameter values increase—or
decrease—and from this recover the usual talk of spatial relations “changing”
from one configuration to the other as the curve is traversed. Note how this
connotation of change is not primitive but given in terms of primitive
distinctness and similarity.

This setting also permits—at least in well-behaved cases—to establish a notion
of “identity over configurations” for both relations and _relata_. Take, for
example, the case of the Newtonian $N$-particle system. In this case, each
shape consists of $N$ particles related through Euclidean spatial relations
but, according to the metaphysics we put forward, there is no fundamental fact
of the matter about the identity of the particles and relations making up a
shape being carried over to the other configurations: Each self-subsisting
structure has “its own” particles and relations. By using the arc-length
parametrization to traverse the curve, we automatically relate particles and
relations in subsequent configurations by establishing that moving from one to
the other represents the change in the spatial configuration of a _unique_ set
of $N$ particles linked through Euclidean spatial relations. Pictorially
speaking, this is similar to animating a picture by quickly showing a series
of frames ordered in terms of their similarity. Obviously, each frame shows a
picture different from the others, but by showing them in the specific order
induced by their similarity, we gain the impression of a unique subject moving
(i.e., changing its spatial configuration in time). One may object that this
animation analogy sneaks in some decidedly temporal notion in the discussion:
After all, animations unfold _in time_. Our reply is that this objection gets
things backwards, so to speak: It assumes that time is required to make sense
of change, whereas our construction is exactly meant to show that nothing over
and above a global and directed ordering among self-subsisting structures is
needed to make sense of change as a succession of “frames.” As Savitt (2002,
p. 163) puts it: “We do not need an animated picture to have a picture of
animation.”

Furthermore, by establishing a measure of “how much structure” a configuration
contains—as represented, in well-behaved Newtonian cases, by the complexity
function (3)—it is moreover possible to establish a notion of duration. This
notion is arrived at by constructing a temporal metric that, in the non-
relativistic case, gives precisely the Newtonian time $t$ (recall the
construction of the ephemeris duration equation (17) sketched in §2.4.2). This
is the level-$2$ domain of time, which accounts for the physical appearance of
clocks marching in step. Also in this case, it is worth noting that there are
no fundamental facts about a global metric of time crafted in the mosaic but,
instead, facts about duration supervene on primitive facts about shapes’
complexity through an arbitrary—albeit natural, in many cases—mathematical
construction such as that of equation (17). The construction of this second
layer is based on the dynamical formation of stable records. The assessment of
the stability of such records—which are nothing but clusters of particles in
the classical case—is made possible by invoking the notion of identity over
configurations that comes at the level-$1$ domain. So, for example, the talk
of clocks in the Newtonian case turns out to be a concise yet informative way
to referring to the dynamics of the Kepler pairs mentioned at the end of §2.3.

Finally, the level-$3$ domain of time, that is, the appearance of a directed
flow of time, is accounted for in terms of the secular growth of the
complexity function (3). More concretely, we can establish a notion of
directed temporal passage by individuating a certain starting self-subsisting
structure and then “trace” its development through the dynamical curve (again,
by exploiting level-$1$ notions). Simply speaking, this “present”
configuration moves away from the past and into the future following the
secular growth of complexity inherent into the curve. This is how an
“A-series” of time is recovered in this context. Of course, such a
construction is entirely arbitrary—in the sense that there is no genuine fact
about something being past, present, or future—and is meant to show how a
tensed description of a physical process can be achieved in our framework.

The layered structure sketched above can be summarized using the following
diagram:

Level-$3$: Arrow of time, temporal passageLevel-$2$: Clocks marching in
stepLevel-$1$: Global change, identity over configurationsLevel-$0$: Facts
about self-subsisting structures$\scriptstyle{(\text{Secular growth of
complexity})}$$\scriptstyle{(\text{Ephemeris duration}\text{, stable records
formation})}$(Parametrization)

As already stressed several times, only the bottom level of the diagram is
concerned with the ontology; the subsequent levels deal with the description
of such underlying reality. Also, note that this reconstruction of change,
duration, and an arrow of time is meant to show how the level-$0$ mosaic
grounds these notions as used in physics (and in scientific practice in
general). Of course, a more detailed story should be provided as to how we get
to _perceive_ change and the directed passing of time. This task goes far
beyond the scope of the present paper and involves explaining how minds emerge
as subsystems of self-subsisting structures and how they get correlated with
the fundamental facts grounding the higher-level temporal notions
characterized above. This is undoubtedly an interesting future line of
research.

To conclude this section, we point out that there can be possible worlds in
which the amount of structure formation inherent into the mosaic is not enough
to ground the emergence of level-$3$, or even level-$2$ temporal structures:
Think, for example, of a universe eternally in thermal equilibrium. In such
worlds, it makes no sense to speak of clocks, let alone a preferred direction
for the passing of time.

##  5 Conclusion: Pure Shape Dynamics in Perspective

PSD represents a natural evolution of the relational framework originally put
forward by Barbour and Bertotti. It promises to deliver a complete
geometrization of the dynamics, which eschews from the physical picture any
remnant of non-intrinsic characterization of the evolution of a system. In
this sense, the most remarkable feature of PSD is that it does not need any
notion of parametrization to make sense of the dynamics. We can hence talk of
_strong_ temporal relationalism, as opposed to the weaker type that does
require a notion of monotonically increasing parameter to establish a way in
which the dynamics unfolds.

Furthermore, the PSD’s framework is all-encompassing in that it can, in
principle, accommodate relational counterparts of physical models pertaining
to classical, relativistic, and quantum physics (this latter case being the
focus of a companion paper). Such inter-theory relation between standard and
relational models is rendered in terms of a “quotienting out” procedure that
eliminates all the non-relational degrees of freedom from the former models,
thus generating the relational ones of PSD.

From a metaphysical perspective, we have argued that the PSD framework
naturally goes hand in hand with an ontic structural realist and Humean take
on the laws of physics, which means that PSD is a theory of self-subsisting
structures—i.e., structures characterized wholly intrinsically. As we have
pointed out before, however, structuralism alone is not enough to fully
establish the ontological nature of the relations and _relata_ entering a
model of PSD. Both relations and _relata_ then have to be specified via a
reconceptualization of the ontology of the starting non-relational models.
This need for a reconceptualization implies that physics does have a say in
our metaphysical framework, differently from the Super-Humean approach that
fixes _a priori_ the ontology of physics. The clearest example of
reconceptualization is represented by the case of material particles moving in
a background space in classical mechanics, which, in our structuralist and
Humean reading of PSD, become a Leibnizian structure with identityless and
propertyless _relata_ individuated solely through the Euclidean spatial
relations they stand in.

We have also clarified that the models of PSD should be taken at face value
only insofar as their (empirically adequate) relational aspects are concerned,
thus considering their complete ontological characterization as physical
_possibilia_ rather than faithful pictures of how the actual world
fundamentally is. This begs the question as to which sector of PSD we should
then take to deliver the set of truly fundamental models, i.e., those that
fully and adequately describe the actual world. An answer to this question
depends on whether PSD may eventually succeed as a “final” physical theory.
This represents a significant challenge to the future developments of the
framework for many reasons, both technical and conceptual.

Let us start by pointing out that such an advanced sector of PSD is likely to
account for the quantum _and_ gravitational aspects of the dynamics. Otherwise
said, the quantum models of PSD and the dynamical geometry ones are expected
to be some sort of low energy and long-range limits or approximations of these
yet-to-be worked out fundamental models of the theory. One may be then tempted
to classify this fundamental sector of the theory as the “quantum-
gravitational” one. However, that would be misleading because, in PSD, there
are no genuinely “quantum” or “gravitational” features of the physical
description of the world: Everything is reduced to geometric degrees of
freedom of dynamical curves on shape space (which in turn are just a simple
yet informative way to refer to the mosaic made of self-subsisting
structures). Indeed, the companion paper will discuss the possibility of
“geometrizing away” the wave function of a system and its associated
transition amplitudes. If such a radical geometrization of physics is viable,
there will then be a clear sense in which this hypothetical fundamental sector
of the theory represents a pre-quantum-gravitational reality from which both
the quantum and the gravitational aspects of physics appear in the appropriate
limit or approximation. This highlights how the PSD framework is a potentially
novel approach to quantum gravity that dispenses with well-known issues
related to the quantization of the gravitational field.

The above remark shows how formidable it is the task of working out such a
fundamental sector of PSD. Indeed, while the non-fundamental sectors of the
framework can be constructed as quotiented out versions of existing non-
relational theories, such a strategy is not available in the fundamental case
for two reasons. First, there is no “final” theory of quantum gravity as yet
and, second, it would be improbable that such a theory comprised
spatiotemporal background structures (to the contrary, such a theory should
exactly be about how it comes that background independence is a fundamental
feature of reality). Hence, the development of this sector of PSD should be
carried out “on its own,” so to speak. This challenge opens up a plethora of
conceptual questions, one of the most pressing being: What would become of
self-subsisting structures in such a fundamental regime if, as it is very
likely, no spatial or also quantum notions were available? In other words,
what would it be the relation making up these structures if they were not
accounted for in terms of spatial or even entanglement relations? Furthermore,
what would make these relations _physical_ as opposed to merely mathematical
constructions if they could not be characterized in any of the above ways?

In short, the PSD framework represents a largely unexplored territory that is
full of promises but, at the same time, gives no guarantees as yet of total
success. However, in the present paper, we hope to have convincingly shown
that the PSD project—in both its physical and metaphysical aspects—is worth
pursuing because it provides new perspectives on current physics and promises
to deliver a novel take on the problem of quantum gravity.

Acknowledgements

We thank an anonymous reviewer for the very helpful comments on an earlier
version of this manuscript. AV and PN gratefully acknowledge financial support
from the Polish National Science Centre, grant No. 2019/33/B/HS1/01772.

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