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# From Humean Laws to a Neo-Kantian Spacetime:A Dynamics-First View of
Topology

Daniel Grimmer  Faculty of Philosophy, University of Oxford, Oxford, OX2 6GG
United Kingdom
Reuben College, University of Oxford, Oxford, OX2 6HW United Kingdom

###### Abstract

Do the spacetime manifolds which feature in our best scientific theories
reflect anything metaphysically weighty in the world (e.g., any fundamental
substances or relations)? Should we extend our notions of space and time
beyond the epistemological roles they play in helping us codify the dynamical
behavior of matter? Kant famously answered “No” to both of these questions,
contra Newton and Leibniz. This paper introduces novel technical and
philosophical support for such a (Neo-)Kantian perspective on the metaphysics
of space and time. To begin, I will make an explicit analogy between broadly
Humean views of laws (e.g., Lewis, Demarest, etc.) and dynamics-first views of
geometry (e.g., Brown). I will then continue this line of analogous views
beyond the metaphysics of laws debate and the dynamical vs geometric spacetime
debate by extending it into the context of spacetime topology. Namely, I will
put forward a dynamics-first view of topology (in answer to Norton’s problem
of pre-geometry).

This dynamics-first view of topology is supported by some powerful new
techniques for topological redescription which I have recently developed in
Grimmer (2023a) and Grimmer (2023b), namely the ISE Method. These new
techniques allow us to remove and replace the topological underpinnings of our
spacetime theories just as easily as we can switch between different
coordinate systems. For instance, a theory set on a Möbius strip might be
redescribed as being set on the Euclidean plane and vice versa. Indeed, as
Grimmer (2023b) has proved, the ISE Method gives us access to effectively
every possible spacetime framing of a given theory’s kinematical and dynamical
content. Given this overabundance of candidate spacetime framings, it is then
conceivable that one can pick out a theory’s topological structure via
something analogous to a Best Systems Analysis.

##  1 Introduction

What, if anything, can help us explain the dynamical behavior of
matter?111This paper considers only metaphysically supported explanations,
i.e., explanations which require that we grant some metaphysical weight to
their explanans. The central issue in this paper is whether or not one should
grant any metaphysical weight to various theoretical structures (e.g., laws,
coordinates, geometry, topology, etc.) in hopes of giving a metaphysically
supported explanation of the dynamical behavior of matter. Dynamics-second
views do this whereas dynamics-first views do not. Instead, dynamics-first
views offer us a metaphysically supported explanation that is rooted in
dynamics, e.g., the laws of nature are what they are ultimately because of the
dynamical behavior of matter, not vice versa. On this view, the laws might
still help us “explain” dynamics but not in a metaphysically supported way.
For instance, Newton’s law of gravity provides us with a unified account of
the motions of apples and planets without giving us a metaphysical picture of
how gravitation works. This would be an explanation by unification, see
Kitcher (1989).  Allow me to first demonstrate several candidate explanans by
considering four example questions about dynamics (whose answers I will later
challenge). Q1: Why did that apple fall? A1: Because of the laws of gravity.
Q2: Why does Foucault’s pendulum rotate? A2: Because the Earth’s lines of
latitude and longitude do not form an inertial coordinate system. Q3: Why do
all kinds of clocks exhibit time dilation no matter what material they are
made of? A3: Because the geometric structure of spacetime is Lorentzian. Q4:
On a spacetime manifold with spherical topology, why do our measuring rods
always measure the average spatial curvature to be positive no matter the
metric field? A4: Because the topological structure of this spacetime manifold
has an Euler characteristic of $\chi=2$ (this then has implications for
curvature by the Gauss-Bonnet theorem). Thus, we can identify four candidate
explanans for dynamics: laws, coordinates, geometry, and topology.

But how have we come to know anything about these candidate explanans? The
story goes roughly as follows: Scientists have carefully studied the
observable dynamics of matter (e.g., the motion of certain needles in a
laboratory). Then, after a significant amount of work, they have codified
these observations into our current best scientific theories. Notably, these
theories will typically refer to matter and dynamics beyond those which were
directly observed (e.g., they may refer to electrons, currents, magnetic
fields, etc.). Moreover, it is common for these theories to feature additional
structures which do not refer to either matter or its dynamical behavior.
Namely, they might feature laws of nature and/or special coordinate systems.
They might also make use of some fixed geometric and topological structures
(e.g., the fixed Lorentzian metric, $\eta_{\text{ab}}$, and spacetime
manifold, $\mathcal{M}$, appearing in special relativity). Notice that these
four additional theoretical structures are exactly our four candidate
explanans.

Hence, we know where our knowledge of these four candidate explanans comes
from. Regardless of whether these theoretical structures can or cannot help us
explain dynamics, we can all agree that we have learned about them by studying
and codifying the dynamical behavior of matter. Let us call this the arrow of
knowledge generation. It is after this epistemological point of agreement,
however, that some metaphysical disagreements may arise regarding the
direction of the arrow of explanation. In each of the four contexts (e.g.,
laws, coordinates, geometry, and topology) one might take the arrow of
explanation to run antiparallel to the arrow of knowledge, see Fig. 1b).
Namely, one might follow A1-A4 in appealing to these theoretical structures in
order to explain dynamics. Alternatively, however, one might reverse the arrow
of explanation so that it runs parallel to the arrow of knowledge, see Fig.
1c).222Some have tried to find a middle ground between the dynamics-first and
dynamics-second approaches by claiming that the arrow of explanation is bi-
directional. For instance, Acuña (2016) claims that Minkowski spacetime
structure and Lorentz invariant dynamics are two sides of a single coin. That
is, one might explain these theoretical structures in terms of dynamics, e.g.,
the laws of nature are what they are ultimately because of the dynamical
behavior of matter, not vice versa.

Figure 1: This figure shows a schematic of Plato’s epistemology and
metaphysics (a), as well as schematics of a dynamics-second view (b) and a
dynamics-first view (c). For each kind of view, this figure shows the
direction of the arrow of knowledge generation as well as the direction of the
arrow of metaphysically supported explanation. Plato’s view has the Forms as
the ultimate source of both knowledge and metaphysical explanation. Dynamics-
second views continue with Plato’s style of metaphysics while rejecting his
epistemology in favor of empiricism. By contrast, for dynamics-first views
additionally reverse the arrow of explanation. As I discuss in the main text,
this move is of an unofficially Humean character and ultimately allows for a
sparser metaphysics.

In the next section, I will unite the pre-existing debates about the
metaphysics of laws, coordinates, and geometry into a single framework, which
I will call the dynamics-first vs dynamics-second debates. While these three
debates are already well-established individually, a detailed discussion of
their similarities is conspicuously missing from the existing literature.
However, the following section does more than just provide a novel perspective
on the relevant literature. My ultimate goal is to extend these debates beyond
the contexts of laws, coordinates, and geometry, into the context of spacetime
topology. In fact, I will put forward a dynamics-first view of topology in
Sec. 5 (with the help of Secs. 3 and 4). This new view is intended to be
closely analogous to the dynamics-first views of laws, coordinates, and
geometry discussed below. Being so analogous, these four dynamics-first views
share in their motivations and lend each other mutual support. Since my
argument for this dynamics-first view of topology relies upon this analogy,
some substantive philosophical work is first needed in unifying these
dynamics-first vs dynamics-second debates.

##  2 Four Dynamics-First vs Dynamics-Second Debates

According to what I will call dynamics-second views (of laws, coordinates,
geometry, topology, etc.), our theoretical investigations of matter’s dynamics
have revealed to us the metaphysical substructure of the world (e.g., its
nomological, geometric, topological substructures, etc.). Moreover, these
revealed substructures are taken to have enough metaphysical weight that we
can then turn around and use them to help give a metaphysically supported
explanation of the very dynamics which we began investigating. See Fig. 1b).
For instance, the governing view of laws is a dynamics-second view:333More
generally, dynamics-second views of laws are roughly what Ott (2022) calls the
“top-down” views in the metaphysics of laws debate: e.g., Cartesian
occasionalism, see Ott (2009, Ch. 9); primitivism, see Ott (2022, Ch. 4); and
the Dretske–Tooley–Armstrong conception of laws as relations between
universals, see Armstrong (1983). It claims that by investigating some
dynamical phenomena we can discover the world’s nomological substructures
(i.e., the laws of nature) which in turn govern those very phenomena.

Relatedly, the following story provides the outline for a dynamics-second view
of geometry:444The dynamics-second views of geometry are on the geometrical
side of the geometrical vs dynamical spacetime debate. This largely orthodox
view is implicit in Friedman’s textbook (1983, Ch 6 Sec 4). It can be seen
more explicitly in Maudlin (2012); Janssen (2009); Norton (2008); Dorato
(2007). By studying the inertial behavior of measuring rods and clocks we can
discover the world’s geometric substructure (e.g., a Lorentzian metric,
$\eta_{\text{ab}}$). In turn, this geometric substructure then serves to
constrain the symmetries of all dynamics and thereby helps us to explain the
very phenomena under consideration. Finally, a dynamics-second view of
topology could be built around the following story: By studying and codifying
the dynamical behavior of matter we can discover the world’s topological
substructure (e.g., a spacetime manifold, $\mathcal{M}$). In turn, this
topological substructure then gives the dynamics a spatiotemporal arena to
happen within and thereby helps us explain this dynamics.

This dynamics-second explanatory strategy can be traced all the way back to
Plato. For Plato, the world’s metaphysical substructures were the Platonic
Forms and these were taken to be the ultimate explanans for the physical
world, see Plato (1997). Hence, like the dynamics-second views, Plato’s arrow
of explanation also points toward dynamics from some metaphysically weighty
substructures. The key difference, however, is that for Plato the arrow of
knowledge also points towards dynamics from these substructures. Namely,
according to Plato, we don’t learn about the Forms by studying the physical
world; Instead, we learn about the physical world by recognizing it as an
imitation of the Forms which we can recall from having previously interacted
with them directly, see Plato (1980). Compare Fig. 1a) and Fig. 1b). Note that
dynamics-second views accept Plato’s style of metaphysics while rejecting his
epistemology in favor of empiricism.

Some philosophers (myself included) find the metaphysical substructures
invoked by such dynamics-second explanations to be excessively heavy. A
metaphysically lighter way forward is to reverse the arrow of explanation,
putting dynamics first, so to speak. See Fig. 1c). On such dynamics-first
views (of laws, coordinates, geometry, topology, etc.) one still codifies the
dynamical behavior of matter into theories that can feature nomological,
geometric, and/or topological structures. At this point, however, one does not
grant the relevant theoretical structures any metaphysical weight of their
own, nor does one attempt to use them as explanans for dynamics. Instead, one
limits these additional theoretical structures to the epistemic roles which
they play in helping us to codify dynamics. To see some examples of such
dynamics-first views, let us first look to the metaphysics of laws
debate.555What I am calling the dynamics-first vs dynamics-second debate about
laws is, of course, a well-established debate within the metaphysics of laws
literature. For an overview of this debate see the SEP article by Carroll
(2020) among many others: Demarest (2017); Lewis (1973, 1983, 1999); Bhogal
(2020); Hall (2015); Ott (2022); Armstrong (1983); Bird (2007), and Schaffer
(2016).

###  2.1 Dynamics-First Views of Laws

Humeanism about laws is a good example of a dynamics-first view of laws. It
claims (among other things) that the laws which appear in our best theories
reflect nothing metaphysically weighty in the world beyond them being a
particularly nice way of codifying the dynamical behavior of matter.666As I
will discuss at the end of Sec. 2.1, different dynamics-first views of laws
are allowed to vary regarding what the term “particularly nice” means exactly.
Reversing the arrow of explanation in this way allows us to metaphysically
deflate the laws and thereby avoid the following tricky metaphysical
questions. How is matter supposed to “know” what the laws are so that it can
behave accordingly? What gives the laws the “nomic voltage” required to govern
matter? How exactly do the laws govern matter? By pushing it around, so to
speak? If so, then why can’t the matter push back on the laws? Such questions
are ubiquitous throughout the metaphysics of laws debate.

As a rejoinder, a proponent of the dynamic-second view of laws might complain
that, sans metaphysically weighty laws, the Humeans now have no explanation
whatsoever for dynamics. Isn’t science supposed to (among other things)
explain the dynamical behavior of matter? This is a valid complaint against
the canonical Humean view since it does take dynamics to be unexplained (i.e.,
it accepts the orderliness of the Humean mosaic as a brute fact). This,
however, is not a necessary feature of dynamics-first views. Indeed, another
notable dynamics-first view of laws is the generalization of Lewis’s (1973;
1999) Best Systems Analysis (BSA) by Demarest (2017). This view allows for
causal powers and dispositions to exist on the Humean mosaic, see Fig. 2. On
such a view the laws are then the theorems/axioms of whichever theory best
systematizes this arrangement of dispositional and categorical properties.
Importantly, on Demarest’s (2017) view, the orderliness of the Humean mosaic
is not accepted as a brute fact; Instead, it is explained in terms of the
causal powers inherent in matter itself. These causal powers are then accepted
as primitive. As this example shows, dynamics-first views can explain dynamics
in terms of yet-more fundamental dynamics (which itself remains unexplained).

Figure 2: As an illustration of a Humean mosaic, this figure shows the
painting “The Great Wave off Kanagawa” by Hokusai (1830) reproduced on a wall.
On a canonical Humean view of laws (e.g., Lewis (1973, 1999)) one imagines
that every bit of paint is completely independent of every other bit of paint
(i.e., there are no necessary connections between distinct existences). By
contrast, Demarest’s (2017) view allows us to imagine that these metaphorical
bits of paint have some causal powers over each other. On either view, the
laws are to be read off of the mosaic via a Best Systems Analysis. Regarding
Regularity Relationism (discussed in Sec. 2.3) one imagines peeling all of the
paint off of the wall in a single topologically connected sheet. The fact that
we can then freely bend and stretch this topological sheet shows us that the
Humean mosaic does not require any geometric underpinnings.

It should be noted that Demarest’s (2017) BSA abandons the canonical Humean
idea that there are no necessary connections between distinct existences. As
Bhogal (2020) observes, however, her view is still Humean in the following
“unofficial” sense:

> One common motivation [for Humeanism about laws] comes from the way that the
> defender of the BSA claims that their view mirrors scientific practice
> (e.g., Demarest (2017)). The BSA sticks very closely to the actual practice
> of science—in fact, one way to describe the view is that it takes the actual
> methodology of science and it mirrors that methodology in the account of the
> metaphysics of laws. Hall (2015) calls this the “unofficial guiding idea”
> behind Humeanism about laws.777Bhogal (2020, pg. 8)

That is, the common feature shared by Demarest’s (2017) BSA and more canonical
Humean views is exactly the above-discussed paralleling of the arrow of
metaphysically explanation with the arrow of knowledge generation. As I will
argue throughout this paper, the benefits of aligning these two arrows extend
beyond the metaphysics of laws debate. All dynamics-first views make use of
this alignment and are therefore “unofficially” Humean. In every context, this
alignment is favorable since we are generally on a more secure footing when we
explain the less-directly-known in terms of the more-directly-known rather
than vice-versa. Isn’t science supposed to (among other things) explain the
world’s structural features in terms of the dynamical behavior of
matter?888Moreover, isn’t science supposed to explain the non-fundamental
dynamics of matter in terms of its more fundamental dynamics? Both here and in
the “structural” question in the main text, by “explanation” I mean
metaphysically supported explanation, see footnote 1.

Before moving on to the other three contexts allow me to make explicit how a
capacity for nomological redescription can be leveraged in support of a
dynamics-first view of laws. A distinction must first be made between axiom-
like laws of nature and theorem-like laws (i.e., derived laws). Throughout
this paper, my focus is exclusively on axiom-like laws since the dynamics-
second views only seek to grant metaphysical weight to axiom-like laws.
(Granting nomological force to theorem-like laws as well as axiom-like laws is
clearly redundant). This focus on axiom-like laws should be contrasted with
other parts of the metaphysics of laws literature. For instance, Lewis (1973,
1999) takes the laws to be (a subset of) the theorems of whichever theory best
systematizes the Humean mosaic.

This brings us to the typical starting place for Humean views of laws: the
Humean mosaic. Given a single Humean mosaic, one’s task is typically to read
the laws of nature off of it by creating a theory about this single mosaic
from evidence available on this single mosaic. To my mind, however, proceeding
in this way conflates two distinct processes: 1) the process of constructing
an initial theory, and 2) the process of selecting the best axiom-like laws
for this theory. For now, let us focus on the second process. Moreover, let us
temporarily adopt a syntactic view of theories such that the axiom-like laws
of a given axiomatized theory are simply (a subset of) the axioms of that
theory. Now suppose that by some unspecified process one arrives at an
empirically successful theory which is stated in first-order logic in terms of
some (perhaps many, perhaps complicated) axioms. What capacity for re-
axiomatization do we have such that we can find some better axioms for this
theory?

First notice that re-axiomatizing a theory does not change any of its logical
consequences. This merely amounts to a nomological redescription of our
theory. Moreover, note that by applying standard deductive logic techniques
one can access every possible nomological description of one’s theory. Given
this overabundance of nomological descriptions, one finds some motivation to
metaphysically deflate the laws and to focus instead on the theory’s law-
independent content (e.g., its models viewed as Humean mosaics). From here one
can then search for a good axiomatization of the theory (i.e., a good
nomological description). Viewed from this law-neutral perspective, whichever
axiom-like laws one ultimately settles on will reflect nothing metaphysically
weighty in the world beyond them being one particularly nice way (among other
options) of codifying the dynamical behavior of matter.

It is important to note that past making this metaphysical claim about the
laws, dynamics-first views of laws are allowed to vary regarding a wide
variety of further epistemological questions about the laws. What role does
the notion of laws of nature play in the initial process of theory
construction? How do we first come to have a nomological picture of the world?
Given our capacity for nomological redescription, what guides us in selecting
the axiom-like laws for our theories? That is, what does the term
“particularly nice” mean exactly? Is the law-selection process objective and
mind-independent, making the laws real (despite them being metaphysically
deflated)? Or, alternatively, is there some non-trivial element of
conventionality to it? As I will discuss in Sec. 3, certain answers to these
epistemological questions may have interesting metaphysical consequences.

###  2.2 Dynamics-First Views of Coordinates

Let us move on from the metaphysics of laws debate to the next context:
coordinate systems. Suppose that after investigating the inertial behavior of
a scalar field, $\varphi(t,x,y,z)$, and a point-like particle with trajectory,
$X(\tau)=(t(\tau),x(\tau),y(\tau),z(\tau))$, one arrives at the following
theory written in some fixed global coordinate system, $(t,x,y,z)$:

$\displaystyle(\partial_{t}^{2}-\partial_{x}^{2}-\partial_{y}^{2}-\partial_{z}^{2}+M^{2})\,\varphi(t,x,y,z)$
$\displaystyle=0,$ (1) $\displaystyle(\partial_{\tau}^{2}t,\
\partial_{\tau}^{2}x,\ \partial_{\tau}^{2}y,\ \partial_{\tau}^{2}z)$
$\displaystyle=F/m,$ (2)
$\displaystyle(\partial_{\tau}t)^{2}-(\partial_{\tau}x)^{2}-(\partial_{\tau}y)^{2}-(\partial_{\tau}z)^{2}$
$\displaystyle=1.$ (3)

Eq. (1) is the Klein-Gordon equation for a free scalar field with mass $M$.
Eq. (2) is the special-relativistic version of Newton’s second law for a
particle with a rest mass of $m$ undergoing a constant 4-force of
$F=(f_{t},f_{x},f_{y},f_{z})$. In the $F=0$ case, it states that force-free
particles travel on straight lines in the $(t,x,y,z)$-coordinate system. Eq.
(3) states that the particle is always moving slower than the speed of light.

A dynamics-second reading of the $(t,x,y,z)$-coordinate system is clearly
inappropriate; This would involve granting some metaphysical weight to this
coordinate system and then using it as an explanans for the inertial dynamics
of $\varphi$ and $X$. This clearly will not work since coordinate systems are
ultimately a representational artifact which we project onto the world.
Indeed, it is nearly trivial to switch between a wide variety of different
coordinate systems. Moreover, as I will soon demonstrate, we can always move
into a coordinate-free formulation of our physics.999Or, at least, almost
always, see Pitts (2012). From here we can then access every possible
coordinate description of our theory. These considerations push us strongly
towards a dynamics-first view of coordinates: The coordinate systems which
feature in our best theories reflect nothing metaphysically weighty in the
world beyond them being a particularly nice way (among other options) of
codifying the dynamical behavior of matter.

Hence, appealing to an inertial coordinate system will not give us a
metaphysically supported explanation of the inertial behavior of matter. If
anything is to explain the inertial behavior of matter, it would be the
inertial structure of spacetime itself (this being the feature of the world
which our inertial coordinate systems have latched onto). But where can this
inertial structure be seen in the above theory? As I will now demonstrate,
this theory’s inertial structure will miraculously appear when we move to a
generally covariant (i.e., a coordinate-independent) representation of our
physics. (Readers unfamiliar with differential geometry need not worry, no
technical know-how is needed to understand what follows.) Written in the
coordinate-free language of differential geometry, Eq. (1) - (3) become,

$\displaystyle(\eta^{\text{ab}}\nabla_{\text{a}}\nabla_{\text{b}}-M^{2})\varphi$
$\displaystyle=0,$ (4)
$\displaystyle\dot{X}^{\text{a}}\nabla_{\text{a}}\dot{X}^{\text{b}}$
$\displaystyle=F^{\text{b}}/m,$ (5)
$\displaystyle\eta_{\text{ab}}\dot{X}^{\text{a}}\dot{X}^{\text{b}}$
$\displaystyle=-1,$ (6)

where $\varphi:\mathcal{M}\to\mathbb{R}$ is a real-valued scalar field on a
spacetime manifold $\mathcal{M}\cong\mathbb{R}^{4}$. Here $\eta_{\text{ab}}$
is a fixed flat Lorentzian metric with signature $(-,+,+,+)$ defined on
$\mathcal{M}$ and $\nabla_{\text{c}}$ is the unique covariant derivative
compatible with $\eta_{\text{ab}}$, i.e., with
$\nabla_{\text{c}}\eta_{\text{ab}}=0$. Here $X:\mathbb{R}\to\mathcal{M}$ is a
particle trajectory with $\dot{X}^{\text{a}}:\mathbb{R}\to\text{T}\mathcal{M}$
being its tangent vector. Finally, $F^{\text{b}}$ is the 4-force which is
being applied to the particle.

The key takeaway from rewriting our theory in this way is the appearance of
the following three mathematical objects: a Lorentzian metric,
$\eta_{\text{ab}}$, a covariant derivative, $\nabla_{\text{c}}$, and a smooth
manifold, $\mathcal{M}$. Each of these three objects was in some sense
“hidden” by our previous use of the $(t,x,y,z)$-coordinate system. We have now
found this theory’s inertial structure. In particular, Eq. (5) still says that
force-free particles travel in “straight lines”. Now, however, rather than
checking the straightness of the trajectory $X(\tau)$ against the
$(t,x,y,z)$-coordinate system, we are instead evaluating its straightness with
respect to a piece of fixed geometric structure, namely $\nabla_{\text{c}}$
(which happens to be uniquely determined by $\eta_{\text{ab}}$). This sets the
standard of straightness on $\mathcal{M}$, and thereby picks out the theory’s
inertial structure.

###  2.3 Dynamics-First Views of Geometry

As I noted above, if anything is going to help us explain the inertial
behavior of matter, it would be the inertial structure of the spacetime
manifold itself (i.e., the Lorentzian metric, $\eta_{\text{ab}}$). This brings
us to the next context for the dynamics-first vs dynamics-second debates:
geometry. On a dynamics-second view of geometry, our above investigation of
inertial dynamics has revealed to us the world’s primitive geometric
substructures. Moreover, these geometric substructures are then taken to be
metaphysically weighty in such a way that we can use them to help explain the
inertial behavior of matter. For instance, one might argue that the law
“force-free particles travel on straight lines” presupposes the existence of
some primitive geometric structures on $\mathcal{M}$ with respect to which the
concept of “straightness” is defined.

In opposition to this dynamics-second explanatory strategy, we have the
dynamics-first view of geometry championed by Brown (2005) and
others.101010The dynamics-first views of geometry are, roughly, the dynamical
side of the geometrical vs dynamical spacetime debate. For a recent overview
of the dynamics-first side of this debate see Brown and Read (2022) and
references therein, especially Brown (2005); Brown and Pooley (2006, 2001);
Menon (2019); Huggett (2006); Stevens (2014). According to this view, the
dynamical behavior of matter does not require any geometric underpinnings. In
fact, the arrow of explanation is here reversed: The Lorentzian metric,
$\eta_{\text{ab}}$, is seen as being merely a codification of the dynamical
behavior of measuring rods and clocks. As in the laws’ context, making the
arrows of knowledge and explanation parallel allows us to adopt a
metaphysically deflated view of geometry and thereby avoid the following
tricky metaphysical questions. How is a force-free particle supposed to “know”
what a straight line is? Must it consult with some fixed geometrical
background structure? How exactly does the geometry guide the matter’s
dynamics? Why can’t the particles/fields push back on the geometry?111111As a
historical note, it was this last line of questioning which led Einstein to
develop his theory of general relativity, see Brown and Lehmkuhl (2020). Such
questions are raised against dynamics-second views throughout the dynamical vs
geometrical spacetime debate, especially in the work of Brown (2005).

This dynamics-first view of geometry bares more than just a superficial
resemblance to the above-discussed dynamics-first view of laws. Indeed, one
can pick out a theory’s geometric structures in roughly the same way that
Humeans pick out a theory’s laws. As a particularly vivid example of this,
consider the view of Regularity Relationism developed by Huggett (2006) and
later extended by Pooley (2013) and Stevens (2014). The following imagery
provides a sketch of this view. Imagine carefully peeling all of the
metaphorical Humean paint off of the wall in one piece, see Fig. 2. Moreover,
imagine that this detached paint sheet is locally bendable and stretchable but
cannot tear or break apart. Now imagine searching for a particularly nice way
of bending and stretching this detached paint sheet before reassociating it
with the fixed geometry of the wall. (As in the laws’ context, different
dynamics-first views of geometry are allowed to vary regarding what qualifies
as “particularly nice”.)

The purpose of this metaphorical imagery is to show us that the Humean mosaic
does not require any geometric underpinnings. Indeed, the fact that we can
imagine freely bending and stretching this detached paint sheet shows that it
has only a topological structure, i.e., no geometric structure. Moreover, our
ability to play so freely with the relationship between the mosaic and the
wall’s fixed geometry suggests that we might be able to pick out the theory’s
geometric structure by something like a Best Systems Analysis. As I will argue
in Sec. 4, the capacity for geometric redescription that Regularity
Relationism gives us is analogous to our above-discussed capacity for
nomological redescription and coordinate redescription. Namely, it gives us a
geometry-neutral framing of our theory’s models (i.e., the detached paint
sheets) from which we can freely explore every possible geometric description
of our theory. These analogous redescription techniques push us towards the
following dynamics-first view of geometry: The fixed geometric structures
which feature in our best theories (e.g., $\eta_{\text{ab}}$ in special
relativity) reflect nothing metaphysically weighty in the world beyond them
being one particularly nice way (among other options) of codifying the
dynamical behavior of matter.

Just as in the laws’ context, past making this metaphysical claim, dynamics-
first views of geometry are allowed to vary regarding a wide variety of
further epistemological questions about geometry. Indeed, the questions we saw
at the end of Sec. 2.1 apply here nearly unchanged. What role do geometric
notions play in the initial process of theory construction? How do we first
come to have a geometrical picture of the world? Given our capacity for
geometric redescription, what guides us in selecting a geometric framing for
our theories? That is, what does the term “particularly nice” mean exactly? Is
the geometry-selection process objective and mind-independent, making the
theory’s geometry real (despite it being metaphysically deflated)? Or,
alternatively, is there some non-trivial element of conventionality to it?
Unfortunately, it is beyond the scope of this paper to discuss these questions
any further.

###  2.4 Dynamics-First Views of Topology

The above discussion has laid the groundwork for us to now make a novel
extension of the dynamics-first vs dynamics-second debates into the context of
spacetime topology. Norton (2008) has already raised the challenge of making
such an extension. He correctly notes that deflating a theory’s geometric
structures as we did in Sec. 2.3 still leaves its topological structure (i.e.,
the spacetime manifold itself, $\mathcal{M}$) intact as a serious candidate
for ontological commitment. This has been called the “problem of pre-
geometry”:121212For further discussion of the problem of pre-geometry, see
Linnemann and Salimkhani (2021); Chen and Fritz (2021); Menon (2019). The
dynamics-first approach to spacetime described above only applies to the
geometric structure of spacetime and not to its topological structure. Indeed,
the above discussion of Regularity Relationism seems to rely upon imagining
the Humean mosaic as a topologically connected sheet.

In light of Norton’s complaint, it is natural to wonder whether the above-
discussed dynamics-first views of laws, coordinates, and geometry can be
extended to cover our theories’ topological structures as well. Indeed as
Norton (2008, pg. 833) notes, “One can certainly conceive more modest versions
[of the dynamics-first approach to spacetime] that presume some spatiotemporal
notions and not others, although the resulting constructivist view becomes
less interesting the more spatiotemporal properties it assumes.” (Note that
Norton calls the followers of the dynamics-first approach constructivists.)
Hence, if such a dynamics-first view of topology could be put forward, then
proponents of dynamics-second perspectives (such as Norton) should rejoice in
having a more interesting and unified position to argue against.

In just the same way, proponents of dynamics-first views (of laws,
coordinates, and/or geometry) should rejoice about the prospects of a
dynamics-first view of topology. While one could evaluate these four debates
independently, there is much to be gained by putting them into close contact
with each other and allowing one’s view to equilibrate across these four
contexts. In addition to the rather banal point that analogous situations
ought to be treated analogously, adopting a unified metaphysical orientation
(either dynamics-first or dynamics-second) allows one’s positions in these
debates to lend each other mutual support and share in their common
motivations. These considerations drive us to search for a dynamics-first view
of topology.

But how should we then proceed? A way forward has already been revealed by the
above discussion. Indeed, as I have stressed above, the dynamics-first views
of laws, coordinates, and geometry are each supported by our existing
capacities for redescription. Hence, one way to achieve a dynamics-first view
of topology would be to develop some analogous topological redescription
techniques (ideally, something as powerful as our existing techniques for
nomological, coordinate, and geometric redescription). To this end, I have
recently developed such a suite of topological redescription techniques,
namely the ISE Method, see Grimmer (2023a) and Grimmer (2023b).

In particular, Grimmer (2023b) has evaluated the degree to which the
topological redescription techniques introduced in Grimmer (2023a) are
analogous to our existing capacities for nomological, coordinate, and
geometric redescription. Sec. 5.1 will summarize the results of these two
papers. For now, it suffices to state their conclusions. The ISE Method allows
us to remove and replace the topological underpinnings of our spacetime
theories just as easily as we can switch between different coordinate systems
(or different axiom systems, or different geometric descriptions).

The purpose of this paper is to assess (and ultimately validate) the claim
that these new topological redescription techniques can be used to support a
dynamics-first view of topology which is analogous to the above-discussed
dynamics-first views of laws, coordinates, and geometry. In direct analogy
with these views, the central claim of the dynamics-first view of topology is
as follows: The topological structure which features in our best theories
(e.g., the spacetime manifold, $\mathcal{M}$) reflects nothing metaphysically
weighty in the world beyond it being one particularly nice way (among other
options) of codifying the dynamical behavior of matter. (As in the previous
contexts, different dynamics-first views of topology will be allowed to vary
regarding what qualifies as “particularly nice”.)

Allow me to briefly spell out some of the philosophical consequences of this
view. Recall from above that dynamics-first views deny the metaphysical
weightiness of our theory’s laws/geometry/topology by aligning the arrows of
knowledge and explanation, see Fig. 1c). Ultimately, this means that our
theories’ nomological/geometrical/topological structures are to be limited to
the epistemological roles they play in helping us study and codify the
dynamical behavior of matter. As I have discussed above, this allows us to
avoid many tricky metaphysical questions. Whatever metaphysical questions
about laws/geometry/topology remain are then reduced to epistemological
questions about the roles these structures play in our theorizing.
Importantly, however, taking a dynamics-first perspective does not obligate
one to adopt any specific views about these further epistemological questions.
Indeed, there are plenty of epistemological issues left on the table for
dynamics-first views to disagree about.

Just as in the previous context, dynamics-first views of topology are allowed
to give varying answers to a wide variety of epistemological questions about
spacetime topology.131313As Dewar et al. (2022) notes, these epistemological
questions about spacetime have recently been neglected in favor of more
metaphysical questions about space and time. Indeed, the questions we saw at
the end of Sec. 2.1 and Sec. 2.3 apply here nearly unchanged. What role do
topological notions play in the initial process of theory construction? How do
we first come to have a topological picture of the world? Given our capacity
for topological redescription, what guides us in selecting a topological
framing for our theories? That is, what does the term “particularly nice” mean
exactly? Is the topology-selection process objective and mind-independent
making the theory’s topology real (despite it being metaphysically deflated)?
Or, alternatively, is there some non-trivial element of conventionality to it?

Despite being open on all of these epistemological points, the dynamics-first
view of topology does present a definite metaphysical picture of space and
time which is of a roughly Neo-Kantian character (contra Newton and Leibniz).
Namely, according to the dynamics-first view of topology, rather than
corresponding to any fundamental substances or relations out there in the
world, the spacetime manifolds which feature in our best theories ought to be
thought of as merely being an aspect of how we represent the world. Compare
this view with the following quote from Kant about the metaphysical nature of
space:

> Space is not something objective and real, nor a substance, nor an accident,
> nor a relation; instead, it is subjective and ideal, and originates from the
> mind’s nature in accord with a stable law as a scheme, as it were, for
> coordinating everything sensed externally.141414Kant (1992, pg. 397)

That is, according to Kant space is not a substance (denying Newton) nor a
relation (denying Leibniz) but rather something which our minds project onto
the world as an organizing principle. Of course, Kant then goes on to make
some further epistemological claims about space and time being matters of a
priori intuition which are necessary for perception and cognition to even
begin. The dynamics-first view of topology agrees with Kant’s metaphysical
claims about space and time but is not committed to his further
epistemological claims. For instance, the dynamics-first view of topology
might instead be taken in a conventionalist/Neo-Kantian direction paralleling
Poincaré’s (2018) conventionalism about geometry.151515For some discussion of
where Poincaré sits on the spectrum from Neo-Kantians to conventionalists, see
Ivanova (2015). Alternatively, the dynamics-first view of topology could be
taken in a realist direction paralleling realism about Humean laws of nature,
see Lewis (1994).

The remainder of this paper is organized as follows. Sec. 3 will highlight one
key way in which dynamics-first views are allowed to vary: They might apply
either at the model level or at the theory level. In Sec. 4, I will then
stress how the dynamics-first views of laws, coordinates, and geometry are
supported by our existing capacities for nomological, coordinate, and
geometric redescription respectively. Sec. 5 will then present some analogous
topological redescription techniques which can be leveraged in support of an
analogous dynamics-first view of topology.

##  3 Dynamics-First Views at the Theory Level and at the Model Level

The previous section has highlighted that dynamics-first views are allowed to
vary regarding a wide variety of epistemological questions. As I noted at the
end of Sec. 2.1, some of these epistemological questions may have metaphysical
consequences. One such question deserves special attention. To what degree are
our best theory’s laws/geometry/topology either directly or indirectly
associated with the actual world? Let us assume that the theory in question
aims to represent the entirety of the actual world with one of its models
(i.e., without the aid of its other models).161616For some critiques of this
assumptions, see Curiel (2020) and Wallace (2022). Without this assumption the
pressure to deliver a model-level understanding of our theory’s
laws/geometry/topology is greatly reduced. Our question then becomes: To what
degree are our theory’s laws/geometry/topology identifiable within each of its
models individually (i.e., without relying the relationships between
models)?171717For commentary on the importance of understanding the
relationships between a theory’s models see Halvorson (2016). Said
differently, given any model of the theory is it possible for us to point to
some structure within that model alone and declare, “that is what the theory’s
nomological, geometric, and/or topological structure has latched onto”? If so,
let us say that we have a model-level understanding of the theory’s laws,
geometry, and/or topology.

Importantly, dynamics-first views may or may not give us a model-level
understanding of the theory’s nomological, geometric, and/or topological
structures. Indeed, as I will now discuss, the default position for dynamics-
first views is that they do not. Recall that the dynamics-first vs dynamics-
second debates are about whether or not certain theoretical structures (e.g.,
laws, geometry, topology, etc.) which appear in our best scientific theories
should be granted any metaphysical weight in order to help us explain
dynamics. What should be noticed about this framing of the debate is that
these theoretical structures are just that, structures which appear in our
theories. Hence, unless some further argument is provided, these structures
should be thought of first and foremost as the laws/geometry/topology of the
theory and only indirectly the laws/geometry/topology of the world.

This indirect association of these structures with the actual world comes
about simply because they appear in a theory in which is (or at least aspires
to be) about the actual world. Namely, the theory in which these structures
feature: 1) was built from real-world evidence, and 2) successfully makes
predictions about real-world experiments. Moreover, if the theory is semantic
then 3) its models may accurately represent parts of the actual world’s matter
and dynamics, or if the theory is syntactic then 4) its terms may successfully
refer to parts of the actual world’s matter and dynamics. These four points of
connection with the actual world automatically give us a theory-level
understanding of the theory’s laws, geometry, and topology. Past this,
additional argumentation must be provided to achieve a model-level
understanding of these structures. As I will now discuss, it is possible
(under certain assumptions) to achieve a model-level understanding of at least
the nomological structures of a certain kind of theories.

To demonstrate this possibility, let us adopt a syntactic view of theories
such that the axiom-like laws of theory are simply the axioms of that theory.
Note that the theory’s laws are axioms and therefore are sentences. If we want
the truth-makers of these sentences at a given model to be contained strictly
within that model, then we need to assume that the theory’s axioms are non-
modal (e.g., ‘all F’s are G’s’). Notably, this is not the case on Demarest’s
(2017) account since her laws are allowed to refer to dispositions (e.g.,
‘this would have happened if that had happened’). Under our non-modal
assumption, however, each axiom-like law reports a pattern (e.g., the fact
that all F’s are G’s) which is realized within each of the theory’s models
individually (i.e., independent of what is happening in its other models). If
the theory’s axioms are true at the actual world, then we can point to the
corresponding pattern and declare, “that is what the theory’s nomological
structure has latched onto”. This gives us a model-level understanding of the
laws of nature, as desired.

It should be clear from the above discussion how relaxing the modal assumption
would seriously challenge our ability to have a model-level understanding of
the laws of nature.181818One might, however, follow Demarest (2017) in
claiming that the theory’s nomological structure has latched onto patterns in
the actual world’s dispositional properties. Accepting such modal properties
into one’s ontology challenges our above assumption that our theories aim to
represent the actual world with exactly one of its models in isolation.
Moreover, as I will now discuss, such challenges also arise if we move to a
semantic understanding of theories. On a semantic view of theories, the laws
of nature are not pattern-reporting sentences. Instead, they are axiom-like
rules for generating models, e.g., the laws are the equations to which the
theory’s models are solutions. Unlike in the syntactic case, it is possible
that a semantic theory’s laws of nature cannot be understood individually or
assigned truth-values individually, see Ott (2022, Ch. 11.4). For instance, it
is notoriously difficult to understand Newton’s three laws of motion in such
an individuated way, see Ott (2022, Ch. 3.2) and Read (2023, Ch. 1). Indeed, a
theory’s laws might work together in non-trivial ways to generate the theory’s
models. Given that such a web of laws is not just a collection of pattern-
reporting sentences, it is much more difficult to point to some concrete
structure in either its models or in the actual world which this web of laws
has latched onto. Fortunately, despite the difficulties we face in
understanding such a web of laws at the model level, we can always understand
them at the theory level (and thereby maintaining their indirect connection
with the actual world).

The same theory-level vs model-level distinction can be applied in the context
of geometry. Let us see how. Recall from Sec. 2 the imagery associated with
Regularity Relationism. Recall that one first imagines peeling all of the
metaphorical Humean paint off of the Humean mosaic as a single topologically
connected sheet, see Fig. 2. One then searches for a particularly nice way of
bending and stretching this detached paint sheet before reassociating it with
the fixed geometry of the wall. The way that I have described this process
makes it sounds as if the theory’s geometric structure is to be read off of a
single Humean mosaic (i.e., a single model of our theory). Indeed, this is the
typical way of understanding Regularity Relationism and would amount to a
model-level understanding of the theory’s geometric structure. As in the laws’
context, however, this is not a necessary feature of dynamics-first views.

Indeed, one’s selection criteria for picking out the theory’s geometric
structure might require that it nicely codifies not just one of the theory’s
models/mosaics but multiple of its models/mosaics simultaneously. That is, one
could search for a particularly nice way of simultaneously bending and
stretching all of the detached paint sheets collectively. In particular,
motivated by Earman (1989), one could search for a geometric codification of
the theory which captures the symmetry relationships which hold between the
theory’s models. Note that a theory’s symmetries are not visible within its
individual models; Instead, they are about certain relationships holding
between the theory’s models. In sum, one can easily imagine applying
Regularity Relationism at a theory level. Even if it is only understood at a
theory level, there is still a significant (albeit indirect) connection
between the theory’s geometric structure and the actual world. Namely, these
geometric structures still feature in a theory which is about the actual
world. This theory: 1) was built from real-world evidence, and 2) successfully
makes predictions about real-world experiments, and 3) its geometry-neutral
models may, in fact, accurately represent the actual world’s matter and
dynamics.

Some commentary on how the theory-level vs model-level distinction applies in
the context of topology will be given at the end of Sec. 5. Indeed, there I
will put forward a dynamics-first view of topology which can likely only offer
us a theory-level understanding of the theory’s spacetime manifold.

##  4 How Strong Redescriptive Capacities Support
Dynamics-First Views

In Sec. 2 I stressed that the dynamics-first views of laws, coordinates, and
geometry are each supported by the existence of certain strong redescription
techniques. In this section, I will elaborate on how analogous these various
redescription techniques are and how they provide analogous support for their
respective dynamics-first views. I do this in anticipation of putting forward
the dynamics-first view of topology in Sec. 5. Indeed, it will be also
supported by some strong topological redescription techniques, namely the ISE
Method.

A first point of analogy is that all three redescription techniques can be
implemented in a broadly subject-neutral way. Namely, the mechanical process
of re-axiomatization can be applied to our theories independently of what
subject matter they are about: indeed, logic is subject-neutral (or at least
it aspires to be). This is ultimately the reason why both Lewis (1973, 1999)
and Demarest (2017) can invoke Best Systems Analyses despite disagreeing
radically on the metaphysical details of the to-be-codified mosaic; The
logical process of re-axiomatization is metaphysics-blind. Similarly, the
mechanical process of changing coordinates applies to our theories in exactly
the same way no matter what the theory is about: e.g., scalar fields, particle
trajectories, bosons, fermions, strings, etc. Finally, note that the paint-
peeling imagery associated with Regularity Relationism can be applied equally
well independent of the metaphysical thickness/thinness of the Humean paint
(e.g., whether it contains dispositions or not).

Indeed, unless one carefully directs one’s effort to redescribe one’s theory
in a perspicuous way, then the resulting description is likely to be
tremendously complicated. Almost all coordinate systems are horribly
complicated; Almost all axiomatizations are horribly complicated, resulting in
laws which are ill-suited to capture the theory’s metaphysics; Almost all ways
of bending and stretching the Humean mosaic result in geometric structures
which are ill-suited to the theory in question. Said differently, each of
these redescription techniques vastly overgenerates candidate descriptions
(most of which will be tremendously complicated).

At this point, one may begin to worry about the overabundance of nomological,
coordinate, and geometric descriptions which are on offer. In each case,
however, our saving grace is that our redescriptive capacities are not only of
wide scope but also complete in the following sense: By using these
redescription techniques we can access every possible
nomological/coordinate/geometric description of our theory. Hence, even though
descriptions will be horribly complicated, if there exists a particularly nice
description, then we can find it. Note that this claim holds no matter what
selection criteria one adopts regarding what qualifies as a “particularly
nice” nomological/coordinate/geometric description.

Considering our strong capacities for nomological, coordinate, and geometric
redescription, one might grow suspicious of any metaphysical weight being
granted to these axioms systems, coordinate systems, or fixed geometric
structures. That is, one might be led towards a dynamics-first view of laws,
coordinates, and/or geometry. Indeed, given these strong capacities for
redescription, one finds some motivation to focus on our theory’s coordinate-
independent, law-independent, and/or geometry-independent content. As I have
discussed already in Sec. 2.2, we can always move into a coordinate-neutral
framing of our theory from which we can search for perspicuous coordinate
systems. Similarly, our nomological redescription techniques allow us to
conceive of our theories in a law-independent way: namely, we can view them as
either a collection of sentences or as a collection of models. Moreover, each
model of our theory can be thought of in an law-independent way by thinking of
it as a possible Humean mosaic, see Fig. 2. Notice also that we can think of
these Humean mosaics in a geometry-independent way by considering the detached
paint sheets which result from the above-described paint-peeling process.

In sum, our nomological, coordinate, and geometric redescription techniques
give us access to a law-neutral/coordinate-neutral/geometry-neutral
description of our theories from which we can search for good
nomological/coordinate/geometric descriptions. In each case, these strong
redescription techniques lend support to the dynamics-first views of laws,
coordinates, and geometry which were discussed in Sec. 2: The laws,
coordinates, and/or fixed geometric structures which feature in our best
theories reflect nothing metaphysically weighty in the world beyond them being
one particularly nice way (among other options) of codifying the dynamical
behavior of matter.

##  5 My Dynamics-First View of Topology

The previous two sections have laid the groundwork for the dynamics-first view
of topology which I will put forward in this section. As I have stressed
above, the dynamics-first views of laws, coordinates, and geometry are
supported by our capacities for nomological, coordinate, and geometric
redescription respectively. My dynamics-first view of topology will be
supported by some analogous topological redescription techniques. Namely, it
will be supported by the ISE Method which I have developed in Grimmer (2023a)
and Grimmer (2023b).

###  5.1 A Quick Overview of the ISE Method

This section will provide an overview of the ISE Method’s topological
redescription techniques. For details beyond what follows, I direct the
interested reader to Grimmer (2023a) and Grimmer (2023b). The ISE Method is a
three-step process: Internalize, Search, Externalize (hence the initials,
ISE). In the internalization step, one first divorces the dynamical behavior
of matter from its topological underpinnings. This divorce is carried out in a
similar spirit to spacetime algebraicism, albeit with some key
differences.191919Roughly put, spacetime algebraicism is the view that the
dynamical behavior of matter (specifically, field states) does not require any
topological underpinnings (specifically, because they can be thought of
algebraically). By contrast, my view drops these parentheticals. For further
discussion of spacetime algebraicism, see Chen and Fritz (2021); Menon (2019);
Linnemann and Salimkhani (2021); Rynasiewicz (1992); Earman (1989); Rosenstock
et al. (2015). Notably, Geroch (1972) uses Einstein Algebras to reconstruct
general relativity in this way. For instance, the internalization process is
similar to the move commonly made in non-relativistic quantum mechanics from a
position representation of the wavefunction, $\psi(t,x)$, to a Hilbert space
representation, $\ket{\psi(t)}$. One key difference, however, is that we here
internalize time as well as space. Moreover, the ISE Method does not require
the theory in question to have any algebraic structure. For instance, the ISE
Method is applicable to theories about particle trajectories which lack any
meaningful algebraic operations (e.g., no sums and products).

Internalization results in a topology-neutral characterization (i.e., a
spacetime-neutral characterization) of the theory in question. One next
searches for the building blocks of a new spacetime framing of the theory. I
call these building blocks pre-spacetime translation operations, or PSTOs. A
technical definition of PSTOs can be found in Grimmer (2023a), but the
following characterization is sufficient for our purposes. Let
$S_{\text{neutral}}^{\text{kin}}$ be the theory’s set of kinematically allowed
models understood in a topology-neutral way. Picking a set of PSTOs involves
picking out a pair of Lie groups, $G_{\text{trans}}$ and
$G_{\text{fix}}\subset G_{\text{trans}}$, which act on
$S_{\text{neutral}}^{\text{kin}}$. To qualify as PSTOs, this pair of Lie group
actions on $S_{\text{neutral}}^{\text{kin}}$ need to be compatible with each
other in two ways: they must have faithful construction compatibility and
injection compatibility, for details see Grimmer (2023a) and Grimmer (2023b).
These compatibility constraints are sufficient to guarantee that our PSTOs
are, in a certain sense, structurally indistinguishable from spacetime
translations. An intuitive example would be translations, not in spacetime,
but rather in old theory’s Fourier space.

Finally, in the externalization step, one uses whichever PSTOs one has chosen
to construct a new topological setting for the theory in question. One does
this simply by taking these PSTOs seriously as spacetime translations. Indeed,
by construction, one’s chosen pre-spacetime translation operations will end up
becoming honest-to-goodness spacetime translations in the new theory. More
concretely, the process goes roughly as follows. One first declares that the
Lie group, $G_{\text{trans}}$, is to be a group of diffeomorphisms which act
transitively on new spacetime, $\mathcal{M}_{\text{new}}$. Secondly, one
declares that $G_{\text{fix}}$ is to be the stabilizer subgroup of
$G_{\text{trans}}$’s action on $\mathcal{M}_{\text{new}}$. Making these two
declarations uniquely specifies $\mathcal{M}_{\text{new}}$ up to
diffeomorphism as $\mathcal{M}_{\text{new}}\cong
G_{\text{trans}}/G_{\text{fix}}$. For details see Appendix A of Grimmer
(2023a). In the above-discussed Fourier example, the new theory’s spacetime is
effectively a copy of the old theory’s Fourier space.

To complete the externalization process, one then must map the old theory’s
states and dynamics onto $\mathcal{M}_{\text{new}}$. As Grimmer (2023a)
discusses, this final stage of the externalization process is tightly
constrained by our choice of PSTOs. For instance, in the Fourier example, the
new theory’s states and dynamics are required to be the Fourier transform of
the old theory’s states and dynamics (up to some rescaling freedom). Grimmer
(2023a) discusses this Fourier redescription example at length in the context
of non-relativistic quantum mechanics where it amounts to a position-momentum
duality.

The exact same topological redescription techniques were also used in Grimmer
(2023a) to redescribe a lattice theory (i.e., a theory set on a discrete
spacetime, $\mathcal{M}_{\text{old}}\cong\mathbb{R}\times\mathbb{Z}$) as
existing on a continuous spacetime manifold,
$\mathcal{M}_{\text{new}}\cong\mathbb{R}\times\mathbb{R}$. This topological
redescription is facilitated by the fact that the discrete spacetime theory in
question admits a set of continuous pre-spacetime translation operations (see
Fig. 1 in Grimmer (2023a)). Moreover, Grimmer (2023b) has used the ISE Method
to implement a Kaluza-Klein dimensional reduction (and to then undo this
dimensional reduction).

Grimmer (2023b) also discusses how the ISE Method can be applied outside of
the scalar field case. Indeed, it can also be applied to theories about a
parameterized particle trajectory, $x:\mathcal{V}\to\mathcal{M}$. For
instance, consider a theory about a point-like particle,
$x_{\text{old}}:\mathcal{V}_{\text{old}}\to\mathcal{M}_{\text{old}}$, moving
on a Möbius strip over time (with $\mathcal{V}_{\text{old}}\cong\mathbb{R}$
and $\mathcal{M}_{\text{old}}\cong\mathbb{R}\times\mathbb{M}$). One can use
the ISE Method to redescribe such a theory as being about an extended
particle,
$x_{\text{new}}:\mathcal{V}_{\text{new}}\to\mathcal{M}_{\text{new}}$, moving
on the Euclidean plane over time (with
$\mathcal{V}_{\text{new}}\cong\mathbb{R}^{2}$ and
$\mathcal{M}_{\text{new}}\cong\mathbb{R}\times\mathbb{R}^{2}$).

Moving beyond these example applications, Grimmer (2023b) has proved the ISE
Equivalence Theorem which states the following. The ISE Method gives us access
to effectively every possible spacetime framing of a given theory’s
kinematical and dynamical content, bounded only by a weak spacetime-kinematic
compatibility condition. This proves that the ISE Method is a completely
general tool for topological redescription (at least, within the realm of
spacetime theories which satisfy this spacetime-kinematic compatibility
condition). Every instance of topological redescription between any two such
theories can be implemented by the ISE Method. As Grimmer (2023b, Sec. 2.5)
has discussed, this spacetime-kinematic compatibility is an extremely weak
assumption. Hence, the scope of topological redescriptions offered to us by
the ISE Method is extremely broad.

###  5.2 The ISE Method Supports a Dynamics-First View of Topology

The ISE Method supports a dynamics-first view of topology in the same way that
our existing capacities for nomological, coordinate, and geometric
redescription respectively support dynamics-first views of laws, coordinates,
and geometry. Indeed, in Grimmer (2023b) I already explored in depth the
analogy between our existing capacities for redescription and the capacity for
topological redescription offered to us by the ISE Method. I will now briefly
review the conclusions of Grimmer (2023b) regarding the strength of this
analogy.

Like each of the above-discussed redescription techniques, the ISE Method can
be applied to a wide range of theories in a broadly subject-neutral way.
Namely, it does not matter whether the theory is about scalar fields or
particle trajectories. Nor does it matter whether the theory’s dynamics are
linear or non-linear. Another mark of similarity is that if applied in an
unguided way, the ISE Method radically overgenerates possible topological
redescriptions of the theory in question. See the above-discussed ISE
Equivalence Theorem. Indeed, just like our other redescription techniques
almost all of the topological redescriptions on offer will be horribly
complicated and ill-suited to the theory in question. Fortunately, however,
these topological redescription techniques are also complete in the following
sense. Under the assumption that any “particularly nice” topological
description of a theory will have spacetime-kinematic compatibility, we have
the following result. We are guaranteed that if there exists a particularly
nice topological framing of one’s theory, then it is accessible to us via the
ISE Method.

The above-discussed points have already given us a strong analogy between the
ISE Method and our existing capacities for coordinate, nomological, and
geometric redescription. The similarities do not stop here. Indeed, as I noted
above, the ISE Method offers us topological redescriptions of the theory in
question by first moving into a topology-neutral characterization of this
theory. In light of our strong capacity for topological redescription, one
might grow suspicious of any metaphysical weight being granted to the
topological structures which appear in our theories (e.g., the spacetime
manifold). That is, one is led towards a dynamics-first view of topology: The
topological structure which features in our best theories (e.g., the spacetime
manifold) reflects nothing metaphysically weighty in the world beyond it being
one particularly nice way (among other options) of codifying the dynamical
behavior of matter.

In closing, one final point should be noted regarding the ISE Method. The way
in which the ISE Method generates new topological settings for our theories
likely forces us to have a theory-level understanding of our theory’s
topological structure (i.e., its spacetime manifold). To see this, recall from
Sec. 5.1 that the theory’s new spacetime manifold, $\mathcal{M}_{\text{new}}$,
is built out of a pair of groups which act on
$S_{\text{neutral}}^{\text{kin}}$, i.e., the theory’s set of kinematically
allowed models. For any given model, $\sigma\in
S_{\text{neutral}}^{\text{kin}}$, the subset of models which are involved in
this construction contains at least the orbit of the Lie group
$G_{\text{trans}}$ at $\sigma$. This reliance upon multiple models in the
construction of a new spacetime setting suggests that it will be difficult to
achieve a model-level understanding of the theory’s topological structure.

That is, given any spacetime-neutral model of the theory, it is unlikely that
we will be able to point to some structure within this model alone and
declare, “that is what the theory’s topological structure has latched onto”.
As I have discussed in Sec. 3, however, there is still a significant (albeit
indirect) connection between the theory’s topological structure and the actual
world. Namely, the theory’s topological structure still features in a theory
which is about the actual world. In particular, this theory: 1) was built from
real-world evidence, and 2) successfully makes predictions about real-world
experiments, and 3) its spacetime-neutral models may, in fact, accurately
represent the actual world’s matter and dynamics.

##  6 Conclusion

This paper has unified certain aspects of the metaphysics of laws debate and
the geometrical vs dynamical spacetime debate by framing them as dynamics-
first vs dynamics-second debates. I have then extended these debates outside
of their original contexts of laws and geometry into the context of spacetime
topology. As I discussed in Sec. 2, in each context, the central issue in each
of these debates is whether or not one should grant any metaphysical weight to
various theoretical structures (e.g., laws, geometry, topology, etc.) in hopes
of giving a metaphysically supported explanation of dynamics.

I take it as a given that the theories in which these nomological, geometric,
and topological structures appear are produced by studying and codifying the
dynamical behavior of matter. That is, I take both the dynamics-first and the
dynamics-second camps to agree upon the direction of the arrow of knowledge,
see Fig. 1. After this point of epistemological agreement, however, these two
camps have a metaphysical disagreement about the direction of the arrow of
explanation. In particular, the dynamics-second views take the presence of
certain nomological, geometric, and/or topological structures in our best
theories to reveal to us aspects of the world’s metaphysical substructure.
Moreover, they grant these nomological, geometric, and/or topological
substructures enough metaphysical weight such that we can then turn around and
use them to help explain the very dynamics which we began by investigating,
see Fig. 1b).

As I have discussed in Sec. 2, some tricky metaphysical questions arise for
dynamics-second views. One way to avoid these questions (and to achieve a
sparser metaphysics) is to reverse the arrow of explanation, putting dynamics
first so to speak. As I have discussed in Sec. 2, adopting this explanatory
strategy gives dynamics-first views an “unofficially” Humean character even
outside of the laws’ context. In particular, dynamics-first views deny the
metaphysical weightiness of our theory’s laws, geometry, and/or topology by
aligning the arrow of explanation with the arrow of knowledge, see Fig. 1c).
Ultimately, this means that these structures are to be limited to the
epistemological roles they play in helping us to study and codify the
dynamical behavior of matter. This maneuver avoids many tricky metaphysical
questions about these structures and subsequently reduces whatever
metaphysical questions remain to epistemological questions regarding the roles
these structures play in our theorizing. Importantly, however, taking a
dynamics-first perspective does not obligate one to adopt any specific views
about these further epistemological questions. Indeed, as I have discussed in
Sec. 2.1, Sec. 2.3, and Sec. 2.4, there are plenty of epistemological issues
left on the table for dynamics-first views to disagree about (some of which
may have metaphysical consequences, see Sec. 3).

The main goal of this paper, however, was to introduce a dynamics-first view
of topology in Sec. 5. In Sec. 4, I stressed how the dynamics-first views of
laws, coordinates, and geometry are respectively supported by our existing
capacities for nomological, coordinate, and geometric redescription. Sec. 5.1
then introduced some powerful new topological redescription which I developed
in Grimmer (2023a) and Grimmer (2023b), namely the ISE Method. As I discussed
in Sec. 5.2, these new topological redescription techniques are analogous to
our existing capacities for nomological, coordinate, and geometric
redescription. Hence, they give analogous support to an analogous dynamics-
first view of topology: The topological structure which features in our best
theories (e.g., the spacetime manifold) reflects nothing metaphysically
weighty in the world beyond it being one particularly nice way (among other
options) of codifying the dynamical behavior of matter. As I discussed at the
end of Sec. 2.4, the metaphysics of space and time put forward by this view
has a notably Neo-Kantian character (contra Newton and Leibniz) while
remaining open regarding Kant’s further epistemological claims. Indeed, the
dynamics-first view of topology could alternatively be taken in either a
conventionalist/Neo-Kantian direction (paralleling Poincaré (2018)) or a
realist direction (paralleling Lewis (1994)).

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