section

# On the Mathematics and Metaphysics of the Hole Argument

Oliver Pooley and James Read

###### Abstract

We make some remarks on the mathematics and metaphysics of the hole argument,
in response to a recent article in this journal by Weatherall ([2018]).
Broadly speaking, we defend the mainstream philosophical literature from the
claim that correct usage of the mathematics of general relativity ‘blocks’ the
argument.

######

  1. 1 Introduction
  2. 2 Indeterminism versus Underdetermination
  3. 3 Weatherall’s Arguments
  4. 4 The Equivocation Argument
  5. 5 The Argument from Mathematical Structuralism
  6. 6 Conclusions

##  1 Introduction

Consider models of general relativity $\mathcal{M}=\left(M,g_{ab}\right)$ and
$\widetilde{\mathcal{M}}=\left(M,\tilde{g}_{ab}\right)$,111In order to guard
against conflating such models with the possibilities they represent (a
distinction that is vital when discussing the hole argument), we do not refer
to the models as ‘spacetimes’. both foliable into spacelike hypersurfaces,
identical up to some such hypersurface $\Sigma$, but differing by a
diffeomorphism (such that $\tilde{g}_{ab}=d_{*}g_{ab}$, where $d_{*}$
represents the push-forward map) which acts non-trivially to the future of
$\Sigma$. Naïvely, if $\mathcal{M}$ is interpreted as representing some
possible world $W$, then $\widetilde{\mathcal{M}}$ is to be interpreted as
representing a distinct possible world $\widetilde{W}$, in which (the images
of—that is, the physical correlates of) both the metric field $g_{ab}$ and
material fields are distributed differently (but isomorphically) on (the image
of) $M$. The following problem now arises: given (the history associated with)
these solutions of general relativity up to $\Sigma$, the theory (it seems)
simply does not determine which of $W$ or $\widetilde{W}$ will be realized.
Thus, the theory appears to be radically indeterministic.

The foregoing is a version of the hole argument, first presented by Einstein
in 1913 as a means of excluding from consideration any diffeomorphism
invariant theory of gravitation222We understand the notion of ‘diffeomorphism
invariance’ to be distinct from that of ‘general covariance’—see Pooley
[2017].—an argument that he would later reject. The argument was revived by
Earman and Norton ([1987]) as a means of rejecting substantivalism about the
manifold $M$ of general relativity—for, as they elaborate, the above problem
can be avoided if the independent reality of (the physical correlate of) $M$
is rejected, so that one can no longer articulate the difference between $W$
and $\widetilde{W}$, and so cannot articulate the distinction that gives rise
to the apparent indeterminism. Earman and Norton’s paper spawned a sizeable
literature, including several defences of the viability of
substantivalism.333For more on contemporary philosophy of physics discussion
of the hole argument, see Norton [2019], Pooley [2013], Pooley [2021], Roberts
and Weatherall [2020] and references therein.

In a recent paper, Weatherall has questioned whether this second,
philosophical wave of writing on the hole argument is well-motivated. His
central thesis is articulated at the beginning of his article:

> Einstein and the generations of physicists and mathematicians after him were
> right to reject the hole argument. It is based on a misleading use of the
> mathematical formalism of general relativity. If one is attentive to
> mathematical practice, I will argue, the hole argument is blocked.
> (Weatherall [2018], p. 330)

Later, Weatherall extols an apparent virtue of his approach: ‘This particular
option is distinctive—and, I think, attractive—because it is essentially
neutral on the metaphysics of space and time.’ (Weatherall [2018], p. 330)

Our purpose in this article is to make some constructive remarks on the role
of mathematics and metaphysics in the context of the hole argument. Broadly
speaking, these lead us to reject Weatherall’s approach: a proper
understanding of the mathematics of general relativity does not suffice to
block the hole argument.

##  2 Indeterminism versus Underdetermination

In the guise presented above, the hole argument raises the spectre of
indeterminism: given initial data on and prior to $\Sigma$, general relativity
is silent on whether the world will evolve as per $\mathcal{M}$, or as per
$\widetilde{\mathcal{M}}$. This is, indeed, the version of the argument most
often discussed in the literature. One may also, however, use the hole
construction to generate (what we call for the purposes of this paper) a
problem of underdetermination.444The distinction between the indeterminism and
underdetermination versions of the hole arguments is closely related to Earman
and Norton’s ([1987]) distinction between the ‘indeterminism dilemma’ and the
‘verificationist dilemma’. Our thanks to John Norton and Neil Dewar for
discussion of how best to understand the underdetermination version of the
argument.

Suppose that model $\mathcal{M}$ is a candidate to represent the actual world.
Now consider the model $\widetilde{\mathcal{M}}$, generated from $\mathcal{M}$
via a hole diffeomorphism $d$. Since all relational quantities (Einstein’s
‘point coincidences’—cf. (Norton [2019], §8.2)) are preserved under such a
diffeomorphism, it appears that the empirical data available to any observer
within a world represented by either $\mathcal{M}$ or
$\widetilde{\mathcal{M}}$ will fail to determine which model represents that
observer’s world.

In particular, suppose that, according to $\mathcal{M}$, the observer at the
salient stage of their trajectory is located at (the spacetime point
represented by) $p$ and that $d$ maps $p$ to a distinct point $q$. According
to $\widetilde{\mathcal{M}}$, therefore, the relevant stage of the observer’s
trajectory is located at $q$. It follows that no measurement that the observer
might perform at that point along their trajectory can determine whether they
are located at (the point represented by) $p$ or at (the point represented by)
$q$, for the outcomes of any measurements are the same according to
$\mathcal{M}$ and $\widetilde{\mathcal{M}}$.

When Weatherall presents the hole argument at (Weatherall [2018], §3), he does
not distinguish clearly this underdetermination version of the argument from
the indeterminism version. As we show in what follows, some of Weatherall’s
criticisms of the hole argument do not survive proper attention to this
distinction.

##  3 Weatherall’s Arguments

Weatherall can be read as providing two distinct lines of criticism of the
hole argument, although they are not individuated explicitly as such in his
paper. (We recognize that by regimenting Weatherall’s criticism in this way,
we may thereby fail to represent it completely faithfully; nevertheless, we
believe that this is a productive way to proceed.)

Central to both arguments is a distinction that Weatherall draws between
$1_{M}$ and $\tilde{\psi}$, two maps between $\mathcal{M}$ and
$\widetilde{\mathcal{M}}$, which he introduces as follows:

> Assertions concerning point $p$ in the context of both
> $\left(M,g_{ab}\right)$ and $\left(M,\tilde{g}_{ab}\right)$ are implicitly
> made relative to the identity map on this manifold, $1_{M}\colon
> M\rightarrow M$. Meanwhile, we have the map $\psi\colon M\rightarrow M$,
> which is a diffeomorphism. It is $\psi$ that gives rise to the isometry
> $\tilde{\psi}\colon\left(M,g_{ab}\right)\rightarrow\left(M,\tilde{g}_{ab}\right)$,
> where I have again used a [tilde] to be clear that $\psi$ and $\tilde{\psi}$
> are different maps—$\psi$ is an automorphism of $M$, whereas $\tilde{\psi}$
> is an isomorphism, but not an automorphism, between the Lorentizan manifolds
> $\left(M,g_{ab}\right)$ and $\left(M,\tilde{g}_{ab}\right)$. (Weatherall
> [2018], p. 336)

After introducing the distinction, Weatherall goes on to consider the
comparisons of two hole-diffeomorphic models under each of the maps:

> When we say that $\left(M,g_{ab}\right)$ and $\left(M,\tilde{g}_{ab}\right)$
> are isometric spacetimes, and thus that they have all of the same invariant,
> observable structure, we are comparing them relative to $\tilde{\psi}$.
> Indeed, we must be because […] there is no sense in which $1_{M}$ either is
> or gives rise to an isometry. In other words, relative to $1_{M}$,
> $\left(M,g_{ab}\right)$ and $\left(M,\tilde{g}_{ab}\right)$ are not
> equivalent, physically or otherwise. The reason is that there exist points
> $p\in O$ [the region on which the diffeomorphism $d$ acts non-trivially] at
> which
> ${\left(g_{ab}\right)}_{|p}\neq{\left(\tilde{g}_{ab}\right)}_{|1_{M}\left(p\right)}$.
> Consider an observer sitting at a point of spacetime represented by the
> point $p\in O$ of $\left(M,g_{ab}\right)$. If one were to attempt to
> represent that same observer’s location by the point $1_{M}\left(p\right)=p$
> of $\left(M,\tilde{g}_{ab}\right)$, one would conclude that the observer
> would see measurably different metrical properties, different curvature, and
> so on. In general, only one of these assignments can be correct. Meanwhile,
> if one only considers $\tilde{\psi}$, no disagreement arises regarding the
> value of the metric at any given point, since for any point $p\in M$,
> ${\left(g_{ab}\right)}_{|p}={\left(\tilde{g}_{ab}\right)}_{|\psi\left(p\right)}$
> by construction. (Weatherall [2018], p. 336)

Weatherall’s reasoning in this passage can be summarized as follows: when
$\mathcal{M}$ and $\widetilde{\mathcal{M}}$ are compared using $1_{M}$ (that
is, when one’s interpretation of these models is sensitive to differences
captured by $1_{M}$ and so to questions concerning the field values at any
given $p\in M$), they should be taken to represent observationally
distinguishable situations. On the other hand, when $\mathcal{M}$ and
$\widetilde{\mathcal{M}}$ are compared using $\tilde{\psi}$ (that is, when
one’s interpretation of these models is sensitive only to differences captured
by $\tilde{\psi}$ and so _not_ sensitive to questions concerning the field
values at any given $p\in M$), then the models are to be taken to be
physically and not just empirically equivalent.

We will call Weatherall’s two arguments against the hole argument ‘the
equivocation argument’ and ‘the argument from mathematical structuralism’.
According to the equivocation argument, while comparisons of models either via
$1_{M}$ or via $\tilde{\psi}$ might be legitimate, the hole argument rests on
an illegitimate equivocation between the two. According to the argument from
mathematical structuralism, models should be compared via $\tilde{\psi}$, but
the hole argument requires an (illegitimate) comparison of models via $1_{M}$.

We are not persuaded by either argument. For convenience, we summarize here
the three central threads of our critique, which are covered in detail in the
following sections. The first two strands focus on the equivocation argument.
The third addresses the argument from mathematical structuralism.

  1. 1.

Weatherall’s claim that the hole argument involves an equivocation leaves the
indeterminism version of the argument completely untouched. This is because
$1_{M}$ alone is used to articulate this version of the argument. Comparisons
via $\tilde{\psi}$, and the related claim that models so compared are
observationally equivalent, play no role in the indeterminism version of the
hole argument. The charge of equivocation is therefore at best relevant to the
underdetermination version of the hole argument.

  2. 2.

Weatherall’s suggestion that hole-diffeomorphic models are observationally
inequivalent when compared using $1_{M}$ is questionable. His position is
either a close cousin of Maudlin’s response to the static shift argument
against Newtonian absolute space or it presupposes a controversial conception
of empirical equivalence. Either way, Weatherall’s alleged dissolution of the
underdetermination problem is contentious.

  3. 3.

Weatherall’s claim that the hole argument is blocked when models are compared
using $\tilde{\psi}$ is also questionable, for the interpretation of models
that such a comparison might mandate does nothing, by itself, to eliminate the
relevant space of metaphysical possibilities.

##  4 The Equivocation Argument

According to Weatherall, in order to generate the hole argument, one has to
compare $\mathcal{M}$ and $\widetilde{\mathcal{M}}$ using $1_{M}$, insofar as
one regards them as representing physically distinct states of affairs, but
also—in an (allegedly) illegitimate manner—compare them using $\tilde{\psi}$,
insofar as one wishes to regard the worlds represented by those models as
observationally indistinguishable. He thus maintains that the argument rests
on an equivocation:

> There is a sense in which $\left(M,g_{ab}\right)$ and
> $\left(M,\tilde{g}_{ab}\right)$ are the same, and there is a sense in which
> they are different […] But—and this is the central point—one cannot have it
> both ways. Insofar as one wants to claim that these Lorentzian manifolds are
> physically equivalent, or agree on all observable/physical structure, one
> has to use $\tilde{\psi}$ to establish a standard of comparison between
> points. And relative to this standard, the two Lorentzian manifolds agree on
> the metric at every point—there is no ambiguity, and no indeterminism. (This
> is just what it means to say that they are isometric.) Meanwhile, insofar as
> one wants to claim that these Lorentzian manifolds assign different values
> of the metric to each point, one must use a different standard of
> comparison. And relative to this standard—that given by $1_{M}$—the two
> Lorentzian manifolds are not equivalent. One way or the other, the hole
> argument seems to be blocked. (Weatherall [2018], pp. 338–39)

Let us assess this equivocation argument. Recall Weatherall’s central
contentions concerning comparisons of models via $1_{M}$:

> [R]elative to $1_{M}$, $\left(M,g_{ab}\right)$ and
> $\left(M,\tilde{g}_{ab}\right)$ are not equivalent, physically or otherwise.
> The reason is that there exist points $p\in O$ at which
> ${\left(g_{ab}\right)}_{|p}\neq{\left(\tilde{g}_{ab}\right)}_{|1_{M}\left(p\right)}$.

So far, this observation is as much an elaboration of the hole argument as a
response to it. Regarding $\mathcal{M}$ and $\widetilde{\mathcal{M}}$ as
physically inequivalent (that is, as representing distinct possibilities) when
compared via $1_{M}$ is central to both arguments. On the indeterminism
argument, the claim is that the laws together with the state of spacetime
outside of $O$ (that is, outside of the ‘hole’) fail to fix whether the state
of $O$ is as described by $\mathcal{M}$ or by $\widetilde{\mathcal{M}}$. On
the underdetermination argument, the claim is that no observation could tell
us whether the state of $O$ is as described by $\mathcal{M}$ or by
$\widetilde{\mathcal{M}}$. That
${\left(g_{ab}\right)}_{|p}\neq{\left(\tilde{g}_{ab}\right)}_{|1_{M}\left(p\right)}$
is precisely the point of both arguments: amongst the things that the laws do
not fix, or that observations cannot disclose, are, _inter alia_ , which
particular geometrical properties get to be instantiated by (the spacetime
point represented by) $p$.

Weatherall, however, goes on:

> Consider an observer sitting at a point of spacetime represented by the
> point $p\in O$ of $\left(M,g_{ab}\right)$. If one were to attempt to
> represent that same observer’s location by the point
> $1_{M}\\!\left(p\right)=p$ of $\left(M,\tilde{g}_{ab}\right)$, one would
> conclude that the observer would see measurably different metrical
> properties, different curvature, and so on. In general, only one of these
> assignments can be correct.

At this point Weatherall’s failure to distinguish clearly between the
indeterminism and underdetermination versions of the hole argument becomes
significant. His claim that at most one of $\mathcal{M}$ and
$\widetilde{\mathcal{M}}$ correctly characterizes the observations made by an
observer inside the hole, far from blocking the indeterminism version of the
argument, in fact makes it more acute! For, if Weatherall is to be believed,
what the matching regions of the two models and laws fail to fix are not
merely (i) metaphysical but physically undetectable facts concerning which
spacetime points have which properties (as many have thought), but rather (ii)
states of affairs that are observationally distinguishable.

In light of Weatherall’s apparent failure to spot this problem, a few features
of the indeterminism version of the hole argument are worth re-emphasizing.
Precisely because the diffeomorphism $d$ acts trivially outside of $O$, it is
uncontroversial that spacetimes represented by $\mathcal{M}$ and
$\widetilde{\mathcal{M}}$ are to be regarded as completely—not just
empirically—identical in the region outside of $O$.555This is not to deny that
one could concoct contexts where $\mathcal{M}$ and $\widetilde{\mathcal{M}}$
are deployed to represent spacetimes that differ on $M\setminus O$. The hole
argument, however, is clearly not such a context. Regarding $\mathcal{M}$ and
$\widetilde{\mathcal{M}}$ as equivalent outside of $O$, but as differing
within $O$, therefore does not involve any allegedly illegitimate equivocation
between $1_{M}$ and $\tilde{\psi}$, for $1_{M}$ and $\tilde{\psi}$ agree
completely outside of $O$; only comparisons via $1_{M}$ ever need be
considered. Weatherall, moreover, appears to concede, at least for the sake of
argument, that comparisons via $1_{M}$ are legitimate in themselves and agrees
that, so compared, the models are not physically equivalent, because of their
different attributions to the points in $O$. He therefore seems to walk
headlong into the indeterminism version of the hole argument.

To the extent that comparisons via $\tilde{\psi}$ are sometimes invoked in the
context of the indeterminism argument, it is after the event, in order to
downplay the apparent problem: perhaps the theory is, strictly speaking,
indeterministic but the facts not fixed are esoteric metaphysical matters
concerning point identities; they are not facts that affect the theory’s
observable predictions (for consider $\tilde{\psi}$!). Weatherall, however,
appears to view $\mathcal{M}$ and $\widetilde{\mathcal{M}}$ as not even
empirically equivalent if compared via $1_{M}$. It is this claim, and the
underdetermination version of the hole argument, to which we now turn.

We start by noting one sense in which Weatherall’s critique fails to engage
directly with what we take to be the natural way to pose the
underdetermination problem. When introducing the problem above, we considered
(like Weatherall) an observer located within the region $O$ where the hole
diffeomorphism acts non-trivially. Idealizing somewhat, suppose once again
that, as it occurs in model $\mathcal{M}$, $p\in M$ represents the observer’s
location at the moment of the relevant observation event. Our contention is
that in model $\widetilde{\mathcal{M}}$ it is then $d(p)=q\neq p$ that
represents the same observer’s location at the same moment of their
trajectory. To assert this is not to switch illegitimately from comparing the
models via $1_{M}$ to comparing them via $\tilde{\psi}$. It is simply to
recognize that the relevant fields representing the observer’s trajectory,
their measuring instruments and the results of the measurements have been
dragged by $d$ from $p$ in $\mathcal{M}$ to $q$ in $\widetilde{\mathcal{M}}$.
$1_{M}$ remains the official map by which we identify points across models.
That does not by itself imply, however, that it can also be used to identify
the perspectives of observers across models.

As Weatherall notes in a footnote to the passage quoted (Weatherall [2018], p.
336, fn. 20), his suggestion that, when compared via $1_{M}$, $\mathcal{M}$
and $\widetilde{\mathcal{M}}$ should count as representing observationally
discernible states of affairs echoes claims made by Tim Maudlin, in the
context of the static shift argument against absolute space. (Recall that
static shifts of a Newtonian world are time-independent rigid translations of
the entire material content of the universe.) It is worth briefly reviewing
Maudlin’s position, for it involves two distinct claims. The first, though
controversial, is one that we are inclined to accept; the second we reject. An
analogue of the first claim can be used to block the underdetermination
argument in the context of general relativity; the second claim, however, is
no more defensible in the context of general relativity than it is in the
context of Newtonian physics.

What, then, is Maudlin’s position regarding static shifts? The claim with
which we are sympathetic is that, even if one acknowledges that possible
Newtonian worlds can differ merely by static shifts, the existence of this
plurality of possibilities does not pose an epistemological challenge.666In
contrast, as Maudlin emphasizes, the case of worlds differing by kinematic
shifts, that is, differing by a time-independent and uniform change in the
absolute velocities of all material bodies, does pose an epistemological
challenge. Prior to Maudlin, variants of the same point were made by Horwich
([1978]), Field ([1985]), and Teller ([1987]). To paraphrase Maudlin, my
acknowledging that there is a genuine distinction between a world in which my
desk777That is, the desk belonging to the coauthor currently typing. is here
and a world in which it is three metres north of here (but in which all
spatial relations between bodies remain just the same) does not mean that
there are any positional facts of which I am ignorant. I cannot contemplate
these two possibilities without knowing full well that the possibility
corresponding to the actual world is the first. To make a closely related
point in terms of a class of statically-shifted models apt to represent these
possibilities: given an arbitrary choice of one of the models from this class
to represent the actual world, I know that the worlds represented by the other
models are not my own. (We stress that the claim that static shifts pose no
epistemological problem has not gone unchallenged—see (Dasgupta [2015]) for
recent discussion.)

A sympathetic reading of Weatherall interprets him as making an analogous
point in the context of the underdetermination argument. Suppose that we
stipulate that $p$ is to be the point of $M$ that represents our observer at
the relevant observation event. Then, relative to that stipulation, it will be
determined that it is $\mathcal{M}$ (say) and not $\widetilde{\mathcal{M}}$
that correctly assigns to $p$ the geometrical properties that the observer
detects. To put the point another way, there is, on reflection, something
suspect about a claim that we made when originally framing the
underdetermination problem, namely, that no measurement that an observer might
perform at the relevant moment could determine whether they are located at
(the point represented by) $p$ or at (the point represented by) $q$. For what
determines which spacetime points are represented by $p$ and $q$? Plausibly,
the only way that this reference relation can be tied down is for the observer
(or us) to choose arbitrarily one such model as representing their world.
Relative that choice, the other models will represent possibilities that the
observer knows to be (for them) counterfactual.

As noted, we are sympathetic to this response to the underdetermination
argument.888It is worth stressing that this Maudlin-style move is ineffective
against the hole argument in its indeterminism form. Even if one can know by
stipulation which model represents my actual future, it remains the case that
the past and the laws do not determine that the future is as represented in
this model rather than as represented, counterfactually, in hole-diffeomorphic
models. On our understanding of this response, although one is not ignorant of
which possibility represented by the models is one’s own, the possibilities
are not observationally distinguishable. Maudlin, however, does claim that
they are and, as we have seen, Weatherall follows his lead. We think that this
is a mistake.

In arguing his case, Maudlin switches from the example of static spatial
shifts to time translations. In order to catalogue efficiently our points of
disagreement, we quote him at length:

> A universe created 15 billion years ago is observationally distinguishable
> from one just like it (that is, having a qualitatively identical total
> history) which began within the last four minutes. Things would look awfully
> different if the big bang had occurred in the last half hour. Of course, if
> the big bang had occurred four minutes ago then in another 15 billion years
> there might be someone who looks just like me writing a sentence that looks
> just like this. But that person would have no difficulty determining that he
> is not alive now, just as I have no difficulty knowing that I will not be
> alive then. (Maudlin [1993], p. 190)

We respond: would things look awfully different if the big bang had occurred
in the last half hour? To whom? Things would look different to someone
located, in this counterfactual universe, at the very time we find ourselves
at right now, but there is no one in this world located at that time (since
the world is hypothesized to have a qualitatively identical total history to
that of the actual world and no observer was around four minutes after the big
bang).

Maudlin concedes that, if the big bang had occurred four minutes ago then, in
another 15 billion years, there might be someone who looks just like him
writing a sentence just like the one quoted. But first note that, since this
universe is qualitatively identical, it is not merely the case that there
might be such a person—in fact, of course, there will be such a person.
Secondly, and crucially, this person who looks just like Maudlin is none other
than Maudlin himself, occupying a moment in time in this counterfactual
possibility different to his actual temporal location.999At least, this is
what haecceitists like Maudlin should say. Even a counterpart theorist will
agree that the Maudlin-like individual of this counterfactual possibility
underwrites _de re_ truths about what Maudlin himself would have observed had
the universe been created only four minutes ago. In other words, the
counterpart theorist agrees that, had things been as envisaged in the
counterfactual scenario, Maudlin’s experiences would have been qualitatively
identical to his actual experiences. So in this counterfactual world his
observations, moment by moment, throughout his illustrious career, are
qualitatively identical to his actual observations. The same goes for every
other observer. In this sense, the worlds are observationally
indistinguishable. Maudlin may know that the world described is not the actual
world, but this is not empirical knowledge; he did not come by this knowledge
by checking experimentally in 1993 that the big bang had not happened only
four minutes previously.

The parallel between Maudlin’s position and Weatherall’s is clear. Just as
Maudlin contrasts the goings on at a fixed time to argue that his two worlds
are observationally distinguishable, so Weatherall contrasts the field values
at a fixed manifold point in the two models to argue that at most one of the
worlds they represent correctly captures what an observer at that point would
measure. There is also, however, an important difference between Maudlin’s and
Weatherall’s positions that is worth dwelling on.

Maudlin’s worlds—the actual world and its rigidly shifted counterparts—include
all relevant observers. They do so because they are complete universes.
Similarly, any mathematical models apt to represent such worlds will model all
observers, and not just the content of their observations. In general,
explicit representation, at least in an idealized manner, of the observer, as
a physical system within spacetime, naturally leads to what one might call an
‘immanent’ conception of empirical (in)equivalence:

> Two models are empirically distinct just in case there are relevant
> relational differences between the field configurations in each.101010Of
> course, identifying the ‘relevant relational differences’ which contribute
> to such empirical differences is a non-trivial matter. See for example (Read
> and Møller-Nielsen [2020]) for discussion.

In our view, this is the notion of empirical equivalence most relevant to
discussions of the hole argument. It is the notion of empirical equivalence
presupposed in (Earman and Norton [1987], pp. 521–22). It is in accord with
Stein’s call to ‘schematize the observer’ ([1995]).111111As Curiel writes (in
the context of an extended discussion of Stein), ‘At bottom, then, what secure
epistemic content a scientific theory has must rest in large part on the
meanings expressed in the sound articulation of experimental knowledge, for
that is the final arbiter of empirical success. This requires at a minimum
that we be able, at least in principle, to construct appropriate and adequate
representations of actual experiments and observations in the frameworks of
our best scientific theories, that is, representations of physical systems and
experimental apparatus in relation to each other as required by actual
experiments, not just representations of physical systems _simpliciter_ , in
abstraction from experimental practice’ (Curiel [unpublished], p. 10). On the
immanent conception of empirical equivalence, hole-diffeomorphic models are
empirically equivalent, even when $1_{M}$ is taken as the standard of cross-
model point identity. To insist on this is not, _pace_ Weatherall, to engage
in an illegitimate equivocation between $1_{M}$ and $\tilde{\psi}$.

When models are compared via $1_{M}$, Weatherall implicitly rejects the
immanent conception in favour of the following criterion:

> Two models are empirically distinct just in case they assign different field
> values to the same manifold points.

(We are here assuming that there is no ‘internal’ gauge redundancy in how
field values represent physical quantities.) Given this assumption, if
$g_{ab}\left(p\right)\neq\tilde{g}_{ab}\left(p\right)$, then $\mathcal{M}$ and
$\widetilde{\mathcal{M}}$ are empirically distinct.

One way in which one might seek to justify this criterion is via (what we will
call) the ‘transcendental’ conception of empirical (in)equivalence, according
to which the ‘observer’ is explicitly recognized to be omitted from the
models. This conception is arguably implicit in Weatherall’s discussion of
what an observer at $p$ would measure according to the different
models.121212As we have effectively already noted, this conception is not
compatible with the completeness of the worlds featuring in Maudlin’s
discussion. It is therefore not available to Maudlin as a defence of his
claims concerning observational (in)equivalence.

Are hole-diffeomorphic models compared using $1_{M}$ to be regarded as not
empirically equivalent according to a transcendental conception of empirical
equivalence? Does this conception licence letting spacetime points or regions
go proxy for the fixed perspective of a potential observer across a class of
hole-diffeomorphic models? It is not clear to us that it does.

According to the interpretation of models that underwrites the transcendental
conception of empirical equivalence, a model represents only a proper
subsystem of the universe; the observer, in particular, is omitted from the
model. It has recently been recognized that this way of thinking about models
promises to provide a systematic way to adjudicate when symmetry-related
models are empirically equivalent or, more generally, cannot be interpreted as
representing distinct physical situations.131313Our thinking here has in large
part been influenced by Wallace ([forthcoming]), whose position we take
ourselves to be endorsing. For example, returning to the static shift case, it
is notoriously controversial whether a substantivalist can always regard
statically-shifted models of a Newtonian theory as physically equivalent. But
all hands should agree that the models can be thought of as empirically
distinct when taken to model a subsystem of the universe that can stand in
different spatial relations to an external observer, who can detect those
spatial differences using physical means (such as light) that fall outside of
the physics represented in the models. A similar story can underwrite the
physical inequivalence of certain models of general relativity related by
diffeomorphisms that satisfy certain non-trivial boundary
conditions.141414That such models should be regarded as physically
inequivalent has been urged, in particular, by Belot ([2018]), albeit not for
reasons to do with empirical (in)equivalence, but rather for reasons to do
with the construction of conserved quantities in general relativity. Note that
Belot (in contrast to Wallace) does not believe that recognizing the models’
physical inequivalence requires that they be regarded as modelling proper
subsystems of the universe.

Does thinking of models related by a hole diffeomorphism as representing only
subsystems of the universe allow one to regard them as empirically
inequivalent? We do not see how it can. Following Wallace ([forthcoming]), the
question to ask is how the symmetry relating the models extends to a set of
more inclusive models that include the observer and their detection of the
relevant differences. Recall that, according to Weatherall’s account of the
models’ empirical distinctness, the unmodelled observer is imagined to be
located at a spacetime $p$ on which the diffeomorphism acts non-trivially. If
we now imagine including fields representing the observer and their ability to
detect differences in the values of the originally modelled fields, it seems
there are two possibilities: either the diffeomorphisms are symmetries only
when acting on all the fields in the expanded model, including those
representing the observer and their measurements, or the diffeomorphisms are
‘subsystem-local’ in Wallace’s terminology, and remain symmetries even when
acting only on the originally modelled fields.

In the latter case, one might try to argue that the original models should be
treated as physically inequivalent (after all, the expanded models differ over
the relative location of physical fields, for example, those representing
spacetime curvature and those representing an observer and their material
measuring devices). However, precisely because the fields representing an
observer are left untransformed when the diffeomorphism acts on the fields of
the original model, the differences are unobservable (so the models remain
observationally equivalent) and, moreover, any theory governing the expanded
models will, very obviously, be viciously indeterministic.

It is far more plausible, therefore, to suppose that any theory governing the
extended models that represent the observer will be invariant under
diffeomorphisms only if they act on all the fields. But now we no longer have
any basis to follow Weatherall in interpreting the map $1_{M}$ as identifying
the fixed perspectives of potential observers across models, for in the
extended models such observers are dragged along together with the fields that
they are observing by any diffeomorphism that is a symmetry of the theory.

To sum up the conclusions of this section: Weatherall’s claim that when models
are compared using $1_{M}$ they count as empirically distinct, and that
therefore the hole argument is blocked, fails because: (1) the claim leaves
entirely untouched the indeterminism version of the hole argument (which does
not presuppose the models are empirically equivalent, only that they are
equivalent in regions outside of the hole); and (2) the claim that models
count as empirically distinct is anyway suspect, even when assuming a
transcendental conception of empirical equivalence.

By considering the indeterminism and underdetermination versions in turn, we
take ourselves to have shown that neither version of the hole argument
involves an illegitimate equivocation between $1_{M}$ and $\tilde{\psi}$. For
anyone who remains unconvinced, however, we close this section by offering the
following simple analogy.

Suppose that two identical twins, Alice and Barbara, cannot be told apart by
visual inspection. Now consider the following two situations, both involving
the twins standing in front of you in plain sight. In the first, Alice stands
to the left of Barbara and wears a red hat, while Barbara wears a blue hat. In
the second, Barbara stands to the left of Alice and wears the red hat, while
Alice wears the blue hat. The twins contrive to look otherwise identical in
the two situations.

We think that the following two claims about this setup are obvious and
uncontroversial. The two situations are observationally indistinguishable, at
least by sight: one cannot tell, as the onlooker, which of the two situations
one is confronted with. Nonetheless, there is a (physical) difference between
the two situations: in one it is Alice in the red hat; in the other it is
Barbara.

Now consider the simple analogues of Weatherall’s $1_{M}$ and $\tilde{\psi}$.
$1_{M}$ corresponds to a map between the two situations that maps Alice to
Alice and Barbara to Barbara. It does not preserve all qualitative (visually
inspectable) features of the situation. In particular it maps someone wearing
a red hat to someone wearing a blue hat. $\tilde{\psi}$ corresponds to a map
between the two situations that maps Alice to Barbara and Barbara to Alice. It
preserves all qualitative features but does not map each twin, as she is in
one situation, to herself, as she is in the other.

According to Weatherall, the two claims that we asserted above to be obvious
and uncontroversial can only be jointly accepted if one engages in an
illegitimate equivocation between the two maps. If one says that the
situations are observationally indistinguishable, one must be comparing them
via the analogue of $\tilde{\psi}$ and so cannot, in the same breath, also say
that they differ because different twins wear the two hats. If one focuses on
the fact that different twins wear different hats, one is comparing them via
the analogue of $1_{M}$, in which case one cannot at the same time recognize
that the situations are observationally indistinguishable.

We hope that the unreasonableness of this position is clear without our
needing to labour the point further. It may be helpful to note that the
observational indistinguishability of the two situations is not a property
that holds of the pair of them only relative to, or when compared via, (the
analogue of) $\tilde{\psi}$. Rather, the existence of this map, which matches
up without exception qualitatively identical features, underwrites an absolute
property possessed by the pair of situations—a feature that holds full stop.
This is why there is no equivocation involved in jointly attending to (i) the
fact that $\tilde{\psi}$ preserves all qualitative features of the spacetime,
because it is an isomorphism, but also (ii) the fact that different points are
assigned different properties, precisely because $1_{M}$ is not an
isomorphism.

##  5 The Argument from Mathematical Structuralism

Up to this point, we have assumed that comparing the models $\mathcal{M}$ and
$\widetilde{\mathcal{M}}$ via $1_{M}$ is legitimate. As the previous section
has made clear, this is the comparison presupposed in standard presentations
of the hole argument, where $1_{M}$ is assumed, tacitly or otherwise, to track
which manifold points represent the same spacetime points across models.
Claims of observational indistinguishability are either irrelevant to the hole
argument (when the issue is indeterminism) or do not require a problematic
switch from comparison via $1_{M}$ to comparison via $\tilde{\psi}$. We turn
now to the question of whether it is indeed legitimate to compare models via
$1_{M}$.

Weatherall’s considered view seems to be that such comparisons are not
legitimate. Towards the end of his article, he canvasses three possible views
that Earman and Norton might be taken to attribute to the spacetime
substantivalist concerning how mathematical models represent spacetime.
According to the first, the substantivalist takes spacetime to be represented
by just the differentiable manifold of the models, not by the manifold plus
metric. According to the second, the substantivalist takes spacetime to be
represented by a structure richer than a Lorentzian manifold. The third view,
which is the one that Weatherall ultimately attributes to Earman and Norton’s
substantivalist, involves representing spacetime by

> a Lorentzian manifold (and no more), such that the manifold is understood to
> represent spacetime points in some sense prior to, or independently of, the
> value of the metric at those points […] The pushforward of the metric along
> an automorphism of the manifold would represent different property
> assignments to the same spacetime points, corresponding to _prima facie_
> different physical situations. (Weatherall [2018], p. 345)

Of this third option, he writes:

> this view, I take it, is incompatible with the background views on
> mathematics described in Section 1. The problem is essentially the same as
> for the hole argument itself: this view depends on taking the identity map
> to provide a prior notion of when the points of two Lorentzian manifolds are
> the same. (Weatherall [2018], p. 345)

More on the background views on mathematics that Weatherall describes in the
first section of his paper in a moment. First, a brief comment on whether
Weatherall is correct to attribute the third view to Earman and Norton’s
substantivalist.

Notoriously, Earman and Norton argued that the metric field should count as a
field contained within spacetime, not as (part of) the characterisation of the
spacetime container (Earman and Norton [1987], pp. 518–19). This suggests that
the first of the views that Weatherall surveys better matches Earman and
Norton’s intentions. Weatherall’s reasons for rejecting this possibility is
that he believes it would commit Earman and Norton’s substantivalist to ‘the
radical view that any two spacetimes are equivalent as long as their
associated manifolds are diffeomorphic—whether the two agree on other
structure’ (Weatherall [2018], p. 344). But the view, of course, has no such
consequence. Earman and Norton’s substantivalist will regard non-isometric
models as representing obviously physically distinct possibilities, but
possibilities that differ in terms of their spacetimes’ having (qualitatively)
different content. This substantivalist will nonetheless regard these
possibilities as involving spacetimes that are, in terms of their intrinsic
characters, qualitatively identical.151515Weatherall’s use of ‘spacetimes’ to
refer to mathematical models is perhaps partly responsible for generating the
appearance of a problem. His allowing the stress-energy of matter to be
implicitly defined by $g_{ab}$, via the Einstein field equations, also serves
to blur the distinction between models differing over their characterisation
of spacetime _per se_ and models that disagree only over its content.

In fact, in terms of Weatherall’s three-way classification, Earman and Norton
substantivalist’s position is best understood as a combination of views (1)
and (3): one represents spacetime itself as a differentiable manifold (‘and no
more’) but one also takes the manifold to represent spacetime points ‘in some
sense prior to, or independently of,’ the value of the metric (and that of any
other field) at those points.

Earman and Norton are in the minority in classifying the metric as content
rather than as an aspect of the spacetime container. A number of authors have
argued against this component of their position.161616Critics of this aspect
of Earman and Norton’s position include Maudlin ([1990], pp. 545–49), Hoefer
([1996], pp. 11–13) and Pooley ([2006], pp. 99–101). Part of Earman and
Norton’s argument rests on the claim that the metric carries (gravitational)
energy and momentum. For recent discussion of this vexed topic, see for
example (Dürr [2019b]; Dürr [2019a]; Read [2020]). The third of Weatherall’s
options is, therefore, a more faithful characterisation of an interpretative
stance implicit or (often) explicit in much of the philosophical literature
that followed Earman and Norton’s paper. What, according to Weatherall, is
wrong with it?

Near the beginning of his article, Weatherall sets out the following two
interpretative principles:

> (1) our interpretations of our physical theories should be guided by the
> formalism of those theories; and (2) insofar as they are so guided, we need
> to be sure that we are using the formalism correctly, consistently, and
> according to our best understanding of the mathematics. (Weatherall [2018],
> p. 330)

The application that Weatherall seeks to make of these principles focuses only
on matters of sameness and equivalence. Principle (2) enjoins us to attend to
our best understanding of the mathematics involved in a theory’s formalism. In
this connection, Weatherall endorses a view that he takes to be ‘a variety of
mathematical structuralism’ (Weatherall [2018], fn. 8): that isomorphism is
the relevant standard of sameness in mathematics.171717Varieties of
mathematical structuralism can differ over whether isomorphic models are
understood to be (a) numerically distinct but in some sense treated ‘as if’
identical, versus (b) numerically identical. Weatherall clearly does not
assume anything as strong as (b) (cf. Weatherall [2018], p. 331, fn. 8). The
‘univalence axiom’ of homotopy type theory affords a natural means of
formalizing mathematical structuralism—see (Dougherty [2018], p. 104). Our
concerns here also speak against any appeal to homotopy type theory as a means
of evading the hole argument. For more on homotopy type theory and the hole
argument, see (Dougherty [2020]) and (Ladyman and Presnell [2020]). Applying
this to the case at hand, if models of general relativity are taken to be
Lorentzian manifolds, isometry is the standard of isomorphism. So, hole-
diffeomorphic models such as $\mathcal{M}$ and $\widetilde{\mathcal{M}}$,
which are isometric, count as ‘mathematically equivalent’.

Turning to principle (1), what does the mathematical equivalence of such
models imply about their physical interpretation? Weatherall’s official answer
is that isomorphic models should be taken to have the same ‘representational
capacities […] if a particular mathematical model may be used to represent a
given physical situation, then any isomorphic model may be used to represent
that situation equally well’ (Weatherall [2018], p. 332). We take this thesis
to correspond precisely to the position that Fletcher—a self-declared ally of
Weatherall on these matters—labels REME (‘Representational Equivalence by
Mathematical Equivalence’ (Fletcher [2020], p. 233)).

We concur with Weatherall and Fletcher that REME is a legitimate principle of
mathematical representation. REME, however, very obviously fails, by itself,
to block the hole argument, or to rule out the allegedly problematic use of
isomorphic models deployed in standard presentations of the argument. All that
this use requires is that if model $\mathcal{M}$ (say) is taken to represent a
particular possible world $W$, then $\mathcal{\widetilde{M}}$ represents (or,
better, may be taken to represent, perhaps by further specifying the
representational context) a distinct (but merely haecceitistically distinct)
possibility $\widetilde{W}$. This is all completely compatible with
$\mathcal{M}$ and $\mathcal{\widetilde{M}}$’s having identical
representational capacities. That merely requires that
$\mathcal{\widetilde{M}}$ might equally well have been chosen initially to
represent $W$ (in which case, some other hole-diffeomorphic model, related to
$\mathcal{\widetilde{M}}$ just as $\mathcal{\widetilde{M}}$ is related to
$\mathcal{M}$, could be taken to represent $\widetilde{W}$).181818Roberts
([2020], p. 258) and Gryb and Thébault ([2016], pp 566–67) make essentially
the same point. There are also parallels here with the ‘semantic
relationalism’ of Fine ([2003, 2007]), according to which there is no semantic
difference between variables such as $x$, $y$, as illustrated by the fact that
one’s choice of variable in formulae such as $x>0$ or $y>0$ is purely
conventional; however, there is a semantic difference between pairs of
variables—as illustrated by the fact that $\left(x,x\right)$ is not
semantically equivalent to $\left(x,y\right)$, for $x>x$ may (after
quantification) express a distinct proposition from $x>y$. (For some critical
comments on Fine’s work, see (Button and Walsh [2018], pp. 13-4).)

REME, interpreted to the letter, therefore fails to block the hole argument.
Nevertheless, it is clear that both Weatherall and Fletcher believe that a
proper understanding of the mathematical formalism of general relativity does
rule out the use of models that the hole argument presupposes. What stronger
constraint on representation are they assuming?

A tempting thought might be that mathematical structuralism goes hand-in-hand
with taking isomorphism equivalence classes of models to be in one-to-one
correspondence with physical possibilities. This, however, is a position that
Weatherall ([2018], p. 332) explicitly rejects, and for good reason. To see
the problem, it is useful to break down this new thesis into two components:
(i) that all mathematically equivalent models represent the same, unique
possibility, and (ii) that any two mathematically inequivalent models
represent distinct possibilities.

These two claims about representation correspond closely (but not
exactly191919While (ii) requires that mathematically inequivalent models
represent distinct possibilities, Fletcher’s RDMI below requires only that
they do not represent the same possibility (equally well).) to two other
theses presented—and criticized—by Fletcher ([2020], pp. 231–33):

RUME: If two models of a physical theory are mathematically equivalent, then
there is a unique physical state of affairs that they represent equally well.

RDMI: If two models of a physical theory are not mathematically equivalent,
then it’s not the case that there is a unique physical state of affairs that
they represent equally well.202020RUME abbreviates ‘Representational
Uniqueness by Mathematical Equivalence’; RDMI stands for ‘Representational
Distinctness by Mathematical Inequivalence.’

One of Fletcher’s central illustrative examples in his case against RUME and
RDMI involves schwarzschild spacetimes—spherically symmetric solutions of
general relativity, differing by the so-called ‘schwarzschild radius’ $R_{S}$,
which can be interpreted as representing appropriately isolated, non-rotating
stellar bodies (for example stars, black holes, _et cetera_). Against RUME,
Fletcher notes that ‘ _each_ mathematical Schwarzschild spacetime can
represent _any_ physical Schwarzschild spacetime (that is, with any
Schwarzschild radius) through an appropriate choice of units’ (Fletcher
[2020], p. 235). Conversely, non-isomorphic schwarzschild models can represent
one and the same black hole in different contexts, via a change of units,
undermining RDMI (Fletcher 2020, pp. 237-8).

We are happy to take Fletcher’s arguments against RUME and RDMI as decisive.
But we note that most of these arguments exploit the fact that representation
is a pragmatic matter, always involving some level of idealisation and
abstraction, as well as some more-or-less arbitrary conventions, which can
shift from one representational context to another. Fletcher correctly shows
that RDMI and RUME are implausible principles if taken to hold across
representational contexts. But that leaves open the question of whether they
are legitimate principles when relativized to a fixed representational
context.

In our view, the interesting questions have always been about the potential
joint representational uses of different mathematical models: whether two
models can be taken to be ‘co-representational’ and, if so, whether they
represent, when so taken, the same or different possibilities.

Occasionally, such questions have been raised explicitly in other discussions
of the hole argument. In assigning a central role to the question of when two
models can be understood as ‘co-intended’, Rynasiewicz ([1994], p. 423) is
investigating precisely the constraints on when two models can be understood
as co-representational. Gryb and Thébault ([2016], pp. 566-7) draw the
distinction between what models can represent ‘taken in isolation’ and what
they can represent ‘taken together’. And Roberts homes in on a closely related
question when he asks whether two isomorphic structures can have ‘co-
representational capacity’ (whether they can represent the same state of
affairs ‘at once’—a thesis he dubs ‘strong Leibniz equivalence’). He contrasts
this question from the REME-like question of whether they have ‘equal
representational capacity’ (‘weak Leibniz equivalence’), where here their
ability to represent the same state of affairs need not be ‘at once’ (that is,
it need not be relative to the same representational context) (Roberts [2020],
p. 252). Focusing, then, on the co-representational capacities of models, what
might plausible, context-relativized variants of RUME and RDMI look like? We
propose the following as natural emendations of Fletcher’s theses:

RUME∗: If two models of a physical theory are mathematically equivalent and if
one model is chosen to represent (to some degree of accuracy) a particular
physical possibility (thereby fixing the representational context), then the
other model, relative to that choice (that is, relative to the same
representational context), also represents that physical possibility (to the
same degree of accuracy).

RDMI∗: If two models of a physical theory are not mathematically equivalent
and if one model is chosen to represent (to some degree of accuracy) a
particular physical possibility, then the other model, relative to that
choice, does not represent that physical possibility (to the same degree of
accuracy).

To these we add a third thesis—‘Mathematical Inequivalence follows from
Representational Distinctness’:

MIRD∗: If, relative to some representational context, two models of a physical
theory represent distinct physical possibilities, then they are not
mathematically equivalent.

Before assessing these theses, the logical relationship between MIRD∗ and
RUME∗ should be noted. MIRD∗ is equivalent to the claim that if two models of
a physical theory are mathematically equivalent and one of the models
represents a particular physical possibility relative to some representational
context, then the other model, relative to that context, either represents the
same possibility or fails to represent a possibility. It therefore corresponds
to a natural weakening of RUME∗, where the original consequent is replaced by
a disjunction of that original together with an alternative whose salience
only becomes apparent when one is considering the context-relativized
principles.

One consequence of this is that someone denying RUME∗ need not, thereby, be
committed to denying MIRD∗. One might deny RUME∗ and affirm MIRD∗ because one
thinks that, if one of two isomorphic models is chosen to represent a
particular possibility then the other, though equally apt to represent that
possibility (as per REME), acquires the status of an ‘unintended
interpretation’ (Rynasiewicz [1994], p. 420). One is again reminded of lost
insights from the discussion that flourished in the immediate aftermath of
Earman and Norton’s paper. Butterfield ([1989]) lays out essentially the same
three options in the following terms. First he asks of two isomorphic models:
do they represent the same possibility? (This is his question ‘(Same?)’
(Butterfield [1989], p. 12).) If the answer is No, one option is that they
represent distinct possibilities (Butterfield’s ‘(Each)’). But one might also
hold that only one model may represent a possibility relative to any given
representational context (cf. Butterfield’s ‘(One)’).

Fletcher, in motivating his discussion of RUME, RDMI and REME, seeks to
attribute to a number of authors a misguided commitment to RUME. In our view,
the more charitable reading of much of the literature to which Fletcher
adverts sees it as committed to the (weaker) starred variant (or to a close
relative thereof). In particular, Earman and Norton’s ‘Leibniz Equivalence’
states that ‘diffeomorphic models represent the same situation’ (Earman and
Norton [1987], p. 522). Surely, once the need to attend to the
representational context is taken into account, this is most plausibly
interpreted as RUME∗, not as RUME. Such a reading makes good sense of the
alleged conflict with substantivalism. We have seen that the interpretation of
hole-diffeomorphic models to which the substantivalist is allegedly committed
to acknowledging as legitimate, that is, as jointly representing possibilities
that differ merely haecceitistically, is not in conflict with REME. It is in
conflict with RUME∗ because (and here we finally get the heart of the matter)
it directly conflicts with MIRD∗, which, as we noted, is entailed by RUME∗.

The starred theses might make better sense of much of the now-canonical
discussions, but do any of them plausibly follow from Weatherall’s two
principles concerning how mathematics should guide interpretation? RUME∗, or
Leibniz Equivalence, corresponds to the thesis which Roberts [2020] dubs
‘strong Leibniz equivalence’.212121In fact, Roberts’ ‘strong Leibniz
equivalence’ states only that isomorphic models _can_ represent the same
physical state of affairs ‘at once’. While weakening ‘represent’ to ‘can
represent’ is well-motivated for a cross-context principle, favouring REME
over RUME, it is less obviously motivated when one is relativizing to a
representational context. This is another reason to take traditional
(unmodalized) statements of Leibniz Equivalence to be implicitly context-
relativized and so akin to RUME∗ (rather than as expressing an obviously
unmotivated principle like RUME). As such, it is vulnerable to the telling
criticisms to which he subjects it. But, as we have seen, while RUME∗ entails
MIRD∗, it is not equivalent to it. Arguably it is this latter thesis that
deserves to be the focus of attention and, indeed, Fletcher comes close to
giving an explicit argument for it. He writes:

> Lorentzian manifolds may not exemplify [all the] properties of the states of
> affairs they represent, but all the properties they _do_ exemplify—those not
> abstracted away—are the same for isomorphic manifolds. This is precisely
> encoded in the mathematical models themselves with the interpretation of
> isomorphic objects in a mathematical category as being equivalent _as
> objects in that category_. (Fletcher [2020], p. 239)

Put slightly differently, two situations are distinct if one possesses a
feature that the other lacks. So for two models jointly to represent two
distinct situations (and to represent them as distinct) one model must
represent one situation as possessing a feature that the other model
represents the other situation as lacking. But, if we are treating the models
as objects of a certain category, then this representational difference has to
correspond to a difference between them considered as objects of that
category. Since isomorphic models are precisely those models that do not
differ as objects of the category, such models cannot (if really being treated
as objects of that category) differ representationally in the manner required.

That, at any rate, is our best effort at reconstructing the argument from
mathematical structuralism. We finish with three critical observations
concerning where we think this leaves the debate.

First, it bears stressing how natural and apparently appropriate is the
particular use of isomorphic models that Weatherall and Fletcher criticize. A
substantivalist who believes that possible spacetimes can differ merely
haecceitistically is after a way to represent the structure of such a
spacetime, which (let us suppose) really is no richer than that of a
Lorentzian manifold, and at the same time talk about two possibilities that
exemplify the very same structure but differ over which individuals possess
which particular properties. How better to do this than to use isomorphic
models of the appropriate structural type that differ merely over which of the
base elements of their sets are assigned the structural properties common to
both models?

The second of the three options Weatherall canvassed for Earman and Norton’s
substantivalist was to represent spacetime as having a structure richer than
that of a Lorentzian manifold. Fletcher talks of including ‘spacetime point
haecceities’ (Fletcher [2020], p. 239). But why should the substantivalist be
saddled with such baroque machinery in order to have two models that can stand
for the different possibilities that they acknowledge when these possibilities
differ merely over which individuals have which properties? Misleading
terminological etymology notwithstanding, haecceitists need not be committed
to haecceities.

Our second observation is that Fletcher himself, in one of his arguments
against RUME, endorses exactly this type of co-representational use of
isomorphic models to represent distinct possibilities! His example involves a
collection of structures containing nothing but a Minkowski metric on a
manifold diffeomorphic to $\mathbb{R}^{4}$, together with an inextendable,
‘jointed’ timelike curve composed of an initial geodesic segment followed by a
segment exemplifying constant proper acceleration. He holds that this
collection of structures can represent a variety of possibilities for a lone
particle that ceases, at some moment, to move inertially and starts to
accelerate. The possibilities in question are supposed to differ merely over
when and in which direction the particle starts to accelerate. After taking a
particular model to represent one such a history, Fletcher writes:

> Users of relativity theory, I claim, would intend for the theory to endorse
> that the particle could have swerved (at the same acceleration) in another
> direction, even at another time, than it did in the above model. Such
> alternative states of affairs could be realized by a spatial rotation or
> time translation acting on the above model. Yet the resulting model is in
> fact isomorphic to the one above. (Fletcher [2020], p. 237)

In order for these models to (co-)represent states of affairs that differ only
over when and in which direction the particle swerves, a standard of cross-
model point identity needs to be assumed (as Fletcher himself spells out)
which, although an isometry of the metric substructure, is not, _pace_
Fletcher ([2020], p. 240), an isomorphism of the relevant category (that is,
that of ‘Minkowski spacetimes with distinguished worldlines’). Fletcher claims
that the inclusion of a distinguished worldline in the example serves to
differentiate the case from that of the hole argument. We, by contrast, take
its inclusion to make the cases precisely analogous. In both cases, the co-
representational function of the models presupposes comparing isomorphic
models via a map that is not (or need not be) an isomorphism. And in both
cases the various physical states of affairs supposedly represented by
different isomorphic models differ merely haecceitistically. (Lone particle
worlds that differ only over when and in which direction the particle swerves
are worlds that differ only over which spacetime points have the property of
being a location of the particle.)

Our first two observations have questioned whether MIRD∗ should be accepted on
the basis of mathematical structuralism. Our third and final observation
disputes that embracing MIRD∗ on that basis blocks the hole argument.

One theme of Weatherall’s paper is that the formalism of general relativity
does not generate a philosophical problem. We agree with this. One is not
forced to recognize a plurality of physical possibilities merely because a
theory admits as solutions a plurality of isomorphic models. But equally, one
does not rule out the existence of such a plurality merely by prohibiting a
particular interpretation of the formalism.

For Earman and Norton, and the literature they unleashed, the problem was not
the formalism of general relativity _per se_ , but substantivalism.222222Which
means that the formalism is responsible for the problem indirectly, to the
extent that it leads to the postulation of spacetime points. For those who
accept Earman and Norton’s ‘acid test’ of substantivalism, possibilities
involving a common pattern of geometrical properties and a common material
content can differ merely over which substantival points possess which
properties.

Suppose that $C$ is a class of qualitatively identical worlds—and make no
assumptions (Earman and Norton’s ‘acid test’ notwithstanding) about the
cardinality of $C$, which might perhaps contain only a single member. Let
$\mathcal{M}$ and $\widetilde{\mathcal{M}}$ both be members of a class of
isometric spacetime models of a type and particular character that makes them
apt to represent members of $C$.232323So, in line with REME, each model can
serve equally well to represent any element of $C$, however many there are. As
we have seen, this does not block the hole argument. Someone who embraces
MIRD∗ for Weatherall’s and Fletcher’s reasons disavows being able to use
$\mathcal{M}$ and $\widetilde{\mathcal{M}}$ to jointly describe different
members of $C$ (if different members there are). But they do not thereby save
general relativity from indeterminism. If $C$ really does contain a plurality
of members differing in the way envisaged by some substantivalists, then
general relativity according to Weatherall and Fletcher does not distinguish
between the possibilities even to the extent of not being able to refer
differentially to them. It therefore (implicitly) regards them as all equally
possible, which is just to say that, for the relevant type of substantivalist,
the theory is indeterministic.

All hinges on whether, on the assumption of substantivalism, $C$ does indeed
involve the relevant plurality. One cannot answer that question without
engaging with the themes laid bare in the orthodox philosophical
literature.242424Cf. (Teitel [2019], pp. 388–89). One could see here a partial
vindication of what Møller-Nielsen ([2017]) dubs the ‘motivational’ approach
to symmetries over the ‘interpretational’ approach.252525See also (Martens and
Read [2020]; Read and Møller-Nielsen [2018]; Read and Møller-Nielsen [2020]).
In accepting MIRD∗ for formal reasons, Weatherall and Fletcher exemplify an
‘interpretational’ approach, on which symmetry-related models of a given
theory can be ruled out _ab initio_ as (co-)representing distinct physical
states of affairs.262626Indeed, they celebrate their lack of engagement with
metaphysics: see (Weatherall [2018], pp. 330, 343–45) and (Fletcher [2020], p.
240). In contrast, the ‘motivational’ approach would hold that MIRD∗ should
only be adopted once it has been secured by what Møller-Nielsen ([2017], p.
1256) calls a ‘metaphysically perspicuous characterisation’ of the theory’s
ontology. Only by engaging with the metaphysics of space and time can one
confront the question of whether there is or is not a plurality $C$ of worlds
corresponding to an isomorphism class of models.

##  6 Conclusions

There is much in Weatherall’s paper with which we agree—in particular, the
claims that (i) hole-diffeomorphic models of general relativity have identical
representational capacities, and (ii) that mathematical structuralism can be
taken to show that the formalism of general relativity does not generate a
philosophical problem. This notwithstanding, these observations are
insufficient to block the hole argument—for the problem can be posed without
equivocation on the use of $1_{M}$ versus $\tilde{\psi}$, and, ultimately, any
serious attempt to grapple with this problem must involve a careful engagement
with metaphysics.

## Acknowledgements

We are very grateful to Neil Dewar, John Dougherty, Sam Fletcher, Tushar
Menon, Tom Møller-Nielsen, John Norton, Bryan Roberts, and especially Jim
Weatherall, for helpful discussions on this material. We also thank the
anonymous referees for helpful comments. J.R. acknowledges support from the
John Templeton Foundation (grant number 61521).

Oliver Pooley
Faculty of Philosophy
University of Oxford
Oxford, UK
<EMAIL_ADDRESS>
James Read
Faculty of Philosophy
University of Oxford
Oxford, UK
<EMAIL_ADDRESS>
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