

Challenging the Spacetime
Structuralist

Christian Wüthrich∗

30 January 2008

Word count: 4989

Abstract

Structural realist interpretations of generally relativistic spacetimes
have recently come to enjoy a remarkable degree of popularity among
philosophers. I present a challenge to these structuralist interpre-
tations that arises from considering cosmological models in general
relativity. As a consequence of their high degree of spacetime symme-
try, these models resist a structuralist interpretation. I then evaluate
the various strategies available to the structuralist to react to this
challenge.

1 Introduction

Most authors who argue for a structural realist interpretation of spacetime
find their motivation to do so in their belief that the hole argument in general

∗First and foremost, I wish to thank John Earman for proposing the topic to me
and for his many suggestions that have invariably led to improvements of the article.
I am also grateful to Craig Callender and Jonathan Cohen for commenting on earlier
drafts of this paper. Thanks also go to audiences at Lausanne and Geneva. Finally, I
wish to thank my fellow collaborators in the Swiss National Science Foundation project
“Properties and Relations” for stimulating intellectual exchange with them. This research
has been funded in part by the Swiss National Science Foundation project “Properties and
Relations” (100011-113688).

1


relativity naturally leads to this view.1 Often, proponents of this persuasion
see their position as a via media, or, in Mauro Dorato’s (2000) words, a ter-
tium quid, between spacetime substantivalism and relationism. Structural
realists about spacetime side with relationists in their conviction that the
fundamental ontology of spacetime consists of relational complexes rather
than individual objects such as spacetime points, but join substantivalists in
their unabashed realist stance about spacetime. Some authors in this liter-
ature, such as Dean Rickles and Steven French (2006), go as far as to claim
that the “sophisticated” substantivalism and the version of relationism most
prominently defended today, both collapse into spacetime structuralism.2

An altogether different motivation can be found in Jonathan Bain (2006)
who infers spacetime structuralism from the fact that general relativity can
not only be expressed in the standard tensor formalism, but can also be cast
in at least three alternative formalisms: twistor theory, Einstein algebras,
and geometric algebra. For Bain, since these four formalisms afford nearly
intertranslatable expressions of general relativity, it would be foolish to be
wedded to the ontological commitments of any one of them. Instead, we
should only ontologically commit to those structures necessary to support
the mathematical representations of the physical systems and their dynam-
ics, while the exact formulation of what this structure is depends on the
formalism one adopts. Bain, thus, identifies what is common to all represen-
tative frameworks of the same theory, the “structure,” with what we ought
to be realists about.

After fixing what I take to be a sensible characterization of structural
realism in Section 2, I will press on in Section 3 to propose a challenge to
a structural realist interpretation of spacetime that arises from highly (spa-
tially) symmetrical models of general relativity. In these cases, since accord-
ing to the structural realist spacetime points only inherit their properties and
even their individuality from the structure in which they partake, it turns out
that the points’ individuality can no longer be grounded. This result is dis-
astrous as it suggests that the (spatial) universe consists of nothing but one
lonely point. Section 4 addresses and discusses various strategies that may
be pursued in resisting this unattractive implication.

1Cf. for instance Dorato (2000), Rickles and French (2006, 4), and Stachel (2006, 56f).
2Not everyone agrees, see e.g. Pooley (2006).

2


2 Defining Structure

If structural realism can be understood, roughly, as the position according
to which one ought to be realist about the “structure” of a scientific theory,
then this characterization remains empty unless some precisification of the
term “structure” is offered. There is considerable disagreement on this point
among philosophers who consider themselves structural realists, but Rickles
and French (2006, 25) are certainly right in their insistence that structural
realism is not a monolithic position, and neither should it be. At a quite gen-
eral level, structural realism demands that an ontological commitment ought
to be made not to individuals as the primary, most fundamental constituents
of reality, but rather a complex of relations. John Stachel (2006, Sec. 3.1)
identifies three basic types of structural realism, ranging from timid (“there
are objects and relations, with the objects being primary and the relations
being secondary”), to traditional (“there are objects and relations, such that
relations are primary and objects are secondary”), to radical (“there are only
relations, without underlying or accompanying relata”). Call the first two
forms of structural realism non-eliminative and the third eliminative as it
eliminates objects from the basic ontology. Taken at face value, elimina-
tive structural realism as formulated by the third Stachelian option is clearly
incoherent, despite the fact that the rhetoric of some British structuralists
such as French and James Ladyman (2003) indeed suggests that they defend
this extreme position. Given the unintelligibility of this position as taken at
face value, I suspect that such an attribution is plainly false and the position
really defended by these British structuralists differs from the one painted
by their rhetoric. More charitably read, it perhaps means that any given
relation’s relata can in turn be fully individuated by relations. Like Stachel,
“I would not want to be bound by the claim that this is always the case.”
(2006, 54)

Apart from those varieties of structural realism listed by Stachel (2006),
there exists another, in my view quite sensible, characterization of the po-
sition. Instead of insisting that either the objects or the relations must be
primary—where the disjunction is understood to be exclusive—, i.e. instead
of defending either timid or traditional structural realism, one can refuse to
award ontological primacy to either side and consider relations and relata
ex aequo necessary constituents of either the structure in the world or the
structural knowledge the structuralist hopes to attain. Michael Esfeld (2004)
and Esfeld and Vincent Lam (2008) have explicitly endorsed this what they

3


dub modest version of structural realism, and Oliver Pooley (2006, 98) has
argued that the same view—he calls it the no priority view—is the most
plausible form of structural realism. I prefer to call it balanced structural
realism exactly because of the ontological balance it strikes between objects
and relations. For present purposes, I shall presume that an attractive ver-
sion of structuralism can be presented without insisting either on privileging
relations over relata or on eliminating objects altogether. More precisely, let
me assume the following definition: A structure S is a pair 〈O, R〉 which
consists of a non-empty set of concrete relations R (“ideology”) as well as a
non-empty set of physical relata O (“ontology”), the domain of S.

With this definition of structure under our belt, balanced structural real-
ism as defended by Esfeld (2004) and by Esfeld and Lam (2008) then amounts
to an endorsement of the following philosophical position. The “structural”
aspects of a scientific theory, toward which we should entertain a realist atti-
tude, relevantly capture the structure of the external world. The fundamen-
tal scientific theories thus reveal the structure of the external world, where
structure is used in the sense of the definition in the previous paragraph.
The objects x ∈ O, i.e. the things which exemplify the relations R ∈ R, do
not have any intrinsic properties, but only relational ones. So what is re-
ally there according to the structural realist is a network of relations among
objects which do not possess any intrinsic properties but are purely defined
by their “place” in S. How these structural aspects are identified in a given
theory is, of course, a highly non-trivial matter and will largely depend on
one’s interpretation of the theory at stake.

In the case of structural realism as the preferred interpretation of gener-
ally relativistic spacetimes, the spacetime is the structure about which the
structuralist wants to be realist. A spacetime in general relativity is a triple
〈M, gµν , Tµν〉 of a four-dimensional Lorentzian manifold M, a metric tensor
gµν , and a stress-energy tensor Tµν that satisfies Einstein’s field equations,
Gµν [gµν ] = 8πTµν where Gµν is a tensorial functional of gµν , the so-called Ein-
stein tensor. Since the triples 〈M, gµν , Tµν〉 are also the models of general
relativity in the model-theoretic sense, interpreting spacetime structurally
really means offering a structural realist interpretation of general relativity.
The ontology O of the structure at stake, in this case, will just consist of
the points of the manifold M. The points in M will then be the relata that
stand in certain physically admissible relations to one another. The struc-
tural realist insists that the manifold points possess quiddity, i.e. some essence
that it is their nature to possess, but certainly no haecceity, i.e. “primitive

4


thisness” or some particular characteristic of what it is to be that particular
point, as such haecceities would surely be intrinsic—whatever else they might
be. More generally, a structural realist will restrict herself to an ideology of
purely qualitative properties. A property is called qualitative just in case its
exemplification does not depend upon the existence of any particular individ-
ual. Haecceistic properties are thus excluded. It is exactly this feature which
makes structural realism such an attractive response to the hole argument.

It is important to stress at this point that the balanced structural real-
ist rejects that members of the ontology exemplify any intrinsic properties.
Although there is considerable debate in metaphysics as to which properties
ought to qualify as “intrinsic,” it will be sufficient for our purpose to assume
that intrinsic properties are all and only those qualitative properties whose
exemplification is independent of the existence or non-existence of other con-
tingent objects. Intrinsic properties can thus be attributed independently of
accompaniment or loneliness. All non-intrinsic properties are either extrinsic
if they are monadic, or else relational. Any structural realist who pays more
than lip service to the structuralist aspect of her position should repudiate
intrinsic properties as a vestige of an object-based ontology, taking to heart
the structuralist dictum that the individuation of the members of O occurs by
virtue of their being embedded in a structure. Consequently, the structural
realist will also reject the thesis of Humean supervenience, which assumes a
geometry of spatio-temporally related points endowed with perfectly natural
intrinsic properties. In fact, it is the renunciation of Humean supervenience
that seems to motivate much of Esfeld’s structural realism.

Let me attempt to capture the idea of structuralism more formally. What
the balanced structural realist as characterized above demands is that the
world is fundamentally described by a structure S with objects which ex-
emplify the intra-structural relational properties, but no other properties.
Any other properties would introduce a unwanted reference to something be-
yond the purely structural. These intra-structural relational properties are
exactly those which are invariant under automorphisms of S, where an auto-
morphism f is defined as a map from a domain A onto itself which preserves
the structure of A (i.e. is an isomorphism of a set onto itself). A property is
invariant under an automorphism when any element a ∈ A has the property
iff its image f (a) has it. Thus, the set R, which figures in the definition
of a structure, and consequently in the definition of structural realism, con-
tains only automorphically invariant relational properties. Furthermore, let
us assume that the set of properties does not range over properties which

5


are in some philosophically intricate sense pathological, such as disjunctive
or gruesome properties. I am painfully aware that ingenious metaphysical
moves will have to be performed before we can hope to attain some robust
criteria that would allow us to exclude pathological properties on a principled
basis. For present purposes, I cowardly assume to have resolved these issues.

3 The Challenge from Cosmology

It can be shown that structural realism as characterized in the previous sec-
tion and as applied to the present spacetime context suffers from serious
difficulties in accommodating highly symmetric spacetimes. For these par-
ticular spacetime solutions with a high degree of symmetry, a devastating
argument in full analogy to the one run by Jukka Keränen (2001) against
structuralism in the philosophy of mathematics can be given. To this end,
consider the highly symmetric cosmological standard model in classical gen-
eral relativity, the so-called Friedmann-Lemâıtre-Robertson-Walker (FLRW)
spacetimes.

Modern cosmology relies on the so-called cosmological principle, accord-
ing to which no position in space—including ours—is privileged in any way.
It is generally interpreted to mean that the universe must exhibit spatial ho-
mogeneity and isotropy, at least approximately so. A spacetime is spatially
homogeneous just in case there exists a one-parameter family of spacelike
hypersurfaces Σt such that for all t and for any points p, q ∈ Σt, there exists
an isometry f of the metric gµν on M with f (p) = q. Σt is thus a foliation
of spacetime, i.e. a partition of M into a union of disjoint subsets Σt. A
spacetime is spatially isotropic just in case, roughly, it is impossible to find a
geometrically preferred spatial direction in any of the spacelike hypersurfaces
Σt. A theorem due to Walker (1944) establishes that there exists exactly one
foliation that preserves the spatial symmetries. This preferred foliation can
be labelled by a time coordinate t ∈ ]a, b[ ⊆ R. A time t thus privileged
is called a cosmological time. An isometry f is an isomorphism of M onto
itself such that the metric structure is preserved. Isometries of gµν on M
form a group called the isometry group of M. Spatial homogeneity thus
means that for every “snapshot” of the universe, there exists a map from
any point in the universe to any other that preserves the structure of the
universe at that time, and in particular the metric structure. In total, these
symmetries—homogeneity and isotropy—are mathematically encoded in the

6


action of a group of isometries on M with spacelike hypersurfaces as the
group’s surfaces of transitivity.3 Any point on any spacelike hypersurface is
thus equivalent to any other point on the same hypersurface. In particular,
the symmetries imply that the spatial curvature of all the spacelike hypersur-
faces Σt of the preferred foliation is constant. Spacetimes 〈M, gµν , Tµν〉 with
exact spatial homogeneity and isotropy are the FLRW spacetimes mentioned
above. The challenge I am about to mount against spacetime structural
realism exploits these exact spatial symmetries.

The symmetries of isotropy and homogeneity thus demand that the uni-
verse be the “same in every location,” spatially understood, with respect to
all physically relevant qualities. This idea can be formalized as the proposi-
tion, valid for all p, q ∈ Σt ⊂ M, that

∀F ∈ P (F p ↔ F q), (1)

where P is the set of admissible physical properties with respect to which
points in Σt must be the “same.” Proposition (1) is valid for any Σt of a
FLRW-spacetime. For the spacetime structuralist described in the previous
section, ∀F must range over all and only over automorphically invariant rela-
tional properties that do not depend in their exemplification on the existence
of any particular individual.

The individuality of the objects in the ontology O must be ascertained by
an identity criterion with whose help objects can be distinguished. The most
prominent such criterion is of course Leibniz’s Principle of the Identity of
Indiscernibles, or a modernized version thereof. The core idea of this family
of principles is to utilize distinction between objects in terms of the properties
they exemplify as a criterion to individuate them. In more formal words,

∀F ∈ Q (F x ↔ F y) → x = y. (2)

Varieties of this Principle of the Identity of Indiscernibles (PII) typically
differ in what is taken to be in Q. French (2006) suggests the following
three basic possibilities: (i) ∀F ranges over all possible properties, (ii) ∀F
ranges over all possible properties except spatio-temporal ones, and (iii) ∀F
ranges only over intrinsic properties. The properties here at stake, even in
the weakest version (i), are all qualitative properties.

3The surface of transitivity of a group G acting on the manifold M is the set Ω ⊂ M
of all points such that the group action G×Ω → Ω possesses only a single group orbit, i.e.
for every pair of elements x, y ∈ Ω, there exists a group element g ∈ G such that gx = y.

7


The strongest version of PII, viz. PII(iii), claims that no two individuals
can possess the same intrinsic properties. This principle is clearly violated
in classical physics, where distinct particles may be regarded as indistin-
guishable as far as intrinsic properties are concerned. Max Black (1952) has
proposed a counterexample against such a strong version of PII by placing
two indistinguishable spheres in an otherwise empty universe. Assuming that
we placed two individual spheres into the vacuum universe, Black’s example
not only violates PII(iii), but also PII(ii). Does it also violate PII(i)? Not as
long as the universe into which the spheres are put is interpreted as a fixed
background of topology R3. In this case, the set of properties attributed to
the two spheres are not entirely identical: at least properties based on their
spatio-temporal location will not coincide. So at least the weakest form of
PII is usually taken to be valid in classical physics, where spatio-temporal
trajectories of rigid bodies do not overlap.

The structural realist about spacetime must re-interpret PII to adapt it
to her purpose. The only acceptable version of PII for her, clearly, holds
that ∀F ranges over the set of automorphically invariant relations, excluding
any intrinsic properties. Now the analogue of Keränen’s argument (2001)
can be derived easily: for an ontology consisting of all points of a spacelike
hypersurface Σt of a spatially homogeneous and isotropic spacetime, we get
from (1) and (2) by modus ponens that x = y iff Q ⊆ P, where Q is the
set of all structurally admissible properties and P the set of all physical
properties as admitted by the cosmological principle. Importantly, both Q
and P only consist of automorphically invariant relational properties, thus
ascertaining Q ⊆ P. But since x and y have been arbitrary elements in
Σt = O, all points of Σt coincide and there is only one point in Σt. In
other words, the universe consists of one point only! Since the group of
automorphisms of Σt are the isometries of Σt, and the group of isometries of
Σt is transitive, i.e. any point in Σt can be moved into any other point in Σt,
all points in Σt must share the same properties. But if all points in Σt share
the same automorphically invariant properties, they can only constitute one
individual according to even the weakest form of the Principle of the Identity
of Indiscernibles, PII(i).

In other words, the structural realist cannot distinguish between the el-
ements of O. But if the objects in O cannot be distinguished, they must
be identified according to (2). However, when every other object that there
might have been must be identified with one particular object, then we say

8


that there is only one object. Formally speaking,

∃!x [x ∈ O] ⇔def ∃x [x ∈ O ∧ ∀y (y ∈ O → x = y)] . (3)

Thus, we can see that the numerical plurality of the elements in the ontology
cannot be grounded by the relata’s being situated in a relational structure
if this structure exhibits a high degree of symmetry, as is the case in the
FLRW cosmological models of general relativity. The point(s?) of Σt cannot
gain their individuality from the relations which they exemplify. Thus, they
don’t fulfill the balanced programme of structural spacetime realism. Call
this result the abyssal embarrassment for the spacetime structuralist.

4 Exit Strategies for the Structuralist

Can the spacetime structuralist recover from the abyssal embarrassment? If
so, what are the strategies at her disposal to cope with this result? The
most immediate possibility would be to simply redefine structural realism
and to hope that the recharacterization does not suffer from the abyssal em-
barrassment. A comprehensive analysis of this move is beyond the confines
of the present essay. Suffice it to say here that I suspect that most inter-
esting versions of structural realist attempt to interpret generally relativistic
spacetime will be subject to this result. Apart from changing the discourse
in this manner, I see at least four defensible strategies:

1. deny the relevance of homogeneous cosmological models (and similarly
symmetric spacetimes);

2. reject PII(i) even for classical physics and urge its replacement with
another criterion of individuation;

3. claim that PII(i) is inapplicable to the case at hand because no criterion
of individuation is needed at all; or

4. argue that we have misidentified the spacetime structure, i.e. a space-
time should not be identified with a manifold of points endowed with
certain physical fields at all.

The first path is motivated by the realization that these cosmological
models are highly idealized. According to this view, structuralism is still

9


true of our complicated world despite its failure in these stylized, simple
models. Consequently, it denies the relevance of these highly symmetric
spacetimes on which the challenge relies. Such denial could occur on the
basis of a measure-theoretic consideration. The symmetric spacetimes of the
type FLRW, this defence will run, are of measure zero in the space of so-
lutions of the field equations of classical general relativity and thus of not
much concern. Since the Einstein equations have not been solved in their
full generality and since this space is therefore not explicitly known, this
line of argument incurs a promissory note recording the debt of producing
this space including a natural measure defined on it. Even if we accept this
promissory note for the time being, however, the contention that the abyssal
embarrassment will only arise in almost no possible worlds is not implied.
The structural realist will still have to establish that the displeasingly sym-
metric spacetimes are in fact of measure zero. Even if we grant this much
this line of defence would be rather inelegant, as it is simply not the case
that e.g. the FLRW models are irrelevant regardless of their measure in the
generally relativistic solution space. An advocate of this response will then
strictly speaking be a spacetime structuralist with respect to an empirically
adequate proper subset of the set of models of the full theory. It would fall
short of establishing that the full theory should be interpreted structurally.
Even worse, this line of defence does not signficantly alleviate the pain that
the abyssal embarrassment inflicts on the spacetime structuralist; after all,
it leaves exposed the complete failure of the structural approach to accom-
modate what is arguably the most important family of spacetimes in general
relativity.

The second strategy rejects PII(i) as a valid criterion of individuation
and seeks to supplant it. The first answer in this vein builds on an inversion
of the objection aired in the previous paragraph and denies that PII(i) is
necessarily true. It insists that its truth is contingent and if its application
in the context of fundamental physics leads to absurd results, it should be
given up as a criterion of individuation and be replaced by a more sensible
alternative. In this sense, the above argument shows that PII(i) leads to
an abyssally embarrassing conclusion for some highly symmetric spacetimes.
This fact can then be taken as evidence that the principle must be rejected
as a criterion of individuation in those possible worlds in which it thus fails.
This is exactly what it means for the principle to be only contingent: while it
is true in almost all possible worlds, it is false in some possible worlds, which
(may) happen to be of measure zero. To find out whether the principle is true

10


in the actual world is thus an empirical matter. As we arguably live in a spa-
tially inhomogeneous universe, it will likely turn out to be true in the actual
world. This escape, however, must explain why a metaphysical principle of
individuation should not be universally valid no matter what. In particular,
if structuralism is offered as a general interpretation of generally relativistic
spacetimes, then it seems unacceptable to entertain different criteria of indi-
viduation with the range of possibilities that are allegedly captured by the
same interpretation.

Simon Saunders (2003) offers another, and more viable, variant of the sec-
ond strategy. He proposes to find, or construct, a symmetric, but irreflexive
relation S ∈ R such as to render the objects “weakly discernible” and hence
save them from being identified. He calls the principle based on weak dis-
cernability the Principle of the Identity of Indiscriminables. The attraction
of this approach is that this irreflexivity can ground the object’s individual-
ity without recourse to some sort of primitive thisness. However, it faces at
least two other difficulties. First, it will not be trivial to find such a relation
defined on a homogeneous spacelike hypersurface Σt without disturbing its
homogeneity, if homogeneity is understood as defined in (1). Second, in or-
der to appeal to such relations, an individuation of objects must already be
presupposed: how can I know that there are at least two objects such that
an irreflexive relation can be exemplified on the elements of Σt? I do not
see how this suspicion of circularity can be dispelled. Thus, the resolution of
the analogous argument by Keränen (2001) as proposed by Ladyman (2005),
which is based on the Principle of the Identity of Indiscriminables, is not
available here.

Of course there remains at least a third possibility for enacting an exit
strategy of type two. Esfeld (2004, 603) explicitly leaves open the possibility
that the objects have non-qualitative properties such as primitive thisness.
These haecceities seem to deliver a last resort if everything else failed. If the
individuation of objects is based on them, they become ineliminable members
of the set of properties that the structural realist must admit. But haecceities
are non-relational properties, and certainly not automorphically invariant.
Therefore, haecceities cannot be an attractive option for the structural realist.
Furthermore, unobservable haecceities re-open the “gap between metaphysics
and epistemology” that many structural realists such as Esfeld are so anxious
to close.

Esfeld (2004) himself endorses an escape along the fourth line of defence,
i.e. denying that any criterion of individuation is needed at all, when he

11


states that “[t]here is no need for the [objects] of which [relational] properties
can be predicated to be distinct individuals.” (p. 611) He reaffirms this
attitude two pages later when he writes that “[t]he argument of this paper
accepts that relations require things that stand in the relations (although
these things do not have to be individuals, and they need not have intrinsic
properties).” (p. 613) Insisting that objects which stand in relations need
not be individuals seems highly counterintuitive: should there not be a fact
of the matter whether two “places” x and y in a structure S are identical or
not? In other words, should the statement x = y not have a definite truth
value? It should, for otherwise the structuralism defended collapses into the
eliminativist version that Esfeld explicitly rejects. A balanced structuralist
cannot deny that there are facts of the matter as to whether x and y are
identical. At the very least, the advocate of the no-criterion-of-identity-is-
needed-for-balanced-structuralism position owes an account of why such a
criterion is not needed even for the balanced version.

In a brief reaction to my challenge, Esfeld and Lam (2008, 32) criticized
what they take to be my metaphysical position as a back-handed way of
demanding that the individuality of the ontologically prior relata must be
grounded in intrinsic properties. I respond with an emphatic “no.” The
balanced spacetime structuralist needs identity criteria to individuate both
objects and relations. In general, no intrinsic properties are required at all in
order to individuate objects. I acknowledge that individuating objects and
relations is in general possible within the structural realist programme, but
runs into trouble in highly symmetric cases.4 The solution proposed by Es-
feld and Lam (2008, 33) is to accept numerical plurality as a primitive, and
thus to acknowledge that this is the only way of individuating fundamen-
tal physical objects. Although I agree with them that primitive numerical
plurality is superior to primitive thisness, I find this answer highly unsatis-
factory. Particularly since I believe there to be a more promising resolution
available to the spacetime structuralist.

4I have not discussed the individuation of properties. It seems obvious to me, however,
that this will be equally necessary for a structure S to be intelligible. But it might
seem less problematic if relata are only related by one relation. For a structural realist,
however, this would resurrect a similar worry as the one exposed by the challenge: will it
still be possible in general to individuate objects in the ontology with the help of only one
relation? The answer is yes, although in order to be able to individuate objects, symmetric
structures must be excluded.

12


The final reply claims that PII(i) does not apply to the points of Σt
because these are perhaps not the individuals that take the place of the relata
in the fundamental spacetime structure. As it maintains that the objects in
O of a structure S must be individuated, and since the manifold points in
Σt cannot be individuals as shown by the abyssal embarrassment, it must
consequently deny that the manifold points in Σt can play the role of the
relata in S. Therefore, manifold points should not be interpreted as places
in the fundamental spacetime structure. In other words, Σt 6= O. Thus, the
structure S has been misidentified in the first place. In order to correct this
error, one might take a hint from Bain (2006) and seek to reformulate general
relativity in terms of Einstein algebras, or of sheaves, or of something else
entirely, and then reidentify the structure, i.e. the ontology and the ideology,
of general relativity, and hope that this time the individuality of the objects
in the fundamental ontology does indeed derive from their embedding in the
relational structure, even in the case of highly symmetric spacetimes. Of
course, this sketch of a proposed resolution is entirely programmatic. Much
more work needs to be done to flesh out how one can identify the individuals
which take the place of the relata in the structure at hand. But this is the
topic for another day.

References

[1] Bain, Jonathan (2006), “Spacetime Structuralism”, in Dennis Dieks (ed.),
The Ontology of Spacetime. Amsterdam: Elsevier, 37-66.

[2] Black, Max (1952), “The Identity of Indiscernibles”, Mind 61: 153-164.

[3] Dorato, Mauro (2000), “Substantivalism, Relationism, and Structural Space-
time Realism”, Foundations of Physics, 30: 1605-1628.

[4] Esfeld, Michael (2004), “Quantum Entanglement and a Metaphysics of Rela-
tions”, Studies in History and Philosophy of Modern Physics 35: 601-617.

[5] Esfeld, Michael, and Vincent Lam (2008), “Moderate Structural Realism
about Space-time”, Synthese 160: 27-46.

[6] French, Steven (2006), “Identity and Individuality in Quantum Theory”, in
Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Spring 2006.

13


[7] French, Steven and James Ladyman (2003), “Remodelling Structural Real-
ism: Quantum Physics and the Metaphysics of Structure”, Synthese 136:
31-56.

[8] Keränen, Jukka (2001), “The Identity Problem for Realist Structuralism”,
Philosophia Mathematica 9: 308-330.

[9] Ladyman, James (2005), “Mathematical Structuralism and the Identity of
Indiscernibles”, Analysis 65: 218-221.

[10] Pooley, Oliver (2006), “Points, Particles, and Structural Realism”, in Rickles
et al. 2006, 83-120.

[11] Rickles, Dean, and Steven French (2006), “Quantum Gravity Meets Struc-
turalism: Interweaving Relations in the Foundations of Physics”, in Rickles
et al. 2006, 1-39.

[12] Rickles, Dean, Steven French, and Juha Saatsi (eds.) (2006), The Structural
Foundation of Quantum Gravity. Oxford: Clarendon.

[13] Stachel, John (2006), “Structure, Individuality, and Quantum Gravity”, in
Rickles et al. 2006, 53-82.

[14] Walker, A G (1944), “Completely symmetric spaces”, Journal of the London
Mathematical Society 19: 219-226.

14