aa r X i v : . [ phy s i c s . h i s t - ph ] S e p The Hole Argument ∗Oliver Pooley † Oriel College, Oxford21 September, 2020
Our best theory of space, time and gravity is the general theory of relativ-ity (GR). It accounts for gravitational phenomena in terms of the curvatureof spacetime. In more mathematical presentations of the theory, solutionsare standardly represented as n-tuples: (
M, g ab , φ , φ , . . . ). The φ s are ob-jects that represent the assorted material content of spacetime (such as starsand electromagnetic fields). M and g ab together represent spacetime itself. M is a differentiable manifold representing the 4-dimensional continuum ofspacetime points. g ab is a Lorentzian metric tensor defined on M . It encodessome of spacetime’s key spatiotemporal properties, such as the spacetime dis-tances along paths in M . In particular, spacetime’s curvature can be definedin terms of g ab .In its contemporary guise, the hole argument targets a natural interpre-tation of this mathematical machinery. According to the spacetime substan-tivalist , spacetime itself, represented by ( M, g ab ), should be taken to be anelement of reality in its own right, on (at least) equal ontological footingwith its material content. John Earman and John Norton’s version of thehole argument (Earman and Norton 1987) aims to undermine this reading ofthe theory. In particular, it seeks to establish that, under a substantivalistinterpretation, GR is radically—and problematically— indeterministic .Earman and Norton’s hole argument set the agenda for the wide-rangingdebate that burgeoned in the late 1980s and early ’90s, and that has rumbledon in the decades since. The hole argument, however, did not originate with ∗ Written for E. Knox and A. Wilson (eds),
The Routledge Companion to the Philosophyof Physics (Routledge, forthcoming). Please cite the published version. † email: [email protected] g ab ,as the mathematical object that would capture gravitational effects. Theremaining task was to discover field equations describing how g ab depends onmaterial “sources”, encoded by their stress energy tensor T ab .Einstein sought a theory that would generalise the relativity principle .The restricted relativity principle of Newtonian physics and special relativityonly asserts the equivalence of all inertial frames: frames in which ( inter alia )force-free bodies move uniformly in straight lines. The inertial frames are,however, physically distinguished in these theories from frames moving non-uniformly with respect to them. Einstein believed that fundamental physicsshould treat all frames of reference as on a par.In 1913, Einstein came tantalisingly close to settling on the famous fieldequations that he would eventually publish towards the end of 1915. Theseequations are generally covariant : they hold in all coordinates systems withina family related by smooth but otherwise arbitrary coordinate transforma-tions. Because such transformations include transformations between coor-dinates adapted to frames in arbitrary states of motion, Einstein initiallybelieved that generally covariant equations embody a generalised relativityprinciple.In 1913, however, he temporarily gave up the quest for general covariance.His original version of the hole argument convinced him that any generallycovariant theory describing how g ab relates to T ab must be indeterministic.(For further details of Einstein’s argument and its role in his search for hisfield equations, see Stachel (1989), Norton (1984) and Janssen (2014: §3).)In this chapter, I focus on the hole argument as an argument againstsubstantivalism. The next section reviews some technical notions standardlypresupposed in presentations of the argument. Section 3 presents the argu-ment itself. The remainder of the chapter reviews possible responses. As characterised above, the general covariance of a theory is a matter of theinvariance of its equations under smooth but otherwise arbitrary coordinatetransformations. In order to make contact with contemporary discussion ofthe hole argument, we need an alternative formulation that dispenses withreference to coordinates.Let g ( x ) stand for some specific solution of a generally covariant theory2 , expressed with respect to some specific coordinate system { x } . Let g ′ ( x ′ )be a redescription of the same situation but given with respect to a newcoordinate system { x ′ } . Since T is generally covariant, g ′ ( x ′ ) will also satisfy T ’s equations (provided that the coordinate transformation x ′ = f ( x ) issmooth).Understood in this way—as a map between descriptions of the very samephysical situation given with respect to different coordinate systems—thetransformation g ( x ) g ′ ( x ′ ) is a passive transformation . The active inter-pretation of the transformation involves asking what the function of coordi-nates g ′ represents when interpreted with respect to the original coordinatesystem , { x } . (Note that this question presupposes something that one mightdispute, namely, that it makes sense to talk of holding a specific coordinatesystem fixed, independently of any solution described with respect to it.)When g ( x ) = g ′ ( x ), g ( x ) and g ′ ( x ) will describe different physical situa-tions (assuming no redundancy in the way the mathematical object g repre-sents physical reality). And because of T ’s general covariance, g ′ ’s status asa solution of T is independent of which coordinate system, { x } or { x ′ } , it isreferred to.The distinction between active and passive transformations first arisesin the context of coordinate transformations. The terminology is now used,however, in more general contexts. In particular, one may distinguish be-tween what are labelled (somewhat misleadingly) active and passive diffeo-morphisms .Recall that solutions of GR are n -tuples, ( M, g ab , . . . ), where M is a dif-ferentiable manifold. To call M differentiable just means that it is equippedwith structure that distinguishes, e.g., smooth from non-smooth curves. A diffeomorphism between differentiable manifolds M and N is a bijective mapsuch that both the map and its inverse preserve such structure (e.g., underthe map, the image of a smooth curve in M will be a smooth curve in N and vice versa ).Let d be a diffeomorphism from M to itself. There is no sense in which d by itself can be said to be active or passive: it simply associates to eachpoint of M a (possibly distinct) point. To get further, we need to: (i) considermaps naturally associated with d that act on structures defined on M and(ii) distinguish between two types of structure.The first type of structure includes coordinate systems: maps from M into R (assuming that M is 4-dimensional) that preserve M ’s differentiablestructure. (In fact, the differentiable structure of M is standardly definedextrinsically, via a set of preferred coordinate systems.) The second type in-cludes objects defined on M that are intended to represent something phys-ical: fields or other objects located in spacetime, or physically meaningful3patiotemporal properties and relations.The definitions of the natural maps associated with d on such objects willdiffer in detail, depending on the type of object. The basic idea, however, isstraightforward. If d maps p ∈ M to q , we can define d ’s action on an object F via the requirement that the image object, d ∗ F , “takes the same value” at p as the original object takes at q . So, for example, given a coordinate system φ : U ⊆ M → R , we can define a new coordinate system d ∗ φ on the openset d − ( U ) via the requirement that d ∗ φ ( p ) = φ ( d ( p )) for all p ∈ d − ( U ).A passive diffeomorphism corresponds to the case where one contemplatesthe action of the diffeomorphism on a coordinate system (or systems) whilstleaving the objects representing physically significant structure unchanged.It is the exact correlate of the passive coordinate transformation describedabove.An active diffeomorphism , by contrast, leaves coordinate systems un-changed but acts on the objects representing physical structure. So, for exam-ple, if ( M, g ab ) is a manifold equipped with a metric tensor field g ab , ( M, d ∗ g ab )is the (in general) mathematically distinct object that results from applyingto g ab the active diffeomorphism d . ( M, g ab ) and ( M, d ∗ g ab ) are mathemati-cally distinct because, in general, when d ( p ) = p , the value of d ∗ g ab at p (Iwrite: d ∗ g ab | p ) does not equal the value of g ab at p .With this machinery in place, a revised, coordinate-free notion of generalcovariance, often labelled (active) diffeomorphism invariance , can be stated.Let the models of a theory T be n -tuples of the form ( M, O , O , . . . ). T isgenerally covariant if and only if: if ( M, O , O , . . . ) is a structure of the rele-vant type and d is a diffeomorphism between M and N , then ( M, O , O , . . . )is a solution of T if and only if ( N, d ∗ O , d ∗ O , . . . ) is also a solution of T .GR is generally covariant in this sense. (Whether this coordinate-free notionof general covariance is equivalent to the notion of general covariance givenearlier in terms of coordinate transformations is a subtle business. For morediscussion, see Pooley (2017), where the definition of diffeomorphism invari-ance is also refined to take account of the distinction between dynamical andnon-dynamical fields.)Finally, I define a hole diffeomorphism . Let M be a differential manifoldand let H (the “ hole ”) be a compact open subset of M . A diffeomorphism d : M → M is a hole diffeomorphism corresponding to H if and only if d is the identity transformation outside of H but comes smoothly to differfrom the identity transformation within H . In other words: d ( p ) = p for all p ∈ M \ H , but d ( p ) = p for some p ∈ H .4 The Argument
The core conclusion of the hole argument is that, under a substantivalistinterpretation, any generally covariant theory such as GR is indeterministic.(To simplify exposition, I focus on GR as the paradigm generally covarianttheory.) There is then the further question of whether this core conclusioncounts against a substantivalist interpretation of GR: is the indeterminismin question problematic?The argument for the core conclusion has three main premises: a claimabout what substantivalism entails, a claim about what general covarianceentails, and a claim about what it takes for a theory to be deterministic. Thecore conclusion can be resisted by calling into question each of these claims.They therefore merit careful individual articulation. Before that, however, abrief initial statement of the argument will help ensure that we do not losesight of the wood for the trees.Let M = ( M, g ab , T ab ) be a solution of GR. Let d ∗ M = ( M, d ∗ g ab , d ∗ T ab ),where d is a diffeomorphism from M onto itself. It seems that a substan-tivalist should take M and d ∗ M to represent distinct possibilities. This isbecause (i) the substantivalist regards the points of M as representing gen-uine entities, namely substantival spacetime points, and (ii) M and d ∗ M assign the very same points different properties. If p is such that d ( p ) = p ,then (assuming g ab possesses no symmetries) M and d ∗ M will ascribe dif-ferent geometrical properties to p ( g ab | p = d ∗ g ab | p ) and may ascribe differentmatter content to (the neighbourhood around) p (if T ab | p = d ∗ T ab | p ).We assumed that M is a solution of GR. It follows from GR’s generalcovariance that d ∗ M is also a solution. This seems to mean that the (accept-ing the above reasoning, distinct) situations that M and d ∗ M represent areboth physically possible.Finally, suppose that d is a hole diffeomorphism. In particular, supposethat M is foliable by a family of achronal (with respect to g ab ) 3-dimensionalsurfaces, let Σ be one such surface, and let H lie entirely to the future ofΣ. (A foliation of an n -dimensional manifold is family of disjoint ( n − m )-dimensional submanifolds ( m < n ) whose union is M . In this case, we take m = 1. A surface is achronal if no two points in the surface lie to the pastor future of each other.) M and d ∗ M represent distinct situations ( d is non-trivial within H ) but, outside of H , they are identical: they assign exactlythe same (spatiotemporal and material) properties to all the points of M \ H .In particular, the possible spacetimes that they represent are identical up tothe instant corresponding to Σ but differ to its future. Since both spacetimesare physically possible according to GR, it seems that GR is indeterministic:fixing the laws and the entire spacetime up to Σ fails to fix what will happen5which points will have which properties) to the future of Σ.That concludes our initial statement of the hole argument. Note thatthe indeterminism is (in a certain sense) radical. H can be freely specified.Similarly, d can be freely specified, so long as it preserves M ’s differentiablestructure and so long as it smoothly reduces to the identity outside of H . Thismeans that, for any compact region of M as small as one likes, completelyspecifying the properties of spacetime and matter outside of that region failsto fix the properties within the region (Earman and Norton 1987: 524).The rest of this section develops a slightly more careful version of theargument, designed to be immune to some of the less telling criticisms foundin the literature.I stated above that the core of the hole argument involved three mainpremises. In the presentation just given, these are: (1) that the substanti-valist is committed to taking M and d ∗ M to represent distinct situations;(2) that GR’s general covariance entails that the possibilities represented by M and d ∗ M are equally physically possible; and (3) that a theory is indeter-ministic if it both regards M and d ∗ M as representing distinct possibilities(in the way described) and regards the possibilities as equally possible. Foreach premise, we should identify the most defensible version that is strongenough to play the required role in the argument.Premise (1) is closely related to what Earman and Norton dubbed the“acid test” of substantivalism. They wrote:If everything in the world were reflected East to West (or better,translated 3 feet East), retaining all the relations between bodies,would we have a different world? The substantivalist must answeryes since all the bodies of the world are now in different spatiallocations, even though the relations between them are unchanged.(521)They then went on to claim that the diffeomorphism “is the counterpart ofLeibniz’ replacement of all bodies in space in such a way that their relativerelations are preserved” and concluded that substantivalists were necessarilycommitted to the denial of Leibniz Equivalence , which they defined as thethesis that diffeomorphic models represent the same physical situation (522).Here two models (
M, O , O , . . . ) and ( N, O ′ , O ′ , . . . ) are diffeomorphic justin case there is a diffeomorphism d : M → N such that, for each object O i , O ′ i = d ∗ O i .Although Earman and Norton conclude by making a claim about howsubstantivalists must interpret diffeomorphic models, the Leibniz-inspiredscenario that they use to introduce their acid test makes no mention ofmodels . Instead, their claim is directly about the physical situations that such6odels represent. They assert that, for any given situation, a substantivalistmust recognise as genuinely distinct the situation where the entire materialcontent of the universe is shifted three feet East relative to its location in thefirst situation. This very natural assumption went unquestioned by both theantisubstantivalist Leibniz and the substantivalist Clarke, in their famous Correspondence (Clarke 1717). If spatial locations are autonomous entitiesin their own right, doesn’t one have to allow that two situations might begenuinely distinct in virtue of differing only in terms of which substantivalplaces serve as the locations of various material bodies, even if everythingelse about the two situations is identical?Premise (1) is, therefore, best thought of as the combination of two theses:one about the plurality of possibilities that it is alleged a substantivalistmust acknowledge; and another about how particular mathematical objectsrepresent those possibilities. Ultimately, it is only the first thesis that doesessential work in the hole argument.Let’s label the two theses
Plurality and
Models and state them morecarefully.
Plurality
Suppose that P is a possible spacetime. The substantivalist iscommitted to a plurality of possibilities distinct from P that (i) involvethe same pattern of spatiotemporal properties instantiated in P andcontain the same material fields as P , but that (ii) differ from P (solely)over which spacetime points have which properties and serve as thelocations of the common material content.Now for Models . Suppose that M = ( M, g ab , T ab ) can be taken to rep-resent a possible spacetime P , and suppose that P ′ is a distinct but relatedpossibility of the type contemplated in Plurality . Further, suppose that,while differing over how the common geometrical and material properties aredistributed over their common set of spacetime points, P and P ′ do not dif-fer over which collections of points count as smooth paths (i.e., they agreeon differentiable structure). The second thesis in Premise (1) is that, forsome suitable choice of diffeomorphism d , d ∗ M = ( M, d ∗ g ab , d ∗ T ab ) must beinterpreted as representing P ′ .It is immediately clear, however, that this claim is needlessly strong. Itis sufficient for the hole argument that d ∗ M may be so used. In other words,it is sufficient that there is a permissible joint interpretation of the models M and d ∗ M according to which M represents P and d ∗ M represents P ′ .The advocate of the hole argument can easily concede that M and d ∗ M are equally apt to represent either possibility, i.e., that they have the same“representational capacities” (Weatherall 2018: 332). No more is required inorder to articulate the argument than the following claim:7 odels If M = ( M, g ab , T ab ) can be chosen to represent a possible space-time P then, relative to that choice , there is a permissible and naturalinterpretation of d ∗ M according to which it represents a distinct pos-sibility P ′ .Let us now turn to Premise (3), before revisiting Premise (2). In the spiritof the emendation to Premise (1), note that whether a theory is deterministicis, in the first instance, a matter of the range of situations that it judges to bepossible and only secondarily a matter of how models might represent thosepossibilities ( cf Brighouse 1994: 118).Consider two possible spacetimes, P and P ′ , differing in the way just con-templated. That is, suppose that P and P ′ involve the same global pattern ofspatiotemporal properties and the same global pattern of material fields butthat the two spacetimes differ, for some of their common spacetime points,over which of those points instantiate which of the properties common to bothspacetimes. Further, suppose that P and P ′ are in every respect identical upto some global spacelike hypersurface and that their region of disagreementis confined to a “hole” to the future of that hypersurface.If P and P ′ are both physical possibilities according to the theory, thenthe theory is, in one obvious and natural sense, indeterministic. A theory willfail to be deterministic if it is consistent with worlds that involve identicalpasts but different futures. In the case at hand, fixing the entire past upto some instant (a region where P and P ′ match perfectly) fails to fix thefuture: according to the theory, spacetime’s continuation might be that of P ,or it might be that of P ′ .Finally, consider Premise (2) again. GR’s general covariance entails that M is a solution if and only if d ∗ M is a solution. What follows concerning thephysical possibility (according to the interpreted theory) of the spacetimesthat M and d ∗ M may be taken to represent?Since we are not naively assuming that M and d ∗ M represent unique pos-sibilities, we should not simply assert that both the spacetime representedby M and the spacetime represented by d ∗ M are equally possible accordingto GR. These definite descriptions do not pick out unique situations. Rather,the natural claim, in light of our reworked Premises (1) and (3), is the fol-lowing: Copossible
Suppose that M and M are both solutions to a theory T . Ifthere is a permissible joint interpretation of M and M according towhich M represents possibility P and M represents possibility P then if P is physically possible according to T so is P .8 Responses to the Argument
Our reworked premises entail the hole argument’s core conclusion: accordingto the substantivalist, GR is indeterministic. Responses to the argumentdivide into those that accept this core conclusion and those that reject it.Responses rejecting the core conclusion can then be classified according towhich key premise they reject.For those who accept the core conclusion, the options are to reject sub-stantivalism or to bite the bullet and accept that GR is indeterministic . Intheir paper, Earman and Norton favoured the first position. Determin-ism, they concluded “may fail, but if it fails it should fail for a reason ofphysics, not because of a commitment to substantival properties which canbe eradicated without affecting the empirical consequences of the theory”(Earman and Norton 1987: 525).According to the traditional, bipartite classification, the alternative tosubstantivalism is relationalism . Relationalists deny the (autonomous) real-ity of spacetime points and analyse facts about spacetime itself as groundedin facts about spatiotemporal properties and relations instantiated by matter.Relationalism evades the hole argument by lacking the plurality of possibil-ities allegedly plaguing substantivalism: if spacetime points simply do notexist in their own right, there can be no differences between possibilities thatconcern only which points have which properties.If relationalists deny that the manifolds in models of GR have a physi-cal correlate, they owe us a positive alternative picture of what such modelsshould be taken to represent. The simplest option is to view physical fields,not as patterns of properties and relations instantiated by the points of sub-stantival spacetime, but as extended objects in their own right, possessinginfinitely many degrees of freedom. The role of the manifold is then to repre-sent the continuity and differentiable structure of the fields themselves, andto encode which pointlike parts of one field are coincident with those of an-other. There are both philosophers and physicists who count as relationalistsin this sense and who, to a greater or lesser degree, endorse the hole argument(see, e.g., Brown 2005: 156; Rovelli 2007: 1309–10).Some remain as sceptical of relationalism as of substantivalism and havesought a “third way” between the two. This was the programme that Earmanhimself tentatively backed (1989: 208) but a genuinely novel reconception ofthe metaphysics of spacetime remains elusive. In particular, various attemptsto articulate a “structural realist” approach to spacetime arguably collapseinto variants of either relationalism or (more frequently) substantivalism (seeGreaves 2011).The other option available to someone who accepts the core conclusion is9o bite the bullet. Should substantivalists be embarrassed at being forced toview GR as indeterministic? In one sense the indeterminism is pernicious inthat, for every possible spacetime, no matter how small a region one considers,the laws and the rest of spacetime fail to fix the state of that region. Inanother sense, however, the indeterminism is anodyne. Any two possibilitiesrepresented by models that differ by a hole diffeomorphism instantiate thevery same global pattern of properties and relations. They are therefore qualitatively perfectly alike. Their differences involve only which particularindividual spacetime points instantiate which properties. In the terminologyof modal metaphysics, the differences between the possibilities are purely haecceitistic (Kaplan 1975; see also [Caulton, this volume]).The substantivalist can urge us to recognise that determinism is not an“all-or-nothing affair” (Earman 1986: 13). Given the past of a spacetime, GRmight not fix which future individuals get to instantiate this or that qualita-tive feature, but it might nonetheless fix which qualitative features get to beinstantiated. The substantivalist can claim, therefore, that, for all the holeargument has shown, GR is qualitatively or physically deterministic: giventhe past and the laws, all future physical facts might be fixed. The merelyhaecceitistic facts that fail to be pinned down do not, this substantivalistargues, count as the kind of features of the world that one should expectphysics to have anything to say about (Brighouse 1997). (Note that the holeargument’s failing to show that GR is physically indeterministic does notentail that GR is in fact physically deterministic. See Earman (2007: §6) fora review of the wider question of whether GR is deterministic in senses otherthan that at stake in the hole argument.)A further step would be to reject the notion of determinism presupposedin the hole argument. One would then block the core conclusion of theargument by denying Premise (3). Leeds (1995), for example, argues thatwhether a theory is deterministic is not a matter of which situations the inter-preted theory classifies as possible. Instead, he claims, it is a matter of whatsentences are provable within the language of the theory. In order for thisstrategy to work, it would need to be shown that the notion of determinismpresupposed in the hole argument is not merely a possibly disfavoured optionamongst several but that it is somehow illegitimate. That seems like a tallorder. Leeds himself concedes that his proposal can be read as offering justone more definition of determinism and, moreover, one that matches otherdefinitions framed in model-theoretic or possibility-based terms (Leeds 1995:435). If the link between substantivalism and indeterminism is to be severed,Premises (1) and (2) are more promising targets.Maudlin seeks to evade the core conclusion on the basis of a positionhe dubs metrical essentialism (Maudlin 1989, 1990). Suppose model M =10 M, g ab , T ab ) is apt to represent a possible spacetime and consider model d ∗ M = ( M, d ∗ g ab , d ∗ T ab ) for some diffeomorphism d . Recall that Premise (1)of the hole argument was split above into two components: Plurality and
Models . Maudlin accepts
Models in at least the following sense: he ac-cepts (in fact insists: see Maudlin 1989: 84) that there is a permissible jointinterpretation of M and d ∗ M according to which they represent (if one as-sumes substantivalism) different ways for the world to be . (I here borrowterminology from Salmon (1989).) But, according to Maudlin, these waysfor the world to be are not both ways that the world might have been ; theydo not both correspond to genuinely possible worlds.Let us stipulate that model M represents a possible world. The defin-ing commitment of metrical essentialism is that spacetime points bear theirgeometrical properties and relations essentially. The value of the curvaturescalar at the spacetime point represented—or “named”—by p ∈ M is, there-fore, one of that point’s essential properties. Now suppose that d ( p ) = p . Inthat case, d ∗ M represents the very same point as having different geometri-cal properties, for the value of the curvature scalar at p in d ∗ M will (in thegeneric case) be different from its value in M . It follows that, according tothe metrical essentialist, d ∗ M represents a state of affairs that is not evenmetaphysically possible.Note that the initial choice of M to represent the genuine possibilityis arbitrary—consistently with their representational equivalence, one mightequally well have chosen d ∗ M . (This answers Norton’s (1989: 63) charge thatthe metrical essentialist has to explain what distinguishes the “real” modelfrom “imposters”.) What the metrical essentialist insists on is that, relativeto that choice and relative to a natural and permissible joint interpretationof the models , d ∗ M represents something metaphysical impossible. Models had to be tweaked so as to be acceptable to metrical essentialists.Something similar is true of
Plurality . In one sense, metrical essentialistsblock the hole argument by rejecting
Plurality . Setting aside cases withnontrivial isometries that are not also symmetries of the matter distribution,the metrical essentialist recognises at most one possible world correspondingto a given pattern of metrical and material properties and relations for anygiven collection of spacetime points.To characterise the metrical essentialist as rejecting
Plurality is, how-ever, in some ways misleading. Maudlin endorses the intuition behind the“acid test”; he agrees with Earman and Norton that the substantivalist mustview a Leibniz-inspired shift of all matter three feet East as generating agenuinely distinct possibility. His dispute with Earman and Norton is overtheir classification of diffeomorphisms as the natural generalisations of suchshifts. Maudlin stresses that Leibniz shifts apply only to the matter of11he universe; they leave the geometric properties of the individual space-time points unaltered. He therefore sees the models M = ( M, g ab , T ab ) and M ′ = ( M, g ab , d ∗ T ab ) as representing the proper generalization of Leibnizshifts when the points of M are interpreted as naming the same spacetimepoints in each model (Maudlin 1990: 552–3). And, of course, if M is a so-lution of GR, then M ′ will, in general, not be (when T ab = ). Since atmost one of the possibilities represented is physically possible (by the lightsof GR), their distinctness does not threaten indeterminism.Although M ′ does not represent a physically possible spacetime, Maudlinwill judge that it does represent a metaphysically possible spacetime. Theatypical case of spacetimes with symmetries is therefore revealing. In suchcases, if d is an isometry, M ′ can represent a physically possible world gen-uinely distinct from that represented by M but one nevertheless qualitativelyindiscernible from it. Maudlin thus accepts the meaningfulness of merelyhaecceitistic distinctions even if he denies that they (invariably) entail a plu-rality of genuine possibilities.This suggests the following representation of the metrical essentialist’sposition. They accept: Plurality ∗ Suppose that P is a possible spacetime. The substantivalist iscommitted to a plurality of ways for the world to be distinct from P that(i) involve the same pattern of spatiotemporal properties instantiatedin P and contain the same material fields as P , but that (ii) differ from P (solely) over which spacetime points have which properties and serveas the locations of the common material content.But they reject: Copossible ∗ Suppose that M and M are both solutions to a theory T and let P and P be ways for the world to be. If there is a permissiblejoint interpretation of M and M according to which M represents P and M represents P then if P is physically possible according to T so is P .With these tweaks, the metrical essentialist counts as someone who acceptsPremise (1) but rejects Premise (2).This regimentation highlights that metrical essentialists evade the holeargument’s core conclusion only by rejecting what might seem like the obviousmoral of the diffeomorphism invariance of the theory. Grant Maudlin thatthe points of the manifold M can be treated like proper names and that, sounderstood, models M and d ∗ M represent distinct ways for the world tobe. What remains to be decided is whether these ways are both genuinely12ossible ways for the world to be. Having gone this far, however, it is a verynatural further step to take the diffeomorphism invariance of GR as tellingus precisely that both states are possible.In postulating the real existence of spacetime points in the first place,the metrical essentialist is likely to be a scientific realist who is happy totake GR as a guide to ontology. Should not GR also be our guide as towhich properties are essential to spacetime points? What the diffeomorphisminvariance of GR appears to tell a haecceitist substantivalist is that the onlyproperties essential to a spacetime point are those that it exemplifies as partof a set with the structure of a differentiable manifold ( cf Earman 1989: 201).A different criticism of metrical essentialism focusses on non-isomorphicmodels. One straightforward way of illustrating the dynamical nature ofspacetime structure in GR is to assert, for example, that, had extra massbeen present close to some point, then the curvature at that point wouldhave been different (Earman 1989: 201). How are metrical essentialists toevaluate such counterfactuals? They have to judge that the idea that thecurvature could have been different from its actual value at this very point isas metaphysically absurd as, for example, the idea that Tim Maudlin mighthave been made of iron girders. Maudlin is forced to concede that “no modelnot isometric to the actual world can represent how this space-time mighthave been” (Maudlin 1989: 89–90) but he insists that dynamically allowedmodels of GR that are not isometric to the actual world can represent gen-uine possibilities: “they are just different possible space-times, not differentpossible states of this space-time” (1989: 90). He goes on to allow that suchpossible spacetimes can be used to give a counterpart-theoretic explanation ofthe truth of Earman’s counterfactual. As Brighouse observes (1994: 119–20),this is a rather unsatisfying blend of essentialism and counterpart theory.In the face of such criticism, what positive reasons can metric essential-ists offer for their position, aside from the ad hoc benefit that it avoids theindeterminism of the hole argument? Maudlin’s answer appeals to Newton.In a somewhat obscure passage, which has inspired almost as many differentinterpretations as commentators, Newton wrote:The parts of duration and space are understood to be the sameas they really are only because of their mutual order and posi-tion; nor do they have any principle of individuation apart fromthat order and position, which consequently cannot be altered.(Newton 1684 [2004]: 25)According to Maudlin’s gloss, Newton is saying that “the parts of space andtime, being intrinsically identical to one another, [have] to be differentiated13y their mutual relations of position. Parts of space bear their metricalrelations essentially” (Maudlin 1989: 86).This is not especially compelling. Why should someone otherwise comfort-able with haecceitistic distinctions think that intrinsically identical objects re-quire (metaphysical?) “differentiation”? Rather than cleaving to haecceitismand avoiding indeterminism by way of an otherwise unmotivated essential-ism, perhaps the substantivalist does better to embrace wholeheartedly the“structuralist” view that others (e.g., Stein 2002: 272) read in Newton’s cryp-tic remarks. If spacetime points are only “individuated” one from another bytheir spatiotemporal relations (i.e., by their positions in the overall networkof spatiotemporal relations), a possible spacetime is exhaustively specifiedby a complete catalogue of the qualitative facts concerning the full patternof spatiotemporal relations that are instantiated by its points. According tothis antihaecceitist (or generalist ) point of view, there simply are no further“individualistic” facts concerning which objects possess which properties. Ata fundamental level, reality, at least concerning spacetime points, is purelyqualitative.Despite the important differences between them, Butterfield (1989 a ), Maidens(1992), Stachel (1993, 2002, 2006), Brighouse (1994), Rynasiewicz (1994),Hoefer (1996), Saunders (2003), Pooley (2006) and Esfeld and Lam (2008) allendorse some kind of antihaecceitism, at least concerning spacetime points,whether on general philosophical grounds (as in Hoefer’s case), or as a per-ceived lesson of the diffeomorphism invariance of the physics (as in Stachel’scase). Whether acknowledged or not, these authors, in their commitment tospacetime points as entities not reducible to matter and its properties, countas substantivalists, albeit of a “sophisticated” variety (Belot and Earman2001: 228).Sophisticated substantivalists reject the core conclusion of the hole argu-ment by rejecting Premise (1). In particular, they reject Plurality . The dis-tinct possibilities countenanced by
Plurality are precisely possibilities thatdiffer merely haecceitistically. That antihaecceitism and
Plurality are in-compatible is therefore immediate. One strand of criticism questions whetherit is coherent to combine acceptance of spacetime points as entities in theirown right with a denial that there are the substantive facts (about whichsuch entities possess which properties) that would generate haecceitistic dis-tinctions. Some argue that a fleshed-out metaphysical story explaining howthis combination is possible is still to be given (Dasgupta 2011: 130–5).In addition to
Plurality , sophisticated substantivalists also reject
Mod-els , but this is a simple consequence of their rejection of
Plurality , which
Models presupposes. This might lead one to wonder whether there is asatisfactory response to the hole argument that rejects
Models while dis-14vowing metaphysics and remaining neutral with respect to
Plurality . Onecan interpret Weatherall (2018) and Fletcher (2020) as defending positionsof this kind.According to
Models , M and d ∗ M can be used to jointly represent phys-ically distinct situations. The essence of both Weatherall’s and Fletcher’sviews is that this use is not consistent with treating them as Lorentzianmanifolds. Weatherall’s starting point is that the physical interpretation ofa theory’s formalism should be consistent with our best understanding ofthe mathematics of that formalism. In particular, the models employed ina physical theory should count as (physically) equivalent just when they areequivalent according to the mathematics used in formulating those models(Weatherall 2018: 331). Since isometry provides the standard of ‘sameness’ inthe mathematics of Lorentzian manifolds, it is a condition on any acceptableinterpretation that it regard isometric manifolds, such as M = ( M, g ab ) and d ∗ M = ( M, d ∗ g ab ), as physically equivalent.In order to bite against Models , this stricture needs to be understood,not merely as insisting that any two isometric models are equally apt to rep-resent any given possibility (something our formulation of the hole argumentwas careful to allow), but as ruling out a joint interpretation of them onwhich they are physically inequivalent in the sense that (so interpreted) theyrepresent distinct physical possibilities.Fletcher is explicit that such a use of the models is in conflict with treatingthem as members of the mathematical category of Lorentzian manifolds. Anyaspect of a state of affairs that is represented by one such model so conceived must, he argues, be similarly represented by each isomorphic model. Thisis because isomorphic models are equivalent “as objects in that category”:their being isomorphic just is a matter of there being a bijective map of aspecific sort that preserves all of the structures constitutive of objects of thattype. The consequence Fletcher draws is that any putative representationaldifferences between such isomorphic models are “not reflected at all in themodels themselves as members of [the] category they are taken to be [mem-bers of]—there is no mathematical correlate of those differences definable inthe category” (Fletcher 2020: 239–40).For the sake of argument, let us concede to Weatherall and Fletcher that,on a natural understanding of GR’s mathematical machinery, it cannot beused to represent haecceitistic differences between possible spacetimes. Whatfollows for the hole argument? It becomes evident that all the heavy lifting inPremise (1) is done by its metaphysical component, namely,
Plurality . Thatthesis was not the outcome of a naive way of thinking about the mathematicsof GR. It arose from an interrogation of the substantivalist’s metaphysics.Weatherall’s and Fletcher’s reflections, therefore, leave it untouched.15n effect, both authors argue that, if there are pluralities of merely haeccei-tistically distinct possibilities, the mathematical formalism of GR, correctlyinterpreted, is necessarily indifferent to differences between them. But thisjust means that GR does not distinguish between any two elements of sucha plurality; both will count as physically possible according to GR or neitherwill. And that, of course, is just to admit that, according to any metaphysicalview committed to such pluralities, GR is indeterministic. The indetermin-ism cannot be avoided by remaining loftily above the metaphysical affray.
Contrast the following two definitions of determinism for a theory T : Det1 T is deterministic just in case, for any worlds W and W ′ that arepossible according to T , if the past of W up to some timeslice in W is intrinsically identical to the past of W ′ up to some timeslice in W ′ ,then W and W ′ are intrinsically identical. Det2 T is deterministic just in case, for any worlds W and W ′ that arepossible according to T , if the past of W up to some timeslice in W is qualitatively (intrinsically) identical to the past of W ′ up to sometimeslice in W ′ , then W and W ′ are qualitatively identical.(On these definitions, determinism is a matter of whether the entire history ofa world up to some time fixes its future, given the laws. Alternative notionsof determinism are easily obtained by considering whether a world at a time(or some other part of a world) fixes the remainder, given the laws. There arealso reasons to focus on the conditions for a world, rather than a theory, tobe deterministic (Brighouse 1997: 468). For reasons of space, I ignore thesecomplications.)According to sophisticated substantivalists, there are no primitive trans-world facts about which objects in one world are identical to which objects inanother. On their view, two possible worlds (or two proper parts of distinctworlds) that are not (intrinsically) identical differ qualitatively. Sophisticatedsubstantivalists therefore interpret Det1 and
Det2 as strictly equivalent.According to straightforward (haecceitist) substantivalists, in contrast,worlds W and W ′ can differ not just by failing to be perfectly qualita-tively alike but by failing to have the very same individuals playing iden-tical qualitative roles. For them, therefore, Det1 describes a criterion fordeterminism that is strictly stronger than that described in
Det2 . Absentsufficiently strong essentialist constraints on what is possible for spacetime16oints, straightforward substantivalists will judge GR to be indeterministicaccording to
Det1 . Det2 corresponds closely to the definition of determinism offered byDavid Lewis (1983: 359–60). A related model-theoretic definition was de-fended by Butterfield (1989 b ), who argued that it captures the notion ofdeterminism implicit in physicists’ discussions of GR’s determinism. A sig-nificant strand of the hole argument literature has targeted definitions akinto Det2 , arguing that they misclassify as deterministic theories that areclearly indeterministic. Such criticism is an obvious problem for sophisti-cated substantivalists, for whom
Det2 is equivalent to
Det1 , but it is alsoa problem for straightforward substantivalists who, as noted above, mightwish to distinguish a notion of physical determinism from determinism toutcourt . Det2 might have seemed to adequately capture the former notion.Problem cases for
Det2 were raised by Wilson (1993: 216) and Rynasiewicz(1994: 418), and have been discussed in detail by Belot (1995), Brighouse(1997) and Melia (1999). Here are two simple illustrative examples. In thefirst (adapted from Melia 1999: 660–1) our theory, T , governs the behaviourof two types of particle: A particles and B particles. Consider a world thatcontains one A particle equidistant from two B particles, all at rest withrespect to one another. Suppose that T determines that, at some fixed andpredictable time, the A particle will move at a fixed velocity towards one ofthe B particles. Intuitively, T is indeterministic because, despite fixing thequalitative evolution of the situation just described, it fails to fix which Bparticle the A particle will move towards.Imagine that we can label the particles in our toy world: a , b and b .There appear to be two possible futures: one where a moves towards b anda second where a moves towards b . Haecceitists will judge that T is in-deterministic according to Det1 , for they recognise the merely haecceitisticdistinctions between its being b or b towards which the A particle moves.According to Det2 , however, the world is compatible with T ’s being deter-ministic: if there are two possible futures, they are qualitatively identical.In our second example ( cf Belot 1995: 191–2, and Melia 1999: 646–7),theory T governs the decay of A particles into B particles. We supposethat everything qualitative about such decays (their spacetime locations, themomenta of the decay products, etc.) is fixed by the qualitative history ofthe world prior to the decay. Now consider a world governed by T involv-ing the simultaneous decay of two A particles each into a B particle. Thehaecceitist will judge that T is indeterministic because, despite fixing thequalitative behaviour of all decays, it fails to fix the identities of the decayproducts. Again, imagine that we can label the four particles and supposethat, in the world we are considering, a decays into b and a decays into b .17 world where a decays into b and a decays into b , but where everythingelse is otherwise held fixed, might seem to be an alternative possibility com-patible with T . While T is therefore deterministic according to Det2 (thequalitative nature of all decays is fully determined by the qualitative natureof the pre-decay state), a haecceitist will judge that the theory fails to bedeterministic according to
Det1 .Some philosophers (e.g. Belot 1995) accept that both T and T manifestgenuine indeterminism. They have reason to reject Det2 and can rest contentwith a haecceitist understanding of
Det1 . They are likely to accept the holeargument’s conclusion that substantivalist GR is indeterministic.A sophisticated substantivalist, on the other hand, will view the allegedindeterminism of T as suspect: the purportedly distinct possibilities involvedin the example are merely haecceitistically distinct. For many, however, theintuition that theory T is indeterministic is harder to dispel. Is there aprincipled way for sophisticated substantivalists to acknowledge that T isindeterministic but to deny that T is indeterministic?Despite doubts recently expressed by Brighouse (2020: §4), it would seemthat this can be done. Consider again the three-particle world governed by T . According to the haecceitist, there are in fact two such possible worlds:one in which a moves towards b and another in which a moves towards b . According to the antihaecceitist, there is only one such world: it containsan A particle that moves towards one but not the other of two previouslyqualitatively identical B particles. But the antihaecceitist can (and, indeed,must) recognise two possible futures for two qualitatively identical properparts of this world. Prior to the A particle’s starting to move, there were twoqualitatively identical but distinct (and overlapping) pairs composed of anA particle and a B particle. For convenience we can imagine labelling them“( a , b )” and “( a , b )” but, note, their distinctness involves no haecceitisticpresuppositions. It is secured by the distinctness of b and b , two particlescoexisting in the same world and situated some distance apart from oneanother. T ’s indeterminism can, therefore, be understood in terms of its failureto fix the (qualitative) future of every part of each world that it governs(Melia 1999: 652). Take our two pairs of an A particle and a B particle. Anexhaustive qualitative specification of such a pair up to the time at whichthe A particle moves will involve the complete specification of the qualitativehistory of the whole world up to that time together with a qualitative char-acterisation of the pair’s situation in this history. Up until the time at whichthe A particle moves, both pairs will satisfy exactly the same qualitiativedescription. Such a specification, therefore, fails to determine whether oneof the particles in the pair will move towards the other or not.18ote that indeterminism is still conceived, as it must be for the antihaec-ceitist, as a matter of the qualitative past failing to fix the qualitative future.In order to recognise T ’s indeterminism, one just needs to attend to properparts of a world, in addition to the world as a whole. It turns out that itis relatively straightforward to provide alternative definitions of determinismthat regiment the intuitions just described (see Belot 1995: 191, Definition2; Melia 1999: §4.1). This need not mean that Det2 should simply be jetti-soned. Following Dewar (2016), one might go on to distinguish “determinism de dicto ” (captured by
Det2 ) from “determinism de re ”:. . . with a little hindsight, it is utterly unsurprising that thereshould turn out to be two concepts of determinism. Determinismis a matter of whether there is one possibility or more consistentwith things being a certain way at a certain time; we have twospecies of possibility, de dicto and de re ; so as a consequence,there are two species of determinism. (Dewar 2016: 53–4)
Acknowledgements
I am grateful to numerous colleagues for discussions of the hole argumentover the years but, for especially pertinent recent discussion, I am gratefulto Eleanor Knox, Sam Fletcher and Jim Weatherall, and, for very helpfulcomments on an earlier draft, to James Read.
References
Belot, G. (1995), ‘New work for counterpart theorists: Determinism’,
TheBritish Journal for the Philosophy of Science , 185–195.Belot, G. and Earman, J. (2001), Pre-Socratic quantum gravity, in C. Cal-lender and N. Huggett, eds, ‘Physics meets philosophy at the Planck scale:Contempory theories in quantum gravity’, Cambridge University Press,Cambridge, chapter 10, pp. 213–55.Brighouse, C. (1994), Spacetime and holes, in D. Hull, M. Forbes and R. M.Burian, eds, ‘Proceedings of the 1994 Biennial Meeting of the Philosophyof Science Association’, Vol. 1, Philosophy of Science Association, EastLansing, Michigan, pp. 117–125.Brighouse, C. (1997), ‘Determinism and modality’,
The British Journal forthe Philosophy of Science (4), 465–81.19righouse, C. (2020), ‘Confessions of a (cheap) sophisticated substantivalist’, Foundations of Physics , 348–59.Brown, H. R. (2005), Physical Relativity: Space-time Structure from a Dy-namical Perspective , Oxford University Press, Oxford.Butterfield, J. N. (1989 a ), Albert Einstein meets David Lewis, in A. Fine andJ. Leplin, eds, ‘Proceedings of the 1988 Biennial Meeting of the Philosophyof Science Association’, Philosophy of Science Association, East Lansing,Michigan, pp. 65–81.Butterfield, J. N. (1989 b ), ‘The hole truth’, The British Journal for the Phi-losophy of Science (1), 1–28.Clarke, S. (1717), A Collection of Papers, Which passed between the lateLearned Mr Leibnitz, and Dr Clarke, In the Years 1715 and 1716 , London.Dasgupta, S. (2011), ‘The bare necessities’,
Philosophical Perspectives , 115–160.Dewar, N. (2016), Symmetries in physics, metaphysics, and logic, PhD thesis,University of Oxford.Earman, J. (1986), A Primer on Determinism , D. Riedel Publishing Com-pany, Dordrecht.Earman, J. (1989),
World Enough and Space-Time: Absolute versus Rela-tional Theories of Space and Time , MIT Press, Cambridge, MA.Earman, J. (2007), Aspects of determinism in modern physics, in J. N. But-terfield and J. Earman, eds, ‘Philosophy of Physics’, Vol. 2 of
Handbook ofthe Philosophy of Science , Elsevier, Amsterdam, pp. 1369–1434.Earman, J. and Norton, J. D. (1987), ‘What price spacetime substantivalism?the hole story’,
The British Journal for the Philosophy of Science , 515–525.Einstein, A. (1914), ‘Comments on “Outline of a Generalized Theory of Rel-ativity and of a Theory of Gravitation”’, Zeitschrift für Mathematik undPhysik , 260–61. Reprinted in Klein et al. (1995), pp. 580–2.Esfeld, M. and Lam, V. (2008), ‘Moderate structural realism about space-time’, Synthese (1), 27–46. 20letcher, S. C. (2020), ‘On representational capacities, with an applicationto general relativity’,
Foundations of Physics , 228–49.Greaves, H. (2011), ‘In search of (spacetime) structuralism’, PhilosophicalPerspectives , 189–204.Hoefer, C. (1996), ‘The metaphysics of space-time substantivalism’, TheJournal of Philosophy (1), 5–27.Janssen, M. (2014), ‘No success like failure ...’: Einstein’s quest for generalrelativity, 1907–1920, in M. Janssen and C. Lehner, eds, ‘The CambridgeCompanion to Einstein’, Cambridge University Press, Cambridge, pp. 167–227.Kaplan, D. (1975), ‘How to Russell a Frege–Church’,
The Journal of Philos-ophy , 716–729.Klein, M. J., Kox, A., Renn, J. and Schulmann, R., eds (1995), The SwissYears: Writings 1912–1914 , Vol. 4 of
The Collected Papers of Albert Ein-stein , Princeton University Press, Princeton, NJ.Leeds, S. (1995), ‘Holes and determinism: Another look’,
Philosophy of Sci-ence , 425–437.Lewis, D. K. (1983), ‘New work for a theory of universals’, AustralasianJournal of Philosophy , 343–377.Maidens, A. (1992), ‘Review, Earman, John S. [1989]: World Enough andSpace-Time: Absolute versus Relational Theories of Space and Time ’, TheBritish Journal for the Philosophy of Science (1), 129.Maudlin, T. (1989), The essence of space-time, in A. Fine and J. Leplin, eds,‘Proceedings of the 1988 Biennial Meeting of the Philosophy of ScienceAssociation’, Philosophy of Science Association, East Lansing, Michigan„pp. 82–91.Maudlin, T. (1990), ‘Substances and space-time: What Aristotle would havesaid to Einstein’,
Studies in History and Philosophy of Science (4), 531–561.Melia, J. (1999), ‘Holes, haecceitism and two conceptions of determinism’, The British Journal for the Philosophy of Science (4), 639–664.Newton, I. (1684 [2004]), De Gravitatione, in A. Janiak, ed., ‘PhilosophicalWritings’, Cambridge University Press, Cambridge, pp. 12–39.21orton, J. D. (1984), ‘How Einstein found his field equations: 1912-1915’,
Historical Studies in the Physical Sciences (2), 253–316.Norton, J. D. (1989), The hole argument, in A. Fine and J. Leplin, eds,‘Proceedings of the 1988 Biennial Meeting of the Philosophy of ScienceAssociation’, Philosophy of Science Association, East Lansing, Michigan„pp. 56–64.Pooley, O. (2006), Points, particles, and structural realism, in D. Rickles,S. French, and J. Saatsi, eds, ‘The Structural Foundations of QuantumGravity’, Oxford University Press, Oxford, pp. 83–120.Pooley, O. (2017), Background independence, diffeomorphism invariance,and the meaning of coordinates, in D. Lehmkuhl, G. Schiemann andE. Scholz, eds, ‘Towards a Theory of Spacetime Theories’, Vol. 13 of
Ein-stein Studies , Birkhäuser, Basel, pp. 105–143.Rovelli, C. (2007), Quantum gravity, in J. N. Butterfield and J. Earman, eds,‘Philosophy of Physics’, Vol. 2 of
Handbook of the Philosophy of Science ,Elsevier, Amsterdam, pp. 1287–1329.Rynasiewicz, R. (1994), ‘The lessons of the hole argument’,
The British Jour-nal for the Philosophy of Science (2), 407–436.Salmon, N. (1989), ‘The logic of what might have been’, The PhilosophicalReview pp. 3–34.Saunders, S. W. (2003), Indiscernibles, general covariance, and other sym-metries: The case for non-reductive relationalism, in A. Ashtekar, R. S.Cohen, D. Howard, J. Renn, S. Sarkar and A. Shimony, eds, ‘Revisit-ing the Foundations of Relativistic Physics: Festschrift in Honor of JohnStachel’, Vol. 234 of
Boston Studies in the Philosophy of Science , Kluwer,Dordrecht, pp. 151–173.Stachel, J. (1989), Einstein’s search for general covariance, 1912–1915., in D. Howard and J. Stachel, eds, ‘Einstein and the History of General Rela-tivity’, Birkhäuser, Boston.Stachel, J. (1993), The meaning of general covariance, in J. Earman, A. I.Janis, G. J. Massey and N. Rescher, eds, ‘Philosophical problems of the in-ternal and external worlds: essays on the philosophy of Adolf Grünbaum’,Vol. 1 of
Pittsburgh–Konstanz series in the philosophy and history of sci-ence , University of Pittsburgh Press, Pittsburgh, pp. 129–160.22tachel, J. (2002), “The relations between things” versus “the things betweenrelations”: The deeper meaning of the hole argument, in D. B. Malament,ed., ‘Reading Natural Philosophy. Essays in the History and Philosophy ofScience and Mathematics’, Open Court, Chicago, pp. 231–66.Stachel, J. (2006), Structure, individuality and quantum gravity, in D. Rick-les, S. French, and J. Saatsi, eds, ‘The Structural Foundations of QuantumGravity’, Oxford University Press, Oxford, pp. 53–82.Stein, H. (2002), Newton’s metaphysics, in I. B. Cohen and G. E. Smith,eds, ‘The Cambridge Companion to Newton’, Cambridge University Press,Cambridge, pp. 256–307.Weatherall, J. O. (2018), ‘Regarding the “hole argument”’,
The British Jour-nal for the Philosophy of Science , 329–350.Wilson, M. (1993), ‘There’s a hole and a bucket, dear Leibniz’, MidwestStudies in Philosophy18