the lessons of the hole argument

8/9/2019 The Lessons of the Hole Argument

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The British Society for the Philosophy of Science

The Lessons of the Hole ArgumentAuthor(s): Robert RynasiewiczSource: The British Journal for the Philosophy of Science, Vol. 45, No. 2 (Jun., 1994), pp. 407-436

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Brit. . Phil.Sci. 45 (1994), 407-436 Printedn GreatBritain

The Lessons of the Hole ArgumentROBERTRYNASIEWICZ

1 Introduction2 TheHoleArgument3 IsomorphismClosure4 'Newton'Equivalence5 ModelSelectivism6 LeibnizEquivalence7 RadicalLocalIndeterminismnd the Inscrutability f Reference8 A Parable9 TheIdentificationGroup10 Comparisons,Autonomy,and Determinism11 Applicationso SpacetimeTheories12 Conclusion

I INTRODUCTIONEarlier n this century substantivalism was commonly viewed as a gratuitousmetaphysical imposition on physical theory-'a mystical philosophical super-structure', as Reichenbach put it. But with the decline of positivism and theemergence of scientificrealism, space and time (or spacetime) came to be seenas no more suspect in principlethan any other 'theoretical'entity. Substanti-valism could be regardedas a physical doctrine to be accepted or rejectedonscientific grounds. And an impressive array of foundational and historicalstudies argued persuasively for such grounds, whether in the context ofseventeenth-century mechanics or twentieth-century relativity.Earman and Norton [1987], however, have presented a subtle andsophisticated argument-the so-called 'hole argument'--endeavoring toshow that spacetime substantivalism is committed to the impossibility ofdeterminism. Determinismmay be false, they say, but: t should not be judgedso a priori because of a commitment to 'substantival properties'. 'If ametaphysics, which forces all our theories to be deterministic,is unacceptable,then equally a metaphysics, which automatically decides in favor of indeter-minism, is also unacceptable' (p. 524).The reactions to the hole argument have been various. Norton [1987]maintains that the argument impugns substantivalism but not spacetimerealism, although he does not explorehow it is possibleto be a spacetimerealist


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408 RobertRynasiewiczwithout in turn being committed to substantivalism.Earman[1989] draws animplicitlyanti-realistconclusion: what is neededis 'a conception of space-timethat fits neither traditional relationism nor traditional substantivalism' (p.208). His own vision is that realityhas the form of an abstractalgebraand thespacetime manifold provides only an indirect representationof this reality intermsof the ringof continuous real-valued functionson the manifold.Maudlin[1989] believes the hole argument shows only that substantivalists shouldadapt Kripke'sessentialist doctrines to the metric properties of spacetime.Butterfield 1987, 1989a, 1989b] argues that substantivalists should apply avariant of David Lewis's counterpart theory to spacetime.As salutary as the hole argument has been in pumping new life into thesubstantival-relational debate, in what follows I am not primarilyconcernedwith its importfor this controversy. The greatervirtue of the argument is thatit forces a more fundamental issue: how can one understand, in terms of themodels of a theory, what the theory alleges to be physically possible?I first setout some of the intricacies of this problem by exploring various difficultiesinherent in the responses to the hole argument mentioned above. Acomparison of the hole argument with permutation arguments given byQuine, Davidson, Putnam, and others for the inscrutability of referencesuggests the problem s intimately relatedto that ofdeterminingwhen a pairofinterpretationsfora language count as co-intended.Proceedingfrom this clue,I set out a solution to the problem, developing the underlying principles indetail sufficientto draw definiteconclusions concerning the success or failureof determinism in the usual variety of spacetime theories.2 THE HOLE ARGUMENTEarman and Norton's argument has two parts. The main argument showsthat a particularthesis regardingthe relation between the models of a theoryand the possible situations they supposedly represent has the unwantedconsequence that determinism fails miserably in any spacetime theory. Asubsidiaryargument then seeks to associate substantivalismwith that thesis.Let me first review the main argument in slightly more general terms thanthose provided by Earman and Norton and then turn to the subsidiaryargument.But first some terminology, a spacetime heory s understoodto have modelsof the general form (M,01, . . .,On>,where M is a differentiablemanifold andthe Oi'sare geometric object fields definedeverywhere on M. If9Y= (M,01, ...,On) and h is a diffeomorphismof M on to manifoldM', let h.el = (M',h*O01,...,h*O,), where h*Oi s the 'carry along' of the objectfieldOiunderh.' If M= M',i.e. if h is a diffeomorphismof M onto itself,then h is called a Leibnizhift.A holediffeomorphisms a Leibnizshift other than the identity map which, for some' The exact definition of i*,varies from object to object. See Waldh1984]. pp. 437-8.


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The Lessonsof theHoleArgument 409proper neighborhood Hc M (called a 'hole'), agrees with the identity on allpoints not in H. If the object fields of 9Yimpose metric structure or establish aconnection on M, then for any proper neighborhood H there is a holediffeomorphismh such that h,I #W I, and so WIand h,I will disagree atvarious points in the interior of H although they necessarily agree at all pointsnot in H. Arguably, any physically interesting spacetime theory meets thiscondition.2The general form of Earman and Norton's main argument is this. LetTbe aphysically interesting spacetime theory. Assume two things about T.IsomorphismClosure:If9J is a model of Tand WIs isomorphictoT9', then WI'is also a model of T.Model Literalism: Each model represents a possible physical situation, anddistinct models represent distinct situations.However stringent the laws of T,it follows that Tcannot be deterministic.Forlet 9 be a model ofT.Chooseany neighborhoodH of the manifold ofWIand leth be a hole diffeomorphism orH such that h*m# WJ. ince hWI s isomorphicto WJ, W s a model ofT.And, since h*9,J#0 , WIand h,*9Jrepresentdistinctphysical situations which agree everywhere but for certainspacetimepoints inthe interior of the spacetime region representedby H. Hence, we have whatEarman and Norton call radical ocal indetermninism:he physical state of anyregion, no matter how small, can never be determinedby the total state of theremainder of spacetime.The threat of radical local indeterminism is avoided if, instead of ModelLiteralism,we assume:LeibnizEquivalence:Isomorphicmodels representthe same physical situation.This appears to be the course followed by Einstein in late 1915 afterlaboringfor several years under the assumption of ModelLiteralism.3 fWI s describedusing some particular coordinate chart, then a coordinate transformationexpressing the diffeomorphismhyields a descriptionof h,*9J n the same chart.Such descriptions, Einstein explained to Ehrenfest that December, 'refer toexactly the same thing' (quoted in Norton [1987], p. 169). Some of the moreprominent recent textbooks on general relativity also contain statementsnaturally read as endorsements of LeibnizEquivalence.4Earman and Norton2 The property in question fails, for example, if each of the Oi'sis a constant scalar field.3 Between 1913 and 1915 Einstein formulated versions of an argument he referredto as the'hole argument' [Lochbetrachtung].he argument of Earman and Norton is a generalizationofJohn Stachel's interpretation of Einstein's original hole argument. See Stachel [1989] andNorton [19871 for details.4 Hawking and Ellis [1973], p. 73; Sachs and Wu [1977]. p. 27: and Wald [1984], p. 260.


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41o RobertRynasiewiczargue, however, that substantivalists cannot follow this path. For no matterhow the doctrine of substantivalism is formulated, they say, the variousformulations

must allagreeconcerning nacidtest ofsubstantivalism,rawn romLeibniz.feverythingn the worldwere reflectedEast o West(orbetter, ranslated feetEast),retainingall the relationsbetweenbodies,would we have a differentworld?The substantivalistmust answeryessince allthe bodiesoftheworldarenow in differentpatial ocations, venthoughthe relationsbetween hem areunchanged. [1987], p. 521}They claim, moreover, that a diffeomorphismof the manifold on to itself is theformal counterpart of such a rearrangement of bodies. Thus, substantivalistsmust maintain that for any Leibniz shift h, 9I and h*WJ epresent distinctphysical situations, in accordance with Model Literalism.The hole argument is formulatedagainst the backdropof a general pictureas to how the notion of 'physical possibility'attaches to theories. That pictureposits, in addition to the models of theories, a realm of possible physicalsituations, together with a map frommodels to physical situations designatingwhich models representwhich situations. The hole argument establishes thatone thesis concerning the nature of this representation relation, viz., ModelLiteralism, eads to intuitively unacceptable consequences. In the next severalsections, I want to show that the alternatives in this framework, includingLeibnizEquivalence, are equally unacceptable.In order to keep the discussion well regimented, I will assume that modelsare purelymathematical structuresliving in the universe ofZermelo-Fraenkelset theory, or some other foundational alternative. Moreover,I'll assume thatsuch principles as Model Literalism and Leibniz Equivalence are to beconstrued as applying not just to the class of models of a given theory, but tothe entire class of structures of the same mathematical type as those models.Thus, if *XT is the class of models of Tand X' the class of all structuresof thesame mathematical type, then ModelLiteralismasserts that distinct structures1, WJ22e- represent distinct physical situations, while LeibnizEquivalencedemands that if9IJ1 s isomorphicto 91J2,hen they representthe same physicalsituation. If 9JI~E'but TJI T,this is interpretedto mean that the physicalsituation representedby9)3, hough logically possible,is not physically possibleaccording to T.

3 ISOMORPHISM CLOSUREA crucial assumption in the hole argument is that the spacetime theories ofinterest to us satisfy the condition of Isomorphism Closure. It might seem,however, this is not a secure assumption. Earman and Norton suggest it iscommon to regardNewtonian and special relativistictheories as dealing with


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TheLessonsof the HoleArgumenlt 411a single spacetime, viz.,Newtonian or Neo-Newtonian spacetime in the formercase, Minkowksispacetime in the latter. Onemight take this to mean that themodels of such theories employ a common manifold together with aninvariant subset of objectfieldscharacterizingthe fixedgeometric structure ofthe spacetime in question. A special relativistic theory, forexample, then hasmodels of the form W=


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412 Robert Rynasiewiczof models to those whose manifolds are connected spaces. Some authorsintroduce other global assumptions, for example, that the models aremaximal,5 or that the spacetimes are temporally orientable (Sachs and Wu[1977], p. 27), or that they are globally hyperbolic" (Penrose [1979]).Moreover,some formulationsstartingfrom variational principles,such as theHamiltonian formulation, requireat the outset that the spacetime is stronglycausal and partitioned into a family of Cauchy surfaces (Wald [1984],Appendix E.2).More to the point, we are left with the false impressionthat whether or notModel Literalism leads to radical local indeterminism depends critically onwhether the theory in question imposes certain global restrictions. Quitethecontrary, all that is required is Isomorphism Closure, and whether or notIsomorphismClosure s met is entirely independent of the presence or absenceof global assumptions. To see this, suppose that T makes certain globalassumptions.LetT'be obtainedfrom Tby closing the class ofmodels of Tunderisomorphism. On the one hand, T' does not remove any of the globalassumptions of T since for any model of T' there is a model of T with the sameglobal characteristics. On the other, since T' satisfies isomorphism closure, itcan be subjected to the hole argument. Conversely, suppose that T is a localspacetime theory. Let T' be a theory obtained from ' by removing all but asingle model fromeach isomorphismclass of the models of T. On the one hand,T'failsto introduce any global assumptions since now forany model of Tthereis a model of T' with the same global features. Onthe other, it is impossibletofind a pair of models of T' related by a hole diffeomorphism.Now contrary to what was suggested about the predecessors of generalrelativity, it strikes me that we should assume as a matter of course thatspacetime theories, whether 'local' or not, satisfy IsomorphismClosure.By 'amodelof a theory'we mean a structuresatisfyingcertainspecifiedequations orconditions, and our notion of satisfaction is standardly one that does notdiscriminatebetween isomorphic structures. Granted,nothing compels us toadhere absolutely to standardusage. Oneis freeto operatewith non-standardmeanings of 'model' and 'satisfaction', and even to permit any arbitrarilyselected class of structuresto count as a theory. ForModelLiteralism,however,this has a price at least as high as that of embracing radical localindeterminism. Let T be a spacetime theory satisfying Isomorphism Closure.Supposewe stripaway fromthe class of modelsof Tsufficientlymany modelssoas to avoid radical local indeterminism, as well as any others that are5 A relativisticspacetime (M,g) is maximal just in case there is no other (M',g') such that (M.g)

is isomorphically embeddablein (M',g'). Local spacetime theories have the propertythat forany model 9WJnd any open neighborhood Uof the manifold,the restrictionof 9Wo U is also amodel. The restrictionto maximal models is assumed if the intent is to describe,not arbitraryspacetime regions, but entire universes.6 I.e. possess a Cauchy surface.


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The Lessonsof the HoleArgument 413unwanted for whatever other reasons. Label he resulting 'theory'T1.Now, foreach manifold M of some model of T1,choose a hole diffeomorphismhMof thesort that leads to radical local indeterminismfor T. Let 'theory' T2be given bythe class of models of the form hM*9J3 uch that 9Wts a model of T1.The'theories' T1and T2 are formally incompatible, since they have no models incommon. According to Model Literalism,they are, moreover, substantivelyincompatiblesince they agree nowhere as to what is physically possible.Butonwhat conceivable grounds could one 'theory' be preferred o the other? Ourinstinct is to deny there is any substantive conflict and to regardthe models ofT1and T2simply as notational variants on one another.

4 'NEWTON' EQUIVALENCEHow then should we understand talk of Newtonian or Minkowskispacetimein the singular? In mathematics, we commonly speak of 'the' two-elementBoolean algebra,or 'the'quaternion group,even though there arenumericallydistinct algebrasanswering the description,even on the same set of objects.Ineach case, the expressionreferspromiscuouslyeither to an entireisomorphismclass or else to an arbitrarilyselected representativefrom the class. Similarly,we can understand 'Minkowskispacetime' to refereither to the isomorphismclass of pairs of the form


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414 RobertRynasiewiczthese G1, .., Gkand the remaining fieldsP1,...., P1. fa Leibniz hift h: M-- M isa geometric symmetry in the sense that h*Gi= Gifor each i, then h preservesthe 'mutual order and position' of the points of M. If h is not a geometricsymmetry, then, under h each point p is 'numericallyconverted into' its imageh(p)in virtue of the way in which h altersthe 'mutual orderandposition'ofthepoints. Assuming the Gi's are the same up to isomorphism from model tomodel, this means that the physical situations represented by the variousmodels of the theory involve the same physical spacetime. The theory permitsdistinct physical situations on this spacetime if it has distinct models 9J1andh*9J such that h is a geometric symmetry. If h is not a geometric symmetry,then 9Yand h*92 represent the same situation, since, for any region U of themanifold, each object fieldPi is 'situated' with respect to U exactly as h*Piissituated with respect to the region h[U] into which U has been 'convertednumerically'.This removes the threat of radical local indeterminism for Newtonian andspecial relativistic theories. The strategy can also be extended to generalrelativity by assuming that the spacetime metric characterizes the 'mutualorder and position' of the points of spacetime. Unfortunately, the proposal isinternally inconsistent if the models of the theory have non-trivial geometricsymmetries. This can be shown with the help of a construction due to Norton([1989], pp. 60-1). Let h be a geometric symmetry of model 9Y(say a spatialtranslation)that is not also an automorphismof Y. Accordingto the proposal,1 and h,*9f consequently represent distinct situations. However, h can bedecomposedinto a pairof Leibnizshiftshi and h2such that h= h2ohl althoughneither hi nor h2 is itself a geometric symmetry. Now, since hi is not ageometric symmetry, M and h1i*91represent the same situation. Similarly,hl,*9l and h2,(hl*9n) represent the same situation. Hence, 9Y and h2,(h1*9jl)represent the same situation. But

h2*(hi*92f) (h2ohl)*931lh*,yielding a contradiction.5 MODEL SELECTIVISMThe contradictions removedif hl*9~1,nsteadof representing he samesituationasrepresented y9I, represents othingat all. LetModel electivismbethethesisthatsomemodelsmayfail orepresent nysituationwhatsoever,although distinct modelswhich do representsituationsrepresentdistinctsituations.Both Maudlin[1989] and Butterfield 1987, 1989a, 1989b]proposeversionsof ModelSelectivism, lthoughon different rounds.Maudlin rgues hat certainof thepropertiesfspacetime ointsareessentialproperties.If model m represents a physical situation and h is a Leibnizshiftthat does not preservethe essential propertiesof the spacetime points of that


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TheLessonsof the HoleArgument 415situation as portrayed by 91, then h*9JIdescribes these very same points ashaving propertiescontraryto theiressences, and hence describesno possibilityat all. For relativistic theories, Maudlin alleges that the spacetime metriccharacterizes these essential properties.Butterfieldproceeds from the premise of David Lewis'sdoctrine of counter-parts that no individualexists in more than one possibleworld. Taking this toapplyto spacetime points, Butterfieldargues that ifthe manifoldsofD1and9)1'have a non-empty intersection, then, assuming 9W1epresentsa possibleworld,WU'ust failto, on pain of populating distinct worlds with the same spacetimepoints.Neither proposal has much appeal for anyone not already sympathetic toeither de reessentialism or modal realism. Maudlin and Butterfield,I suspect,would likeus to see theirsuggested resolutions of the hole argument as offeringadditional motivation for these metaphysical doctrines. Each view doesprovide a way of avoiding radical local indeterminism: if h is a holediffeomorphism of 9W,then 91 and h*91 cannot both represent physicalsituations. But each raises other worries about determinism.Maudlin's position risks ruling in as deterministic certain clear cases ofindeterminism. Imagine a relativistic theory with globally hyperbolicmodels(M,g, ... ) and (M,g',... ) agreeing to the letterup through some particulartime slice but diverging thereafter. If there is no global isometry carryingg onto g', then the 'past'does not determine the 'future'even up to isomorphism.Applying Maudlin'smetric essentialism, however, we are forced to concludethere is no violation of determinism since at least one of the models portraysthe spacetime points represented as having properties contrary to theiressences.8On Butterfield'sproposal, no two situations can have a common past, andthus he must worry that determinismis true apriori.Butterfieldadmitsthis is a'disadvantage'but suggests that it can be minimizedby construing the basicidea of determinism in terms of a comparison of counterparts. But whatcounterpart relation is the appropriateone to use? If regions R1and R2cancount as counterpartsifthere is a diffeomorphism romthe one on to the other,then we can resuscitate the hole argument by composing a hole diffeomor-phism with an isomorphismonto another model with a differentmanifold. Ifitis requiredthat there exist a complete isomorphism of the object fields on R1and R2as well, radicallocal indeterminism is avoided,but at the cost ofmakingthat particulardistributionoffieldson the regions a necessaryone. As Maudlinpoints out, the position then flunks the Earman-Norton 'acid test' forsubstantivalism ([1989], p. 90). And if it is only necessary that R1and R2beisometric,then the worriesofmetricessentialism mention above are inherited.Apart from these, there is a difficulty common to any version of Model

8 This objection is due to Earman [1989]. p. 202.


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416 RobertRynasiewiczSelectivism that avoids radical local indeterminism. Let # be the class of thosemodels that representsome physical situation orother. For each manifoldMofa member of X, choose a hole diffeomorphismhM.Let Sk' be the class ofmodels of the form hM*9J1such that 9JIWe.Ifradical local indeterminismis tobe avoided, no member of X' represents even a metaphysically possiblesituation. In virtue of what do the members of t rather than the membersof,' represent genuine possibilities?9It might be answered that this is anunreasonable demand for explanation-there is simply a surd metaphysicalfact of the matter. At this point, however, it is fair to compare the ModelSelectivist to the Model Literalist who proposes to avoid radical localindeterminism by giving up IsomorphismClosure.Take T1to be the 'theory'with models *1 and T2 o be the 'theory'with models *2. The Literalistwantsto insist that T1and T2are not just notational variantson one another, but aresubstantively rival theories. The sole differencebetween the Literalistand theSelectivist, though, is that what for the former is a representation of merephysical impossibility is for the latter a representation of metaphysicalimpossibility.Tothe extent that it is intolerable to thinkthat T1and T2 re rival'theories', it is intolerable to think there is a surd metaphysical fact of thematter as to whether it is XI or S2 whose models represent genuinepossibilities.6 LEIBNIZ EQUIVALENCEIt may seem that LeibnizEquivalenceis the ineluctable conclusion of the holeargument. 'But,' as Earmanwrites, 'drawing circles around groups of space-time models and labeling them equivalence classes does not show that there isa viable alternative to substantivalism' ([1986b], p. 237). In particular, itneeds to be shown how LeibnizEquivalenceis a consequence of certain morebasic assumptions. If these assumptions involve an anti-realism concerningspacetime, then the models of a spacetime theory will be seen to contain acertain amount of descriptivefluff.Distinct models which agree modulo thisfluffcan then be claimed to representthe same physical situation. Although itremains to be establishedjust what is fluffand what is not, it is at least clear inoutline why distinct models may fail to represent distinct situations.It is not so clear how LeibnizEquivalencecould be derivedwithout giving upspacetime realism. Let me mention two potential difficulties.First,suppose ais the physical situation represented by model 9m. A spacetime realist ispresumablycommitted to holding that this involves a one-to-one correspon-dencef between the points in the manifoldof M21nd the spacetime points of asuch that the way that the object fields of 9NJre distributedon its manifold9 This is essentially what Norton calls the problemof the 'real' model and the 'imposter' [1989],pp. 62-3).


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TheLessonsof the HoleArgumenlt 417reflects under this correspondence the way the physical fields of a aredistributedon its spacetime. Now the ModelLiteralist and the Selectivist canagree that for any Leibnizshift h, h*9,Jalso representsa in the sense that thedistributionof object fields in h*9J matches the distributionof physical fieldsin a under the correspondence function foh-'. If this is all that LeibnizEquivalence demands, then the thesis is completely trivial. In order to havebite, it must be interpretedto demand that there is no other possiblesituationwhich h*J can represent. In particular, it must be denied that under thecorrespondence functionf, h*,l9 represents anything at all. Hence, the realistwho wants to subscribe to a non-trivial version of LeibnizEquivalencemusteither argue that the situation a' which the Model Literalist magines h*9Jl orepresent under f is nothing at all, or else must explain why h*931 ails torepresenta' under . The latter is clearly hopeless. And to take up the former sto revert to ModelSelectivism.Second, if U and V aredistinctopen neighborhoodsof the manifoldof9NU,herealistis presumablycommittedto holdingthat the substructures9~Ilv nd9.1Vbtained by restricting 9) to U and V, respectively, represent distinctsubsituations of a. Suppose, however, that 9)N as non-trivial automorphismsor that its objectfields are uniformon some region of the manifold.Then U andV can be chosen so that 9NIJus isomorphicto .J1.'"1 eibnizEquivalencethendemands that they represent the same situation. 1Earman[1977, 1986b, 1989] opts foranti-realism,although of a formquitedifferentfrom traditional relationism. Any spacetimemodel can be character-izedup to isomorphism in terms of an algebra starting with an algebraic ringisomorphic to the ring of continuous real-valued functions on the manifold.Earman suggest that these Leibnizalgebras,as he calls them, corresponddirectly to physical reality. A spacetime model only indirectly represents apossible physical situation in virtue of realizing, in a precise mathematicalsense, the Leibniz algebra directly representing that situation. Since eachLeibnizalgebra is realizedby each of the members of an isomorphismclass ofspacetime models, isomorphic spacetime models represent the same physicalsituation.I have argued elsewhere (Rynasiewicz [1992]) that this programdoes notyield a viable version of anti-substantivalism. The basic reason is thatisomorphic spacetime models have the same algebraic structure only in thesense of 'same up to isomorphism'. Thus, the proliferation of isomorphic1i If is a non-trivial automorphism, let U be any neighborhood on which ( differs from theidentity map and let V= h[U]. If the fields of3M re uniform on region R.let Uand V be any pairof distinct neighborhoods contained by R." Note that if the theory in question is a 'local' spacetime theory, these substructures are alsomodels of the theory. The difficulty cannot be overcome by restricting the applicability ofLeibnizEquivalence to maximal models. The hole argument applies whether the models inquestion are maximal or not.


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418 RobertRynasiewiczspacetimestructureson which the hole argument feedsis matched by an equalproliferationof isomorphicLeibnizalgebras,and the argument can be directedagainst the latter as easily as the former.In fact, forthe category of theories ofphysical interest, the algebraicand the spacetime approachesare mathemati-cally equivalent. The programof Leibnizalgebras is substantivalism cast in adifferentlanguage.I also argued there that the very motivation for the program, namely,LeibnizEquivalence,is flawed. Leibnizhad a second demonstration to 'confutethe fancy' of the Newtonians. If time were something 'distinct from thingsexisting in time', it could be asked 'why God did not create everything a yearsooner' (Alexander [1956], pp. 26-7). Now suppose we interpret the act ofcreation not as the implantationof anything into spacetime,but as a transitionfroma static 'groundstate' to one of variedactivity.Furthermore,remove Godfrom the scene, so that the question is not why he startedup the universe attime torather than at tominus a year, but why the universe turned itself on atthat time. Let Y2 epresent this universe. I expect widespread agreement thatany spacetime theory having M fora model is to be judged an indeterministictheory. Let T be the theory whose only models are those isomorphic to M.According to LeibnizEquivalence, these other models all represent the samesituation as 9N, and hence T admits only this one situation as a physicalpossibility. But how can T fail to be deterministic if it recognizes no otherphysical possibilities?By treating this example as a Cauchy initial value problem, relativitytheorists would conclude that determinism breaks down. This shows that,despitewhat they might say, in practice they employ a criterion subtler thanLeibnizEquivalence.We will see what this is in due course. By now the readershould have sufficientdoubts that there is a resolution to the problemraisedbythe hole argument within the general framework commonly presumed. Weneed a fresh start.

7 RADICAL LOCAL INDETERMINISM AND THE INSCRUTABILITY OFREFERENCEThe problemthe hole argument presents is not that substantivalism involvesan aprioricommitted to indeterminismas Earmanand Norton have charged.To be sure, the argument succeeds in showing that Model Literalism(conjoinedwith IsomorphismClosure) mpliesradical local indeterminismforphysically interesting spacetime theories. But even if it is granted thatsubstantivalism requires Model Literalism (something I would hesitate togrant), it follows that substantivists are a prioricommitted to indeterminismonly if they hold a priorithat some spacetime theory or other is true.Now this may seem a bit of a quibble. For what would substantivalismamount to if one didn'tbelieve the propositionthat some spacetime theory or


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TheLessonsof the HoleArgument 419other is true? My guess is that few today who would call themselvessubstantivalistsdo believe apriori hat some spacetimetheory is true. Rather,Ithink most would urge that the case for substantivalism depends on theavailable evidence for this or that particular theory or narrower class ofspacetime theories and that their belief that there exists a true spacetimetheory is based on a posterioribeliefs about the truth or approximatetruth ofspecifictheories (as opposedto, say, theological considerations).And ifso, theircommitment to indeterminism(ifsubstantivalismreally so committed)is also aposteriori.The hole argument can be used to pose a dilemmaonly to the extent there iswide agreement that radicalocalindeterminismis obviously absurd,and thusfalse a priori. Otherwise it might be suggested that we should weigh theexperimental evidence for or against it and begin looking for crucial tests, inanalogy with the correlation experiments targeted at Bell's Inequality. Butunlike the genuine failure ofdeterminismin quantum mechanics, radicallocalindeterminism strikes us as physically phony. The hole argument shows thatModelLiteralism s suspect, not because it is committed a priorito indetermi-nism, but because it saddles us with a bogus sort of indeterminism.Nor is the involvement of spacetime theories essential. The dominant decaymode of the neutral pion is nro-*2y,where the two photons are usuallyobservedonly indirectlyvia pair productiony-* e++e-. A specificinstance ofthis chain of events was observed in the first detection of the Q- particle.Theparticular electrons and positrons produced are identifiable from the bubblechamberphotograph reproduced n the publishedreport (Barnesetal. [1964]).Call one of these electrons 'Murray'and the other 'Yuval'.Now, according toModel Literalism,determinism would fail even if all the relevant probabilityamplitudeswere zero or one. Takeany descriptionof the cascade such that themodels of the description contain elements corresponding to each of thefundamentalparticles.Let %be one of these modelsand let 0 be a permutationof the domain of

%.which interchanges the elements corresponding to theelectrons Murray and Yuval. This defines a model 0*,I of the descriptiondistinct from %, which coincides with % up through some time slice, butthereafterdiverges. Thus assuming these two models representdistinct statesof affairs,determinism fails.These sorts of permutation arguments are familiarto the readersof Quine,Davidson, and the 'new' Putnam in connection with the so-called inscrutabi-lityofreference.Thestrategyofthese writers,however, is to argue not that anycomplete description admits distinct but indiscernible states of affairs, butratherthat, given any complete descriptionof a single state of affairs, here arehopelesslymany distinctbut indiscernibleways ofconstruing the extensions ofthe descriptiveterms of the language on the domain of discourse in question.More specifically, a schemeof reference r interpretationor a language is anassignment of extensions to the primitive predicates and names from some


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420 RobertRynasiewiczdomain of individuals.The semantics for a language is presumedto derivefroma scheme ofreference n a recursivefashion.Ifa is a scheme ofreferenceand 7rpermutationof the domain of discourse, then 7r an be used to definea schemenr*ain general distinct from 0.12 Given certain general and plausibleassumptions, it can be shown that forany sentence 0, the truth conditions foron the scheme a are equivalent to the truth conditions for / on the schemen7r7.Thus, the referenceof the descriptive vocabulary is radicallyunderdeter-mined by the truth conditions for the sentences of the language.The hole argument can be recast as a special instance of this. Assume thetruth conditions forthe sentences of the language of relativityare fixed.Take ato be an interpretationon the set of actual spacetime points consistent withthese truth conditions. Considera permutation7rwhich happens to leave fixedthe interpretation of such predicates as 'open neighborhood' and 'smoothcurve' and agrees with the identity map everywhere except for some smallregion. The interpretation 7r*( yields equivalent truth conditions for thelanguage but assigns differentextensions to such predicatesas 'x is space-likeseparated from y'. Thus, even keeping fixed the interpretation of the'topological' vocabulary, the truth conditions leave underdetermined theinterpretation of the remaining vocabulary, even if it has been specifiedeverywhere but for some small 'hole'.It might be claimed that there could be grounds fordistinguishing betweeninterpretations more finely grained than the truth conditions for wholesentences of the language. Whatever these grounds might be, though, ourintuitions about radical local indeterminism set an upper limit on howdiscriminatingthese grounds can be. For if we have some means to distinguishthe interpretationa from 7r*a, hose very same means allow us to distinguishbetween a situation conforming to ocand one conforming to 7*,a.Thus, to theextent we think we can discriminate between interpretationsof a spacetimetheory related by a hole diffeomorphism,we should be inclined to acceptradical local indeterminism as a genuine violation of physical determinism.Now, a mathematical model can be converted into a scheme of abbreviationby a bijectionof the domain of the model onto a set of actual individuals.Thus,the class of cases in which Model Literalism leads to an objectionablepostulation of distinct but indiscerniblephysical situations coincides with theclass of cases in which there exist underdetermined schemes of referenceaccording to the inscrutability arguments. Put slightly differently, in thosecases in which we want to deny that a given pairof models representdistinctsituations, each model has the status of an unintended interpretationrelativeto the other. If we take the pair to be co-intended,hen we are committed to12 If c is an individualname and a is the designation of c underx. then mr(a)s the designation ofunder r*x. IfP is an n-adic predicate,then the tuple (a ..... n) is in the extension of P under

,*x iff (l-'a. ...., [-'a,,) is in the extension of P'under t.


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TheLessonsof the HoleArgument 42Iregarding the difference between them as indicative of a real difference inpossibilities.8 A PARABLEWhere, it may be asked, does this get us? Assume we deny that radical localindeterminism is a genuine species of physical indeterminism because themodels in question represent, not rival situations, but variant underdeter-mined schemes of reference for describing the same situation. How, then, dowe, or could we, ever know which scheme is the one we are using?The problem raised is a pseudo-problem.If the question as to which is the'correct' interpretation is posed as a factual one, then the various competingschemes must be describable n a way definiteenough to say which is which.Otherwise,the question as to whether i, forexample, is the 'correct'scheme iscomparable to the question 'Is P true?'If the specificationsof the schemes areindependent of the language under interpretation, as is the pretense in theinscrutability arguments, then a user of the language may very well be in thedarkas to which specification s the 'correct'one. However, ifthe specificationsare given in terms of the language being interpreted,then one of the schemescan be uniquely identifiedas the 'correct'one simply in virtue of the way it hasbeen specified.Consequently, if there is no way of identifyingwhich scheme iswhich independently of the resources of the language under consideration,then either it is obvious which scheme is 'correct'or else there is nothing to beknown.An illustration may help. Imagine a logic aptitude experiment set up asfollows. The subject is told that three items will be shown in succession andthat each item is either red or green. After the items are displayed,the subjectis to write down a description of the situation whose deductive closure iscomplete in the first-order ragmentof Englishcontaining the predicates'red','green',and 'is after'.The intelligent subjectknows beforehandthat, given theconstraints of the experimental setup, most of the work can be done ahead oftime. One will have the sentence which expresses that there are exactly threethings, the axioms fora total linear ordering,and a dichotomy axiom for 'red'and 'green'.All that remains to be determinedby the experimentalrun is howto fill in, for n= 1, 2, 3, the first-orderequivalent of the schema

The nth item iswith either 'red'or 'green'.Since there areeight ways to do this, there areeightpossiblecomplete descriptions.The intelligent subjectalso knows that for eachcomplete descriptionthere are six distinct interpretations on a given domainsatisfying the description.So, when the sequence of items in the experimentalrun turns out to be a reditem followed by a red item followed by a green one, the subject fills in the


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422 RobertRynasiewiczinstances of the schema accordinglyand remarks to the experimenter, 'IknowI gave the correct first-orderdescription, but what is the correct scheme ofreferencefor the domain of individuals shown to me?''What do you mean?' she replies.'Letal, a2,and a3be the itemsdisplayed.Onone scheme ofreference,aI is redand is the firstitem, a2 is red and is the second item, and a3is green and is thethird. On another scheme, al is red and is the first item, a3 is red and is thesecond, and a2 is green and is the third ... Which is the correct scheme?'The experimenter is a bit baffled.'Well, if a1is the firstitem, a) the second,and a3 the third, then the first scheme is correct. If aI is the first item, a3 thesecond, and a) the third, then the second scheme is correct ...Unsatisfied,the subject presses. 'But is al the firstitem, a) the second, and a3the third;or ... ?''You tell me!' she blurts back.The subjectrealizeshe has yet to pose a factual question. Had he done so tobegin with, say by identifyingaI, a), and a3as the first,second, and thirditemsrespectively, he would have answered for himself which of the six interpreta-tions is the correct one.But, it may be asked, since the subject can identify a unique intendedinterpretation satisfyingthe descriptionin question, does it follow that there isreally only one possible experimental outcome admitted by the description?We should resist the notion that there is an absolute fact as to the variety ofpossibilities that is not correlative to some assumption about the meansavailable for distinguishing the items in question one from another. If it isassumed that the experimenteras well can identifythe displayeditems only interms of their ordinal positions, say, because the items in question arecomputer generated images, we conclude there is one possible outcome perdescription. If we suppose, however, that she can identify the itemsindependently of their orderand color, and thus can formulatefor herself sixdistinctco-intended interpretationsperdescription,we will maintain there aresix correspondingpossibilities n each case. Assuming the subjectcontinues tobe restrictedto identifyingthe items ordinally,we might speak in terms of thenumberofpossibilitiesrelativeto the subjector experimenter,respectively.Butthe number of possibilitiesrelative to the subject is an uninteresting one if hismeans of identification are artificiallyconstrained. In the conduct of scienceour verdict as to how many possibilitiesthere 'really' are is guided by whatmeans of identificationare conceivably available to the experimenter.Keepinmind, though, that in earlier centuries it was common to liken ourselves tosubjects of a single Experimenter.This vignette might leave the false impression that one has always eitherfeast or famine in the number of co-intended interpretations satisfying acomplete description. Suppose the items displayed are computer generatedimages, so that there is no question of identifyingthem apartfromtheirordinal


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TheLessonsof the HoleArgument 423positions. Suppose, however, that the language of description contains thetriadicpredicate'x is between y and z' instead of the dyadicpredicate'x is aftery'. Given the same experimental run, both subject and experimenter canproduce a pair of co-intended interpretationssatisfying the complete descrip-tion in this new language of the sequence produced. One of these, theinterpretation-first item red, second item red, third item green-correspondsto the actual experimental run. The other interpretation--first item green,second item red, third item red--corresponds to an alternative possibilityadmitted by the same maximal description.Our goal now is to develop a general account of how the number of co-intended interpretations per maximal description derives from the means ofidentification assumed, and to adapt this account to the determination of theco-intended models of a physical theory.9 THE IDENTIFICATION GROUPIn what follows, I will limit myself to those contexts in which interpretationsare 'underdetermined'at best only up to isomorphism. I will also need somepreliminarytechnicalities.Fora given language Y and a scheme ofreference i for Y, let P(i) be the setof all permutationsof the domain of discourse of x.Define ~i"(~) o be the set ofall interpretations for [ on the domain of i isomorphic to i. Call 4"('x) thepermutation lass of i in virtue of the fact that

~"'(7c)='71{*,c:CEP(l)'.Let Y+ be the expansion of Y obtained by introducing a new individualconstant a for each element a in the domain of i. For each flcP"(i), let #/3 bethe expansion of p which assigns to each new constant a the element a. Thediagramof fl, denoted A(#), is the set of atomic sentences and negations ofatomic sentences of ~+atisfiedby pf. Foreach permutation rceP(i),defineAfto be the corresponding permutation of the set of new constants such thatf7(d)= 7o(a).A n turn induces a permutation of the set of sentences of f + suchthat foreach sentence ,, 7,, is the result of uniformlysubstituting i(a) ford,foreach new constant. Note that forany PE3j CX), he image of A(#) under f7 sA(r*fl).It should be evident from the examples of the preceding section that if P isadoptedas an intended interpretation,this mandates that a certain subset I(P)of the diagram of # be satisfied (in #f) simply in virtue of the way in whichthe individuals of the domain are identified so as to make # an intendedinterpretation. Call this subset the identification iagramof Pf.In our experi-mental examples, if the ordinalpositions of the individualsexhaust the meansof identifying them, then in the case of the first language considered, I(/) isthe restrictionof A(/) to the language of the orderpredicate.In the case of the


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424 RobertRynasiewiczsecond language, I(f) is the restriction of A(P) to the language of thebetweenness predicate.If the individuals are identifiable ndependentlyof theirorder and color, then I(f) is empty in each case. Whatever the means ofidentification,the basic sentences of Y + satisfiedin consequence must be thesame up to permutation across interpretationsof the same permutation class,i.e. for any ne7'o'I"(X),the image of I(fl)under 7f s I(71:*3).Now a pairof interpretations# , f2bef (x) will count as co-intendedifthereis no disagreement between them concerning which individual has beenidentified as which. In other words, Pi and #2are co-intended just in caseI(]/1)= I(f2). Thus, the relation of co-intendedness on .?"(cx) s an equivalencerelation and partitions '"(() into subclasses of co-intended interpretations.The relation of co-intendedness can be equivalently definedin an alternateway which will be more useful as we proceed. Obviously, for each PfElk"(a),P(#)= P(i). Note also that P(P) forms a group under the composition ofpermutations. Define G(P)to be the set of those permutations in P(#) whichpreserve the identificationdiagram of fi, i.e.

G(=) !rcfEP(fl): I(*r./) = I(fl)Since G(fl) s a subgroup of P(f), call it the identification roupof Pf. fAut(fl)isthe automorphism group of fl, thenAut(fl)c G(fl)cP(P). (1)

Although P(f)= P(i), it need not be the case that G(l)= G(c).None the less,the identificationgroups of the members of fi"'(c) are systematically related.For each nmP(-),the conjugate group of G(Jf)by 71 s definedG"(fl)= df EP(a): iooon- '1G(f)}.

It follows from the definition of the identificationgroup thatCG-1(7,) = GW(), (2)or equivalently,

G(*) = G"-'().(This parallels the behavior of the automorphism group.) Now let R be therelation on X'"(x)such that PfI P2ust in case there exists a ncG(fl)such thatn*#f1 #2. From (1) and (2) alone it can be shown that R is an equivalencerelation. Fromthe definitionof G,. t follows that R is the relation of being co-intended.The identificationgroup of an interpretationis also linked to the identifica-tion groups of its subinterpretations, although in a less complete fashion.SupposeP is a subinterpretationof a. Since the domain of # is a subset of thedomain of aand ftagrees with on this subdomain, the identificationdiagramof f is a subset of the identificationdiagramof c. Consequently,forany nmeG()


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The Lessonsof the HoleArgument 425if the restriction of7no the domain offl is a permutation tn'eP(fl),hen n'eG(fl).We want now to extend thisanalysis to the mathematical modelsof physicaltheories.There are two hurdles.First,although we often have tolerably precisecharacterizations of the models themselves, we seldom have a preciselycharacterized language for which those models serve as interpretations.Second, we do not always have available an appropriate set of actualindividuals on to which we can map the domains of the models. Theoriesexpressing putative laws of nature, however, are addressed to certaincategories of entities genericallyconceived, andimplicitlyassociated with suchare generic identification criteria.Hence, we can proceed by supposing that tothe extent that a theory has a definiterange of intendedapplications,we tacitlyassociate with each model 91of the theory an identificationgroup G(91) ubjectto the following minimal constraints.

(I) Aut(9I) e G(91)e P(91).(II) Iff is an isomorphism of 91 onto 3, thenG(%1) { meP(1):fonof-I eG( 3)}.

(III) IfI%s a substructure of 3 and tc'eP(g1)s a subpermutation of n~eP(3),then ''eG(91)f nG(93).Structures 1,and $2 fromthe same permutation class k"are co-intendedif and only if there exists a ~n~G(931)uch that 7*.31 =0 2. As we saw above, (I)and (II)entail that this is an equivalence relation on t(". Call the equivalenceclasses of the relation co-intentionclasses, or co-classesfor short. Any pair ofdistinct models from the same co-class are to be counted as 'representingdistinct physical situations'.'1Considernow a permutationclass ,"' ofstructures in isolation fromthe rest.If the identification group of each structure in t"' coincides with thepermutation group, then all the members of t" are co-intended. At the otherextreme, if the identification group coincides with the automorphism group,then each co-class is a singleton set and the members of t" count asnotational variants on one another, representingthe same physical situation.Suppose, however, that the identification group is strictly intermediatebetween these two cases. Then there aremany co-classes with a multiplicityofstructures per co-class. Foreach e-X,'",et $(gI) denote the co-class of I91. f93e-"P, then there exists a permutation neP('L) such that n*=ZI 3. Further-more,

Thus, any pair of distinct co-classes can be put into a one-to-one correspon-13 The quotation marks are used to disavow any commitment to an ontology of possiblesituations. Since the repeateduse of quote markscan be clumsy, I will frequently dropthem inwhat follows.


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426 RobertRynasiewiczdence by some permutation.Such a permutation providesa ruletelling us howthe members of the one class can be systematically interpretedas notationalvariants on the members of the other. Officially,a resolutionof 9I to 93 is apermutationeP($21)uchthat(i)p*9l= ~ and(ii)eitherpoG(9I)r elsepis theidentitypermutation.nresolving 1 o93,p alsoresolves achmember fH(%I)to a memberof #(3), and so I shallsay thatp resolves he one class to theother.If~#(U = (5), then there s a uniqueresolution, amely heidentitypermutation. If ?(I) $ W(j) and the identification group is strictly largerthan the automorphism group, then (9I) can be resolved to (Q) inmultiple ways. Each resolution is an alternate convention for establishingnotational equivalence.A riggingof a permutation class *k"is a function 91 which assigns to eachordered pair (f~1, 2) of co-classes a resolution pw. , 2 of *2 on to suchthat for any 42, #2, and V3(i) poI-. y,= - t2. 1(ii) PwPl,.W2OP#,2.3= P.#1.?3"Any rigging91givesrise to an equivalence elationa:, on fk", uchthat

Thesignificance f91= 93 s that,in the old mannerof speaking, 91 nd93representhe samesituationaccordingo 91.Ihastento emphasizehatif the identificationroup sstrictlyntermediatebetween heautomorphism roupand thepermutation roup, hentherearedifferentriggings yielding distinctequivalencerelations,and the choicebetween hem s anarbitraryne. Thismeans hat,except nlimiting ases, heproject froundingupmodels nto fixedequivalence lasses san idleone. Theimportantnterprises to understand nderwhatconditionst ispermissibleoconstruea givenpairof modelsas 'representinghe samesituation'. f t is notpermissible o construe them as such, i.e. if we are obliged in thosecircumstanceso take hemas'representingistinct ituations', shallsaythatthe two modelsseparate.IO COMPARISONS, AUTONOMY, AND DETERMINISMConsidering permutation lass in isolation,distinctmembersof the classseparate ustin casetheyareco-intended. inceI am (artificially)estrictingmyself o thosecases n whichthe 'underdetermination'f references at bestup to isomorphism, on-isomorphicmodelson the same domainshouldbeassumed o separate.But what if the domainsof models%91nd 912do notcoincide?f91,and912arenotisomorphic,bviouslyweshouldexpect hemtoseparate. However, if they are isomorphic, there is no way to answer the


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TheLessonsof the HoleArgument 427question except with referenceto some assumedcomparisonof theirrespectivedomains. Now, in order to do this it is not necessary that we postulate anontology of possible individuals and set out to resolve issues of transworldidentity. How the domains of %I1nd %92ompare is a matter of what we wantto imagine 91, and 912 as representing. And since we have no handle on'sameness' or 'difference'of representation except in terms of structures on acommon domain, this means we need to embed the domains of 91Iand 912 ntosome common domain. Specifically,a comparison f 91, with 912 involves fourcomponents: a pair of structures 31and 32 sharing the same domain and apair of isomorphic embeddings f, and f2 of 91 into 01 and 912 into 32respectively.911and 912separateundera given comparison just in case there isno resolution of the image of 91t underfi to the image of 912underf2.This definition entails that if 911and 912are not isomorphic, they separateunder any comparison. If 9t1 is isomorphic to 912,then whether or not theyseparatedependson the purposesand details of the comparison. If we want tocompare 91 with 912 apart from their representational relation to otherstructures, we can embed 91Wsomorphically onto itself and 912onto somemember of the permutationclass of 911.Whether or not WI1nd 912are therebyrequiredto representdistinct situations dependson whether the image of 912 isco-intended with 911.We may, however, want to compare %1 and 92 asrepresentations of fragments of certain larger situations. This is done byembedding %I1 nd 912into 01 and 1S2respectively as propersubstructures.The definition of separation entails the following for particular cases. If wechoose 1= 32, then %1and 9I2 separate just in case the comparison mapsthem onto distinct substructures. If 31 and 932are isomorphic but not co-intended, then, even though 31 and 932can be taken to represent the samesituation, 911and 912eparateif no resolution of 31to 23when restrictedto thedomain of the image of 911resolves 911to 912. Or, if 3I and 02 separate, andthus represent distinct situations, 911and 112may fail to separate, and hencemay be taken to representa common subsituation of the two. Finally,note thateven if121, nd 912are co-intended models of the same permutationclass, theycan always be compared in such a way that they fail to separate.The general strategy of performing comparisons is first to fix in mind adefinite spectrum of situations by focusing on the separation relation forstructures on a common domain, considered in isolation. Then we specifyforthe structures to be compared, independently of one another, what subsitua-tions we intend them to representby means of the embedding functions. Therelation of separation for structures on the host domain then informsus as towhether or not these substructures must be regarded as distinct.The order ofcomparison can also be done in an inverse fashion. Supposewefix a spectrumof situations involving some set of imagined individuals in termsof the separationrelation on a class ofmodels on a given domain. We then seekto determine, for some largerset of individuals,the range of situations of which


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428 RobertRynasiewiczthe initially given situations are subsituations. It may seem that the way to dothis is to select some super-domainof the domain of the originallygiven class ofstructures and then simply to examine the separation relation on the class ofextensions to that super-domain.There are two hitches, however. Suppose91is a structurein the originally given class. First,the procedurepresupposes hatwhat 9W.represents is to be imagined antecedently and independently ofwhatever the various extensions of 91might be imagined to represent.Second,it is presumed that, whatever situation 91 has been imagined to represent, itcontinues to representthat very same situation quasubstructure of its variousextensions.Take the second of these first.Suppose93 and 932areboth extensions of 9t tothe super-domainin question. If01 is not resolvable to 02, then evidently 93and 93 must be taken to representdistinct situations. Supposehowever that presolves 03 to 0-2. Unless the restriction of p to the original domain is aresolution of I1 to itself, 91 fails to represent qua substructure of 93 what itrepresentsquasubstructure of 0), and hence, in one of the two cases, fails torepresentwhat 9W as antecedently imagined to represent.So, in general, if9BIand 912are members of the originally given class of structures,and ~ and 32are extensions of 931 and 32 respectively, we must demand that 31 and 02 failto separate only if there is a resolution of 931 to 932 whose restriction to theoriginal domain resolves I9, to IN2.The first hitch mentioned above involves considerations about the identifi-cation of individuals that go beyond what was needed to motivate theintroduction of the identification group. If 91 is a substructure of 0, then, inorder to imagine the situation represented by WfLua substructure of 93antecedently and independentlyof whatever situation we might imagine 93 torepresent, the individuals of the situation that 9W.represents must beidentifiable antecedently and independently of any additional individualspresent in the situation that 0 is taken to represent. We cannot assume thatthis will automatically be the case. Reconsiderthe domain of items presentedin the experimentalparableabove. Ifthese items are identifiableonly in virtueof their ordinalpositions, the subdomain consisting of the firstand third itemsfails to meet the requirement,since the identification of the third item dependson the identificationof the second item, which lies outside the subdomain inquestion.Let us say that a (sub)domainof individuals is autonomousust in case eachmember of the domain is identifiablewithout presupposingthe identificationof any individual not in the domain. For interpretations a and # of a givenlanguage on some actual set of individuals, we can then say that a is anautonomous ubinterpretationf f if a is a subinterpretationof# and the domainof a is autonomous. It should be clear that the identification groups of theinterpretationsfora given language need not suffice to determine the relationof subinterpretationautonomy. For example, taking the language of identity


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TheLessonsof the HoleArgument 429(containing no predicates other than the identity symbol), the permutationgroup and the automorphism group of any interpretationcoincide, and hencethe identification group yields no information as to which (sub)domains areautonomous.Now, the notion of an autonomous (sub)domain cannot be carried overdirectlyto theories with mathematical models, since a given subset ofelementsof a domain may hang together in one structure so as feasibly to represent asituation whose members constitute an autonomous domain, but in anotherstructuremay be scatteredabout in a way inappropriate or the representationof an autonomous subdomain. None the less, we suppose, in analogy withwhat we did with the identificationgroup, that, to the extent that a theory hasa intended range of application, it is tacitly specifiedwhich substructures of amodel are autonomous substructures. This is also subject to certain minimalconstraints. Writing II-


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430 RobertRynasiewiczsub-situation s' of s, s' is a subsituation of no other situation admitted by Tinvolving the same individuals as s.Purging this of reference to a Pickwickian realm of possible situations andindividuals, the physicallypossiblesituation s is replacedby a model I~of Tandthe subsituation s' by an autonomous initial condition substructure '. Thequestion as to whether there is no 'other situation' of the sort in question issettled by examining whether 9I separates from some other model 0' on thedomain of 0 under the inverse comparison involving ' and the identityembedding.In other words, T s deterministicjust in case forany model 9I andany initial condition 3~ia 1,there is no model I91' f Tsuch that (a) the domainof 1I' coincides with the domain of 9I and (b) there is no resolution of I91'o 'Iwhose restriction to the domain of is the identity permutation.By invoking the appropriate comparisons, this definition can be statedequivalently, although less perspicuously,without demandingeither that 3 bea substructureof I or that V' have the same domain as I. Suffice t to say thatif the identification group coincides with the automorphism group, thendeterminismdemands: if 3 is an initial condition structure compatiblewith Tandf andf2 are isomorphicembeddingsof into models L91nd 912of T,thenthere is a model 93and isomorphicembeddingsgI and g2of IIIand 912 nto 0respectivelysuch that glofi = 9g2f2. If we have a notion of maximal models aswell, this in turn is equivalent (assuming Zorn'sLemma) to the requirementthat any extension of 91 o a maximal model of T s unique up to isomorphism.11 APPLICATIONS TO SPACETIME THEORIESThe identificationgroups implicitlyassociatedwith the models of the standardcategories of spacetime theories, as customarily used, are as follows.* General relativistic theories: the isometry group.* Special relativistic theories: the isometry group.* Newtonian theories: the groupofdiffeomorphismswhich leave invariant (a)the temporal separationofevents, (b)the spatialseparationof simultaneousevents, (c) the class of inertial frames,and (d) the absolute rest frame,ifoneis posited.In the firstcase, the metric uniquely determines the stress-energy tensor. Sonormally, the identificationgroup coincides with the automorphismgroup. Inthe other two cases, the metric and affine featuresof the spacetime usually failuniquely to determine the other object fields posited. Consequently, theidentificationgroup is strictly intermediate between the automorphism groupand the permutation group. In all three cases, the autonomous substructuresare those having topologically connected domains. Before discussing therationale behind these determinations, let us see how the quandariesraisedinthe firsthalf of the paper are now resolved.


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TheLessonsof the HoleArgument 431First, reconsider the hole argument. Let h be a hole diffeomorphismformodel 9N1. he argument assumes that his not an automorphismof Mand thatthe 'hole' H in question is not the entire manifold. In the case of generalrelativity, the firstassumption entails that hoG(921).orspecial relativisticandNewtonian theories, the second assumption yields the same conclusion. Nowthe complementH of the 'hole' is either topologicallyconnected or it is not. Ifitis not, then we cannot use the common restrictions of 9J3 nd he*Wo H foraninverse comparison. So the question as to whether 931and h*2 representdistinct situations is to be answered by asking whether 'S and h*9YJre co-intended members of their permutation class. Since hoG(T9R),he answer is no.IfH is connected, then we can ask whether 9JYnd h*9J separate under theinverse comparison. Since h was defined to be the identity on H, itautomatically resolves 9N to h*92 while resolving the common restriction to Hto itself.Thus, M9Ind h*WIailto representdistinctsituations under the inversecomparison to be used in judging whether determinism holds.Consider next the construction used to show the incoherence of 'Newton'equivalence. Recall that that construction began by taking an isometry h ofMinkowskispacetime and then decomposing h into diffeomorphismsh1and h12such that h-ohl = h although neither hi nor h2 is itself an isometry. Aninconsistency resulted since 'Newton' equivalence demands that W9 nd h,*9JI2representthe same situation and that hi*92fand h2a(h ,*9M)epresentthe samesituation, although 9J3 nd h*,2f (i.e. h2*(hl*9.*I))epresentdistinct situations.Given the principlesdeveloped above, though, the paradox is only apparent.Since the identificationgroup in this case is strictly intermediatebetween theautomorphism group and the permutation group, there are a number ofriggings of KP(MJR)ielding distinct equivalence relations, and whether or nottwo models from distinct co-classes represent the same situation is relative tothe rigging chosen. 9JIand hl*9il represent the same situation relative to arigging resolving Y(h1,*9)3)o $K(MJT)ia h -', while hI*91 and 11*91 epresentthe same situation relative to a rigging resolving (hl*91) to IJ(931)ia h_1.s a challenge to the consistency of spacetime realism with LeibnizEquivalence, I raised a merological problem. A model 9M!may have distinctisomorphic submodels 9)1 and 9312.Realism demands that 9MI1nd 9J)12represent distinct situations. LeibnizEquivalencedemands they represent thesame situation. For general relativistic theories, I have endorsed LeibnizEquivalenceto the following extent. Distinctisomorphicmodels with the samedomain represent the same situation, considering their permutation class inisolation. Isomorphic models with disparate domains represent the samesituation under any comparison embeddingthe one on to any member of thepermutation class of the other. Compared n this way, 39.1l nd 9JiZ1epresentthe same situation. Butunder such a comparison,931,and 21are comparedasindependent models of the theory. Comparingthem quasubstructures of 9.under the identity embeddings, they represent distinct situations.


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432 RobertRynasiewiczI also argued that Leibniz Equivalence, even in the context of generalrelativity, cannot be maintained without qualification,since we can generateexamples of genuine failures of determinismeven if the models are determinedup to isomorphism. The example I extracted fromLeibniz'ssecond argumentagainst the Newtonians posed a theory with a model 9Mlwhich persists in astatic 'ground state' up through a given time slice. We can now see whydeterminism fails for such a theory. Take any model M' in the permutationclass of 9W ifferingfrom Wby a time translation. The two models share acommon substructure3 consisting of an initial segment of the static 'groundstate' of each. Fixing first the initial condition situation 3 is imagined torepresent,it follows that 9)1and 9V.'cannot in consequence representthe sameglobal situation. Underthe inverse comparison embedding 3 into 9Nand9IW'via the identity map, 9N1nd 2' separate.As a final application, consider a case not yet mentioned. Relationists aresupposed to insist that all motion is merely the relative motion of bodies. Inparticular,a body cannot be saidto be in a state of acceleration absolutely,butonly relative to some other body or system of bodies. Stein [1977] argues,however, that without an absolute criterionof interialmotion, it is impossibleto formulate a set of deterministiclaws of particlemotion. The argument is avariant on the hole construction. Suppose 0U1s a model of whatever laws areproposed.Ifthere is no criterionof absolute acceleration, then, forany time to,we can definea diffeomorphism0 agreeing with the identityup through tobutdivergingthereafterin such a way as to preserverelative temporaland spatialseparations. If V is the set of world lines of the particles,then 4 maps V on to anew set *)* of trajectories.The resulting model 4*91Ds isomorphictoTI, andhence satisfies the same laws. Thus, the histories of the particlesup throughtime to fail to uniquely determined their future behavior. Earman [1986b]notes that the relationist can avoid the apparent failure of determinism byclaiming that M2)nd 4*92fl eally representthe same possibleworld, as urgedby LeibnizEquivalence. Do we or do we not have here a genuine failure ofdeterminism?The answer dependson what we take the identificationgrouptobe. If it is the group of diffeomorphismswhich preservespatial and temporalseparations, determinism fails. If it is the automorphism group, there is nofailure of determinism.How do we determine the appropriate dentificationgroup for a spacetimetheory? The question deserves more attention than I shall give it here,although my remarksshould make it clear that the above assignments are notadhoc.The most obvious means of identifyingspacetimepoints is in termsof acoordinatesystem or chart. So the question becomes one of how we establishadefinitecoordinate system. In the Newtonian and special relativisticcases, wecan appeal to the sorts of operational procedurescommonly cited. One relies,for example, on test particles to pick out inertial motions, and thencoordinatizes an inertial frame starting from a given point by means of an


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TheLessonsof the HoleArgument 433ensemble of comoving rigid rods and ideal clocks. If we assume that the testparticles,rods, and clocks are not themselves partsof the system modeled, it isclear that the identification group consists exactly of those transformationswhich leaves this chart adapted to the inertial and metric structure posited.This yields just the groups cited above. I take it that, in the ordinaryapplications of these theories, the assumption is realistic enough.If,however, these theories are applied cosmologically, the devicesemployedfor the spacetime identifications are themselves part of the systems to bemodeled, and the models themselves fully portrayhow each point is identified.Transformationsother than automorphismsgenerate conflicting portrayalsofhow the points are identified, and the identification group collapses to theautomorphism group. Thus, under the supposition that these means ofidentification are exhaustive, Leibnizis correct in concluding that reflectingeverythingEast to West or translating everything three feet to the East yieldsnot a differentworld, but the same world describedalternatively. Newton andClarke,though, would have denied the supposition. Even if these means areexhaustive for us. they are not forGod: Heendures forever,and is everywherepresent;and by existing always and everywhere, he constitutes duration andspace' (GeneralScholium, Book III,Principia).According to Newton, becausespace and time are 'as it were an emanent effect of God'(DeGray. n Hall andHall [1962], p. 132), the partsof space and time are identifiableto him merelyin terms of 'their mutual order and positions'.In general relativity, coordinate charts are established in physical termswith the aid of the spacetime metric. Forexample, given a point and the set oftangent vectors at that point, the exponential map yields the Reimanniannormal coordinates for any neighborhood on to which the map is one-one.Similarly, given a spatial hypersurface. the metric can be used to define asystem of Gaussian normal coordinates (synchronous coordinates). (See, e.g.Wald [1984], pp. 41-2, for details.) Hence, diffeomorphismswhich are notisometriesundo the chart established and lead to a variant identification of thespacetime points. It should be noted, however, that in certain applicationstheisometry group, and hence the identificationgroup, is treated as strictly largerthan the automorphism group, forexample, if the stress-energy tensor of thesystem modeled is dominatedby that of a background system which is taken tofix the metric. In this way, one might do, say, neutrino physics in curvedspacetime.The notion of spacetime 'immutability'popsup frequentlyin the philosophi-cal literature as a feature which distinguishes earlier spacetime theories fromgeneral relativity. One explication of 'immutability' is that in each modelthe geometry of the spatial hypersurfaces (assuming the spacetime can bepartitioned nto such) remains constant. Another is that both the topology andthe metric of spacetime is the same (up to isomorphism)from model to model.Neither of these quite captures what is intended 'in the sense that the object


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434 RobertRynasiewiczfields which characterize the structure of spacetime are not given abinitiobutare regarded as dynamic objects on a par with the other fields' (Earman[1986b], p. 235). How is this to be understood? Barring theologicalhypotheses, the structure of spacetime even in Newtonian and specialrelativistic physics cannot be entirely independent of its 'contents' since theidentification means depend on at least some of the 'contents'. Whatdistinguishes general relativity from it predecessors is that the 'contents'cannot be neatly separatedinto subsystems, one of which suffices to identifythe spacetime points while the others are allowed to vary. The spacetime ofthese earlier theories is 'immutable' in the sense that it is imagined that it isfeasible to have such an 'immutable' system of identification.Finally, let me remark on the determination of the relation of substructureautonomy for spacetime theories. In general relativity, the points of aconnected region can be identified from a given point without reference topoints outside the region by covering the region with a set of convexReimannian normal charts beginning with the initialpoint. If the region is notconnected, then, from a given point, the points in a disconnected componentcannot be identified without the identification of intermediate points of thespacetime in which the disconnected components are embedded. Since theexponential map requiresfor its definitiononly the existence of a connection,these remarks suffice for the Newtonian and special relativisticcases as well.12 CONCLUSIONIfspacetimesubstantivalism is assumed to hold that the identificationgroupofa spacetime theory is the diffeomorphismgroup of the manifold, then the holeargument demonstratesthat the priceof substantivalism is indeedhigh. This isan outlandish view to hold even apart from the hole argument. It might beclaimed that general relativity nevertheless forces substantivalism into suchan awkward position since the metric, a.k.a. the gravitational fieldpotentials,is partof the contents of spacetime,and the substantivalistmust maintain thatthe partsof spacetimeare identifiable ndependentlyof its contents. The secondhalf of this claim is a familiar relationistcomplaint in disguise:space and timeare ontologically suspect because they are not directly observable.As for thefirsthalf ofthe claim, it should be askedwhy we must imposefrom the outsideadualism of 'contents' and 'container'. If it is said that the substantival/relational controversy cannot be formulated without it, I suggest we re-examine why we think that that debate deserves to be carried over intotwentieth-century physics.

1)epartmert f PhilosophyJohnsHopkinsUniversity


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The Lessons of the Hole Argumnent 435REFERENCESALEXANDER,. G. (ed.) [ 1956]: TheLeibniz-Clarkeorrespondence.ew York:Barnes &

Noble.BARNES,. E.et al. [ 1964]: 'Observationof a Hyperonwith Strangeness Minus Three',PhysicalReviewLetters,12, pp. 204-6.BUtTTERFIELD,. [1987]: 'Substantivalism and Determinism', International Studies in thePhilosophyof Science,2, pp. 1(- 32.BuTTERFIELD,J. [1989a]: 'The Hole Truth', BritishJournalor the Philosophli f Science,40, pp. 1-28.BITTERFIELD,. [1989b]: 'AlbertEinstein Meets David Lewis',in A. Fine and M. Forbes(eds.):PSA 1988, Vol. 2, pp. 64-81.DAVIDSON,. [19791: 'The Inscrutability of Reference', SouthwesternJournal ofPhilosophy. 10, pp. 7-19. Reprinted in D. Davidson: Inquiries nto Truth andInterpretation.Oxford:OxfordUniversityPress, 1984.EARMAN,.[19 77]: 'LeibnizianSpace-Timesand LeibnizianAlgebras', n R.E. Butts andJ. Hintikka (eds.):Historicaland PhilosophicalDimensionsof Logic:Methodology ndPhilosophyof Science.Dordrecht:D. Reidel.EARMAN,. [1986a]: A Primeron Determinism.Dordrecht:I). Reidel.EARMAN,J. [1986b]: 'Why Space is Not a Substance (At Least Not to First-Degree)',PacificPhilosophicalQuarterly,67, pp. 225-44.EARMAN, . [1989]: WorldEnoughandSpace-Timte:bsoluteversus RelationalTheories fSpaceand Time.Cambridge,Mass.:MIT.EARMAN, . and NORTON, . [1987]: 'What Price Spacetime Substantivalism? The HoleStory', BritishJournalor the Philosophyof Science,38, pp. 51 5-25.HAWKING,S. W. and ELLIS,G. F. R. [1973]: The Large Scale Structure of Space-Time.Cambridge:CambridgeUniversity Press.HALL,A. R. and HALL,M. B. (eds.)[1962]: Unpublished cientificPapersofIsaacNewton.Cambridge:CambridgeUniversity Press.MAUDLIN,T. [1989]: 'The Essence of Space-Time',in A. Fine and M. Forbes(eds.):PSA1988, Vol. 2, pp. 82-91.MAUDLIN,. [1990]: 'Substances and Space-Time:What AristotleWould Have Said to

Einstein', Studies n the HistoryandPhilosophyof Science,21, pp. 531-61.NORTON,. [1987]: 'Einstein,the Hole Argument and the Reality of Space', in J.Forge(ed.):Measurenment,ealismandObjectivity.Dordrecht:Reidel.NORTON,. [1989]: 'The Hole Argument', in A. Fine and M. Forbes (eds.):PSA 1988,Vol. 2, pp. 56-64.QUINE,W. V. [1969]: 'OntologicalRelativity', in OntologicalRelativityand OtherEssays.New York: Columbia University Press.PENROSE,. [1979]: 'Singularities and Time Asymmetry', in S. W. Hawking and W.Israel (eds.): GeneralRelativity:An EinsteinCentenarySurvey. Cambridge:Cam-bridge University Press.

PIUTNAM, . [1981]: Reason,TruthandHistory.Cambridge:CambridgeUniversityPress.RYNASIEWICZ,. [1992]: 'Rings,Holes, and Substantivalism:On the ProgramofLeibnizAlgebras', Philosophyof Science,59, pp. 572-89.SACHS,R. K. and Wu, H. [1977]: GeneralRelativity or Mathematicians.New York:Springer-Verlag.


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436 Robert RynasiewiczSTACHEL, J. [198911: Einstein's Search for General Covariance, 1912-191 5', in D.Howard and J. Stachel (eds.): Einstein Studies, Vol. 1, pp. 63-100. (Einstein and tilheHistory of General Relativity. Boston: Birkhtiuser.)STEIN, H. [19 77]: 'SomePhilosophicalPrehistoryofGeneralRelativity', n J.Earman,C.Glymour,andJ.Stachel (eds.):MinnesotaStudies n thePhilosophy fScience.Vol. 8,Foundations fSpace-Time heories.Minneapolis:Universityof MinnesotaPress,pp.3-49.WALD, R. M. [1984]: General Relativity. Chicago: University of Chicago Press.

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