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Geometry and Motion

Gordon Belot

August 14, 2005

1 Introduction

I will discuss only one of the several entwined strands of the philosophy of

space and time, the question of the relation between the nature of motion andthe geometrical structure of the world.1 This topic has many of the virtues of the best philosophy of science. It is of long-standing philosophical interest andhas a rich history of connections to problems of physics. It has loomed largein discussions of space and time among contemporary philosophers of science.Furthermore, there is, I think, widespread agreement that recent insights herehave lead to a genuine deepening of our understanding.

My focus will be on motion and geometry in prerelativistic physics, begin-ning in the next section with a discussion of our topic’s historical roots, andcontinuing in the following three sections with a review of the new approachto these issues which emerged about thirty years ago and an assessment of thecurrent status of the debate. These sections will cover familiar ground, but withgreater emphasis on the structure of the space of dynamical states than has been

usual. Only at the end of the paper will I make some remarks about cognatequestions in the relativistic context. One reason for this unbalanced treatment:I contend that the correct framework for thinking about the relativistic versionsof our questions is still in the process of emergence.

2 Roots

Our story begins with Galileo’s juxtaposition of two deep insights in his Dia-logue Concerning the Two Chief World Systems:

SALV. Then a ship, when it moves over a calm sea, is one of thesemovables which courses over a surface which is tilted neither up nor

down, and if all external and accidental obstacles were removed, itwould thus be disposed to move incessantly and uniformly from animpulse once received?

1It is with regret that I set aside conventionalism, the other great theme of twentiethcentury philosophy of space and time.

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SIMP. It seems that it ought to be.

SALV. Now as to that stone which is on top of the mast; does it notmove, carried by the ship, both of them going along the circumfer-ence of the circle about its center? And consequently is there not init an ineradicable motion, all external impediments being removed?And is this motion not as fast as that of the ship?

SIMP. All of this is true, but what next?

SALV. Go on and draw the final consequence by yourself, if by your-self you have known all the premises.

SIMP. By the final conclusion you mean that the stone, moving withan indelibly impressed motion, is not going to leave the ship, but willfollow it, and finally will fall at the same place where it fell whenthe ship remained motionless. (Galilei [1967], p. 148.)

Here we have something like the principle of inertia leading to a principle of relativity—experiments with a falling body cannot distinguish between a ship inuniform motion and one at rest because the falling body shares the ship’s state of regular circular motion.2 Later, we are told that this indistinguishability holdsfor a wide class of shipboard experiments—involving, amongst other things,cavorting butterflies, hopping experimentalists, and burning incense—‘so longas the motion [of the ship] is uniform, and not fluctuating this way and that.’(Ibid., pp. 186–187.)

The principle of relativity and the principle of inertia played a significantrole in the development of mechanics—both in the solution of particular prob-lems (most notably, in Huygens’ solution to the problem of impact) and in theconstruction of overarching frameworks. But it was some time before the twoprinciples were to sit as easily side be side as they did in the Dialogue . FromDescartes’s Principles of Philosophy onwards, the principle of inertia was takento decree that bodies move uniformly and rectilinearly unless acted upon byexternal causes ([1991], §§II.37 and II.39). This would seem to require thatmotion be absolute—that there be an objective fact about the state of motionof a body, simpliciter , as well as facts about its state of motion relative to otherbodies. The principle of relativity eventually bifurcated into a stronger and aweaker form, now known as the principle of general relativity and the principle of Galileian relativity . According to the latter, no experiment or observationcan distinguish uniform motion from rest; according to the former, there is nomeans of distinguishing any state of motion from rest, or, indeed, any state of motion from any other. Taken on its own, neither principle of relativity con-

tradicts the claim that motion is absolute. But either will do so if adjoined tobackground philosophical views which imply the unintelligibility of distinctionsbetween indistinguishables.

2Neither principle is given a canonical statement by Galileo. For the principle of relativity,see Galilei ([1967], pp. 114–116, 171, 248–250, and 255). For the principle of inertia, seeDrake ([1957], pp. 113–114) and Galilei ([1954], pp. 181, 215, and 244).

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It is perhaps worth briefly reviewing the positions adopted by Descartes,Newton, and Leibniz—and their criticisms of one another—since much of the

recent literature draws upon their classic arguments.

In the Principles , Descartes identified matter with extension (Ibid., §§I.53,II.4, and II.11); and took the change of its immediate neighbors as his officialcriterion of the motion of a body. Despite its foundation upon relative motion,this account renders motion perfectly absolute—there is an objective fact of thematter whether a body’s immediate neighbors have changed.3 As noted above,Descartes took the principle of inertia as a fundamental law; but relativity,Galileian or general, played no role in his system. Indeed, his laws of impactquite clearly violate Galileian relativity (Ibid., cf. §§II.46 and II.51).

Newton thought the Cartesian account of motion incoherent.4 In the un-published De Gravitatione , he observed that ‘there are no bodies in the worldwhose relative positions remain unchanged with the passage of time,’ hence, onDescartes’s account, ‘there is no basis on which we can at the present pick out aplace which was in the past, or say that such a place is any longer discoverablein nature.’ (Hall and Hall [1962], p. 130.) From this it follows that Descartesis not entitled to the notion of velocity:

Now as it is impossible to pick out the place in which a motionbegan (that is, the beginning of the space passed over). . . so thespace passed over, having no beginning, can have no length; andhence, since velocity depends upon the distance passed over in agiven time, it follows that a moving body can have no velocity. . . .

The alternative preferred by Newton was, of course, ‘that the definition of places,

and hence of local motion, be referred to some motionless thing such as extensionalone or space in so far as it is seen to be distinct from bodies.’ (Ibid., p. 131.)

Just such an account was developed in the Philosophiae Naturalis Prin-cipia Mathematica. There, space consists of parts which maintain their identityover time, allowing the absolute motion of a body to identified with its mo-tion relative to space, thus providing the objective notions of uniform motionand acceleration required by Newton’s first and second laws of motion. In theScholium to the definitions, Newton claimed empirical content for this notion of absolute motion: ‘absolute and relative rest and motion are distinguished fromone another by their properties, causes, and effects’ (Newton [1999], p. 411);‘The effects distinguishing absolute motion from relative motion are the forcesof receding from the axis of circular motion’ (Ibid., p. 412). This claim isillustrated via the bucket and globes examples (Ibid., pp. 413–414).

At the same time, Newton was, of course, well aware that his system alsoplaced limitations on our ability to distinguish motion from rest. The principle

3This is over-stated; see Garber ([1992], pp. 175–81).4As did Leibniz and Euler, for interesting reasons, which are, however, tangential to our

theme. See §13 of ‘On Nature Itself’ (in Ariew and Garber [1989]) and §§VIII and IX of ‘Reflections on Space and Time’ (Euler [1748]).

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of Galileian relativity appears as the fifth corollary to the laws of motion; ‘ When bodies are enclosed in a given space, their motions in relation to one another

are the same whether the space is at rest or whether it is moving uniformly straight forward without circular motion.’ (Ibid., p. 423.) Here we see our

tension quite clearly: Newton’s laws demand a notion of absolute motion whileat the same time implying that there exist states of absolute motion which areindistinguishable one from another.5

The most famous criticisms of this framework are due to Leibniz. Early in hiscorrespondence with Clarke (1715–16), Leibniz asserted that it was metaphysicalrather than mathematical principles which were truly opposed to materialismand impiety, and claimed to have discovered an axiom of metaphysics from whichall of natural philosophy followed—the principle of sufficient reason, accordingto which ‘nothing happens without a reason why it should be so, rather thanotherwise’ (L.II.1).6

Clarke objected that this principle was false unless amended by the provisothat ‘this sufficient reason is oft-times no other, than the mere will of God’,since unless so-amended no reason could be given ‘why this particular system of matter, should be created in one particular place, and that in another particularplace; when, (all place being absolutely indifferent to all matter,) it would havebeen exactly the same thing vice versa ...’ (C.II.1).

Leibniz retorted that Clarke presumed ‘space to be something in itself, be-sides the order of bodies among themselves’; Leibniz’s view was that ‘space isnothing else, but that order or relation [of bodies]; and is nothing at all withoutbodies, but the possibility of placing them;’ from which it follows that ‘thosetwo states, the one such as it is now, the other supposed to be the quite contraryway, would not at all differ from one another.’ (L.III.5.)

Clarke introduced a new example, having rejected Leibniz’s reading of the

previous one:

If space were nothing but the order of things coexisting; it wouldfollow, that if God should remove in a straight line the whole materialworld entire, with any swiftness whatsoever; yet it would still alwayscontinue in the same place: and that nothing would receive any shockupon the most sudden stopping of that motion.7

5Note that the sixth corollary also expresses a relativity principle, closely related to theprinciple of equivalence: ‘If bodies are moving in any way whatsoever with respect to one

another and are urged by equal accelerative forces along parallel lines, they will all continue

to move with respect to one another as if they were not acted on by those forces. ’6Here ‘L.II.1’ refers to the first paragraph of Leibniz’s second letter to Clarke; I quote from

Alexander ([1956]).7C.III.4. The idea was not entirely original. In 1277 the Bishop of Paris condemned 219

propositions, ‘excommunicating all those who shall have taught some or all of said errors, orshall have presumed to defend or support them in any way whatever, and also those who listento these things, unless it be disclosed that within seven days they have come forward to us orto the Chancellor of Paris.. . .’ Among the propositions condemned were: ‘That the only wisemen of the world are philosophers’; ‘That by nutrition a man can become another numericallyand individually’; and ‘That God could not move the heavens with rectilinear motion; and

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Leibniz remained unconvinced: ‘To say that God can cause the whole uni-verse to move forward in a right line, or in any other line, without making

otherwise any alteration in it; is another chimerical supposition. For, two statesindiscernible from each other, are the same state; and consequently, ’tis a changewithout a change.’ (L.IV.13.) Clarke reiterated his argument, appealing thistime to Newton’s authority:

Two places, though exactly alike, are not the same place. Nor is themotion or rest of the universe, the same state; any more than themotion or rest of a ship, is the same state, because a man shut upin a cabin cannot perceive whether the ship sails or not, so long asit moves uniformly. The motion of the ship, though the man per-ceives it not, is a real and different state, and has real and differenteffects; and, upon a sudden stop, it would have other real effects;and so likewise would an indiscernible motion of the universe. . . . Itis largely insisted upon by Sir Isaac Newton in his Mathematical Principles , (Definit. 8.) where, from the considerations of the prop-erties, causes, and effects of motion, he shows the difference betweenreal motion, or a body’s being carried from one part of space to an-other; and relative motion, which is merely a change of the order orsituation of bodies with respect to each other. (C.IV.13.)

Leibniz’s reply is both fascinating and obscure. On the one hand, ‘motiondoes not indeed depend upon being observed; but it does depend upon beingpossible to be observed. There is no motion, when there is no change that canbe observed.’ (L.V.52.) This is precisely the sort of verificationist doubt aboutthe empirical content of Newton’s absolute space and absolute motion whichhas driven the philosophical debate. On the other hand, while Leibniz finds

‘nothing in the Eighth Definition of the Mathematical Principles of Nature , norin the Scholium belonging to it, that proves, or can prove, the reality of spacein itself,’ he does grant that there is indeed ‘a difference between an absolutetrue motion of a body, and a mere relative change of its situation with respectto another body. For when the immediate cause of the change is in the body,that body is truly in motion’ (L.V.53).

Each time that Clarke put forward a situation where, putativley, only appealto the will of God could settle the choice between two indifferent alternatives,Leibniz appealed to the principle of sufficient reason (or to its close relation, theprinciple of the identity of indiscernibles) to show that Clarke had miscountedpossibilities—‘To suppose two things indiscernible, is to suppose the same thingunder two names.’8 Clarke and Leibniz agreed about this much: if Newton’s

the reason is that a vacuum would remain.’ There has been considerable dispute amonghistorians concerning the import of this last for the development of medieval thought. In anycase, beginning in the fourteenth century it was observed (by, e.g., Buridan and Oresme) thatthe denial of this proposition implied the existence of absolute motion. See Grant ([1974], pp.45–50, [1979], pp. 230 and 243).

8(L.IV.6.) See Belot ([2000]) for a discussion of the advantages and limitations of thisstrategy for the interpretation of physical theories admitting symmetries.

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absolute space exists, then there must be many pairs of distinct but indiscerniblesituations; the one in which the centre of the world is here , and the one in which

it is there ; the one in which the world is at rest, the one in which the world movesthus quickly in that direction. It is clear enough that if one follows Leibniz inrejecting distinctions between indistinguishables, then one must also follow himin rejecting absolute space.

What Leibniz was able to offer as an alternative to Newton’s system isconsiderably less clear.9 Many of the themes touched upon in the dispute withClarke had already been more systematically elaborated in earlier writings of Leibniz, especially the second, unpublished, part of the Specimen Dynamicum (1695). There we find that ‘force is something absolutely real in substances. . . while space, time, and motion are, to a certain extent, beings of reason, andare true or real, not per se , but only to the extent that they involve eitherthe divine attributes. . . or the force in created substances.’ (Ariew and Garber

[1989], p. 130.) We may, perhaps must, accept one or another hypothesis aboutthe true motions—‘we speak as the situation requires, in accordance with themore appropriate and simpler explanation of the phenomena.’ (Ibid., p. 131.)

However, in the absence of dynamical considerations, ‘we must hold thathowever many bodies might be in motion, one cannot infer from the phenomenawhich of them really has absolute and determinate motion or rest.’ Leibnizsketches his derivation of this principle of general relativity from the principleof inertia and assumptions deriving from his metaphysics: ‘since only forceand the nisus [striving] arising from it exist at any moment . . . and since everynisus tends in a straight line, it follows that all motion is either rectilinear or composed of rectilinear motions ’ (Ibid., p. 135);

I cannot agree with certain philosophical opinions of certain impor-

tant mathematicians, who, beyond the fact that they admit emptyspace and don’t seem to shrink from attraction, also take motion tobe an absolute thing, and strive to prove this from rotation and thecentrifugal force which arises from it. But since rotation also arisesonly from a combination of rectilinear motions, it follows that if theequivalence of hypotheses is preserved in rectilinear motions . . . thenit will also be preserved in curvilinear motions.10

The next paragraph—which is, perhaps significantly, the last of this unpub-lished manuscript—opens with the emphatic statement that ‘From what was

9For sympathetic and subtle readings of Leibniz’s dynamics, see Bertoloni Meli ([1993])and Garber ([1995]).

10(Ibid ., pp. 136–37.) The equivalence of hypotheses appears as a theorem in the Dynamica ,

where the proof turns upon these same considerations (see §II of the Appendix of Stein [1977]for a translation).

Late in his life, Huygens, too, came to reject rotation as a criterion of motion, and ex-pressed to Leibniz his hope that Newton would retract the claim in the next edition of thePrincipia (see Bernstein [1984] and Stein [1977] for very different accounts of Huygens on theprinciple of general relativity; see §§I and III of Stein’s Appendix for translations of relevantcorrespondence and notes of Huygens).

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said one can also understand that motion common to many bodies does not change their actions with respect to one another , since the speed with which

they approach one another, and therefore the force of impact by which they actupon one another, is not altered.’ Leibniz then restates the position, this timewith a surprising qualification: ‘it is certain that if some people were carriedon a large boat (enclosed, if you please, or, at least, set up in such a way thatthings outside of the boat could not be seen by the passengers), then, howevergreat the speed of the boat might be, as long as it moved peacefully and evenly ,they would have no criterion of discerning . . . whether the boat was at rest or inmotion. . . ’ (emphasis added). This hedge seems to amount to something like aretreat from general relativity to Galileian relativity.

3 Modern Times

It follows from the symmetries of Newtonian mechanics that boosted and shifted situations are indistinguishable; as Leibniz and Clarke saw quite clearly, this implies that the notions of absolute position and absolute velocity are without empirical content; surely, then, we must regard Newton’s postulation of absolute space as mere metaphysics, and denounce it as a blemish on mechanics. Thestories that we tell each other about our discipline have it that stern empiricistanalyses along these lines dominated philosophical discussions of the foundationsof Newtonian mechanics in the late nineteenth and early twentieth centuries; andthat the tide began to turn only with the publications of Stein’s ([1967]) essay‘Newtonian Space-Time’.11

One of Stein’s goals is to evaluate Newton’s warrant for the postulation of absolute space. Stein observes that it is helpful to think of Newton as havingin fact postulated Newtonian spacetime : a stack of three dimensional Euclideanspaces, together with a natural way of identifying points of space which occurat distinct times (absolute space). The fibration of Newtonian spacetime by theworldlines of the points of absolute space allows us to define the required notionof absolute acceleration—but it also supports an empirically otiose notion of absolute velocity.

This is distasteful. We would prefer to have absolute acceleration—and New-ton’s laws—without absolute velocity. And from our present perspective we cansee how this is possible. Suppose that we set out to calculate the acceleration of a body relative to absolute space and its acceleration relative to some inertiallymoving frame of reference. The two relative accelerations will coincide—sinceour two frames have vanishing relative acceleration. So absolute acceleration

is just acceleration relative to any inertial frame. We want the affine struc-ture of Newtonian spacetime—the structure which tells us which worldlines are‘straight’ in the sense of corresponding to inertial motion—without a privileged

11See Mach ([1960], pp. 271–305), and Reichenbach ([1924]) for influential empiricist com-mentaries; see Stein ([1967]) and Earman ([1970b]) for further references.

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frame of absolute rest. So we construct neo-Newtonian spacetime : we take astack of Euclidean geometries carrying a time function, and introduce an affine

structure isomorphic to that of Newtonian spacetime.12

Stein concludes that there is indeed a sense in which Newton erred in postu-lating absolute space—since this meant recognizing a notion of absolute velocitywhich was both empirically otiose and avoidable. At the same time, Stein urgesus to recognize that there is a sense in which Newton was entirely justified, giventhe vast margin by which his great theoretical proposal overshadowed those of his contemporaries.

In making this argument, Stein gives free play to two themes. One de-rives from the insight that many philosophical issues—even those with quiteancient roots—are best addressed by studying the possible structures of space-time , rather than the possible structures of space and time . The other is thatphilosophy of science ought to adopt what might be called a realistic orienta-

tion —as opposed to an intrumentalistic or phenomenalistic one—in its endeav-our to elucidate the content of physical theories. This latter might be explicatedas follows: physicists put forward theories as hypotheses concerning the struc-ture and constitution of the world; often, these admit of conceptual clarificationand ontological precisification at the hands of philosophers of physics; these lat-ter ought to treat entire theories as attempting to describe a possible structureof the world (rather, than, say, treating theories as contentless instruments of prediction, or attempting to locate content piecemeal in individual axioms ordefinitions); we ought to give credence to these accounts of the structure of the world to the extent that their relevant theoretical virtues outstrip those of their competitors; it is the task of general philosophy of science, rather than of philosophy of physics proper, to referee this final inference.13

Stein’s themes and conclusions have been immensely influential. They were

have been taken up, amplified, and generalized—first of all in a pair of funda-mental papers of Earman ([1970a], [1970b]), then more widely (see especiallySklar [1974], Friedman [1983], and Earman [1989]). It is safe to say, I think, thatthey are today universally agreed to be the source of a considerable deepeningof our understanding of the nature of space, time, and motion.

Much of the succeeding literature has been taken up with two ancillary ques-

12Much is gained in moving from Newtonian to neo-Newtonian spacetime: we jettisonabsolute velocity, and adopt a spatiotemporal geometry invariant under all of the symmetriesof equations of motions of classical mechanics (including bo osts). Notice, however, that partof the critics’ point was that these symmetries are themselves problematic—they imply thatthe notion of physical possibility normally associated with Newton’s mechanics far outrunsdistinguishability.

13See Stein ([1967], pp. 190 and 197) and Earman ([1970b] pp. 299 and 305). Calling thisapproach realistic is perhaps misleading, since it is shared by the most influential anti-realist

about physics. Van Fraassen has divided his labor within philosophy of science between elabo-rating the content of theories and arguing that, for reasons of principle, the balance of relevanttheoretical virtues can never compel belief (see his [1991] and [1980] respectively). (I maintainthat the distinction that he draws between Earman’s and Friedman’s discussion of substan-tivalism in §II of his [1995] is ill-considered—the matter-space and observable-unobservabledistinctions simply do not line up so neatly.)

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tions, which happen to have the same verbal form—Is relationalism tenable? The first version of this question concerns the nature of motion: the ultimate

fruitlessness of the relationalist ambitions of Leibniz and Huygens notwithstand-ing, can motion be understood as being relative? In its second version, thequestion concerns ontology: in the context of either spatial or spatiotemporalgeometry, is it possible to resurrect Leibniz’s deflationist ontological posturetowards geometry? These questions will be the topic of the next two sections.

4 Relationalism about Motion

Relationalism about motion is the thesis that all motion is relative motion.This doctrine appears to be inconsistent with what Newtonian physics teachesus about physical possibility. Let us stipulate that two massive bodies are ini-tially at rest a certain distance from one another. Is this information sufficient

to determine the future relative distances and velocities between the bodies?No, for their future behavior also depends on their state of rotation; the bal-ance between the gravitational forces and the ‘inertial forces’ due to rotationdetermine whether the objects move in a collapsing, stable, or expanding orbit.Inertial effects matter to dynamics but do not supervene on initial relative dis-tances and velocities, so these relational quantities do not form a predictivelyclosed set.

This point has often been put in the following terms: the existence of in-ertial effects shows that spacetime must have some non-trivial affine structurewhich ‘connects’ slices of simultaneity and which provides a sense of absolutemotion.14 The chief hope for relationalism would appear to be the programwhich many readers find in Mach ([1960]): to attempt to reduce inertial effectsto interactions between bodies, in such a way that the averaged-out mass of theuniverse determines what is in essence an inertial frame. The consensus amongphilosophers seems to be that a number of substantial—if not insuperable—difficulties stand in the way of this strategy: the complete lack of any accountof the mechanism of these forces (Euler [1748], §XII; Sklar [1972], p. 307, [1974],p. 201, [2000], pp. 46–7); the difficulty in defining such a frame (Stein [1977],p. 17); the daunting problem of reducing the affine properties of spacetime torelations between material particles (Friedman [1983], p. 67). The verdict seemsto be: relationalism about motion is not quite dead as a conceptual possibil-ity; but it remains far from clear how this program could be carried out; theexistence of inertial effects strongly suggests the necessity of an affine structurefor spacetime, and it is difficult to see such structure could reduce to strictlyrelational facts.15

I propose to shift focus away from this daunting and poorly defined rela-tionalist program, and to consider in its stead a strain of relationalism about

14See Stein ([1967], p. 187), Earman ([1970b], p. 313), Earman and Friedman ([1973], pp.343–4), Sklar ([1976], p. 15), and Friedman ([1983], p. 226).

15For a more cautious assessment, see Earman ([1989], p. 108). See also Wilson ([1993]) fora related and interesting form of relationalism.

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motion which is rather more successful but which has received less attentionfrom philosophers. The idea is to focus on the variables employed in a dynam-

ical theory—to examine the structure of phase space rather than the structureof spacetime.16 In this framework, it is easy to give a sharp criterion for thesuccess of relationalism: the formulation of an adequate theory of motion whoseonly dynamical variables are the relative positions and velocities of bodies.17

The argument sketched above shows that this is impossible if we require ourtheory to capture the full content of Newtonian mechanics. But notice that ouruniverse appears to have vanishing angular momentum (Barrow, Luszkiewicz,and Sonoda [1985]).18 So the sector of Newton’s theory in which the total angu-lar momentum of the universe is constrained to vanish is empirically adequate(in the domain of classical mechanics). And there is a straightforward recipewhich shows how to capture the relational content of this sector in terms of relational variables. For any relational initial data (i.e., relative distances and

velocities), find a set of Newtonian initial data which correspond to the givenrelational quantities and according to which the system has vanishing angularmomentum. Now evolve the Newtonian initial data, reading off the values of the relational variables at each instant. For a given set of relational initial data,the choice of Newtonian initial data is determined up to a rigid motion and aboost—that is, up to the action of an element of the Galilei group. The Galilei-invariance of Newtonian physics guarantees that this freedom does not affectthe evolution of the relational variables.

Now, this is arrant knavery: a ‘cheap instrumentalist rip-off’ of Newtoniantheory—rather than a genuine competitor—if there ever was one.19 What makesthis ploy interesting is that it can be given a number of elegant, autonomousformulations—each a fitting competitor to the standard formulation, rather thana parasite.20 These seem to have been discovered independently a number of

times in the course of the twentieth century; Barbour has done more than anyoneto clarify their philosophical import.21 I will sketch an approach that dependson the removal of symmetries from Hamiltonian systems, which I believe allows

16Within the philosophical literature this approach derives from Barbour ([1982]). Brownalso makes a case for shifting attention towards dynamics and away from the geometry of spacetime (see, e.g., Brown [1993] and Brown and Sypel [1995]).

17As Barbour ([1999]) notes, this form of relationalism, at most hinted at by Mach ([1960],pp. 283–4), is suggested quite clearly by Poincare ([1952], Chapter VII).

18Frank Arntzenius has pointed out to me that Newton’s theory ought to lead us to expectthat the rotation of the universe should be undetectable, since as N → ∞ , the measure of the set random distributions of N particles with discernible angular momentum goes to zero(angular momentum requires correlations between velocities).

19The phrase is Earman’s ([1989], p. 127).20In the construction described below, the relational theory results from factoring out a

symmetry from the standard formulation. This no more renders the resulting relational theory

parasitic on the Newtonian theory than the standard construction by identifications makes atorus parasitic on R

2.21For an interesting range of formulations, see Zanstra ([1924]), Barbour and Bertotti

([1982]), Iwai ([1987]), Lynden-Bell ([1995]), and Gergely ([2000]); see Barbour and Pfister([1995]) for further references; see Barbour ([1982], [1999]) for philosophical discussion. Theapproach described below is closest to that of Iwai. The criticisms of this theory in Earmanand Friedman ([1973], pp. 341–4) and Earman ([1989], p. 95 n. 5) are far from conclusive.

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us to clarify the status of both relationalism about motion and relationalism asan ontological posture.22

Recall that in the Hamiltonian approach to classical mechanics our primaryobject of study is the phase space of the theory—the space of dynamically pos-sible states of some physical system. This space carries a geometrical structure,which requires only the choice of a Hamiltonian function on phase space (en-coding information about the total energy of each possible state) in order todetermine a unique dynamically possible trajectory through each state.23 Wewill be interested in the class of simple mechanical systems . Such a systemis specified by selecting a configuration space (typically taken to represent thepossible dispositions of some set of particles or fields relative to an inertialframe) carrying a Riemannian metric (which determines the kinetic energy byattributing magnitudes to vectors representing velocities) and a function (whichdetermines the potential energy). We take the cotangent bundle of the configu-ration space as our phase space—so that a state of the system is represented bya pair consisting of a point in the configuration space and a (co)vector encodingthe rate of change of the configuration variables at that point. The cotangentbundle carries a canonical geometrical structure, which determines the dynam-ics once we stipulate that the Hamiltonian function on our phase space is justthe sum of the kinetic and potential energies.

In the case of N gravitating point particles moving in a three dimensional Eu-clidean space, our configuration space is 3N dimensional space (since it requiresthree coordinates to specify the position of each particle in physical space);this space carries a flat metric which it inherits from the metric on physicalspace; note that the gravitational potential energy depends only on the rela-tive distances between particles. The resulting simple mechanical system hasa 6N dimensional phase space (three position coordinates and three velocity

coordinates for each particle).We seek a theory of gravitating point particles formulated as a simple me-

chanical system whose configuration space is the relative configuration space parameterized by the relative distances between the particles.24 This is a spaceof dimension 3N − 6; the associated relative phase space (the space of possi-ble relative distances and relative velocities) is of dimension 6N − 12. Since theNewtonian gravitational potential energy depends only on the relative distances

22Although my treatment will be superficial, there is interesting mathematics in play here;see, e.g., Guillemin and Sternberg ([1984]) or Marsden ([1992]). Note that a number of other topics in philosophy of physics—e.g., those surrounding indistinguishable particles andgauge invariance—can also be thought of as turning upon the removal of symmetries fromHamiltonian systems.

23Typically a phase space carries a symplectic structure—a closed, non-degenerate two-form. The dynamics are determined by solving for the vector field whose contraction with the

symplectic form is equal to the differential of the Hamiltonian.A symplectic structure endows the space of functions on the manifold with a Poisson struc-

ture (i.e., the set of functions becomes a Lie algebra obeying Leibniz’s rule). The dynamicscan be expressed in terms of the Poisson brackets that arbitrary functions have with theHamiltonian of the theory.

24For N > 4, the relative distances will not independent, since the number of relativedistances will be much larger than the dimensionality of the relative configuration space.

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between particles, we can view this function as living on the relative configu-ration space. It turns out to be possible to equip the relative configuration

space with a metric in such a way that the resulting Hamiltonian theory on therelative phase space determines a dynamical evolution for the relative distancesand velocities which coincides with that of the knavery sketched above.25

There is a second way of approaching this construction. The relative con-figuration space is the result of identifying points in the standard configurationspace that correspond to particle configurations that differ by a rigid motionof R3—we factor out the action of the group of Euclidean isometries on thestandard configuration space.26 Now the Euclidean group also acts on the stan-dard phase space —since we know how it acts on points of configuration space,we know how it acts on curves in configuration space, and hence how it actson tangent vectors and covectors. This action leaves invariant the kinetic andpotential energy, and the geometric structure of the phase space. In order toconstruct our theory, we first restrict our attention to a subspace of the stan-dard phase space by fixing an arbitrary value for the linear momentum of thesystem (killing three dimensions) and setting the angular momentum equal tozero (killing another three); we then identify points of this subspace which arerelated by the action of a Euclidean rigid motion on the system (killing anothersix dimensions, since the group is six dimensional); projecting down the geomet-rical structure and the Hamiltonian to the resulting quotient space yields thetheory of the previous paragraph.27 The invariance of the standard theory un-der boosts ensures that the arbitrariness involved in fixing the linear momentummakes no difference to the final theory. On the other hand, fixing a value forthe angular momentum and identifying those points related by elements of thesubgroup of the Euclidean group which fixes the resulting subspace, or (equiv-alently) fixing a non-zero value for the magnitude of the angular momentum

and factoring out the action of the full group, results in a phase space which isof no interest to the relationalist about motion, since it includes two variables(parameterizing a sphere) in excess of the relative distances and velocities (see§5.3 below for an interpretation of this theory).

The lesson of all of this is that while the relative distances and velocities donot form a predictively closed set within Newtonian dynamics, they do so withinthe empirically adequate rotation-free sector of the theory. Relationalists maywant to claim this as a victory. Some will want to add that they have all alonghad their suspicions about the cogency of the idea that the entire universe could rotate.28

There is not, in fact, much to choose between the standard theory and the

25The metric in question is non-flat (Littlejohn and Reinsch [1997], §IV.C).26Here and throughout I take the group of isometries to include improper isometries. The

quotient spaces constructed in this section and in §5.3 below are stratified manifolds but notmanifolds, since the group actions in question fail to be free (see Cushman and Bates [1997],Appendix B.5).

27Notice that the relative phase space is twelve dimensions smaller than the standard phasespace—so merely factoring out the action of the (six dimensional) Euclidean group on thestandard phase space will not yield our relationalist theory. See §5.3 below.

28I believe that Barbour ([1982], pp. 257–61) is sympathetic to this line of thought.

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relationalist alternative, even if we resort to judgments of elegance. The kineticenergy of the relationalist theory, being encoded by a non-flat metric, is perhaps

less pleasing; on the other hand, the relationalist theory manages without thedynamically irrelevant degrees of freedom associated with the symmetries of thestandard theory (position and velocity of the center of mass of the system andorientation about that center of mass).29

Relationalists about motion who would like to be able to remain true to thefull content of the standard theory have no option but to recognize dynamicalvariables beyond the relative distances and velocities. Sklar has suggested thatrelationalists admit a primitive notion of absolute acceleration ([1974], §III.F,[1976], §IV.F). The most promising version of this proposal would assign (vectoror scalar) quantities—closely related to the usual absolute acceleration—to eachparticle as primitive dynamical properties, not to be thought of as having theirfoundation in the geometry of spacetime.30

In fact, we need only add three dynamical variables—corresponding to thetotal angular momentum of the standard theory—to the our relative phasespace: the standard phase space is only twelve dimensions larger than the rel-ative phase space, and nine of the twelve additional variables describe dynam-ically superfluous quantities. It is possible, as we will see in §5.3, to constructsuch a theory living on a 3N − 9 dimensional phase space which is naturallyviewed as an extension of the relative phase space and which captures the invari-ant content of Newtonian dynamics. It remains to be seen, however, whetherit is possible to plausibly interpret this theory as vindicating a variant of tradi-tional relationalism about motion.

5 Relationalism about Ontology

When we turn to the second sense of relationalism—a deflationary attitude to-wards the ontological aspect of physical geometry—we meet some familiar dif-ficulties. It is never easy to demarcate opposing positions concerning ontologyin such a way that the boundaries are clear, the stakes non-trivial, and the con-test interesting. In addition, our dispute—like many in metaphysics—has hoaryroots, to which it is difficult to remain true while updating the topic. Nonethe-less, we will want to evaluate ontological theses concerning spatiotemporal aswell as spatial geometry.

29What is the connection between this brand of relationalism, and the sort more often con-sidered by philosophers? How does this theory account for inertial forces in terms of relationalvariables? As in any simple mechanical system, here the inertial effects derive from the metricon the configuration space, which is inherited from the metric on the standard configuration

space, which is induced by the Euclidean metric on physical space. We can talk about thestructure of the reduced configuration space, and its relation to the structure of the physicalspace in which the particles move, without referring to an affine structure on spacetime. Thisis one of the reasons why questions about classical mechanics are sometimes better posed interms of the structure of Hamiltonian systems, rather than in terms of ‘spacetime theories’.

30See Friedman ([1983], pp. 232–6) and Sklar ([1990], p. 70) for the proposal; see Huggett([1999]) for the details. See also Poincare ([1952], pp. 121–2).

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The first discussions of relationalism about spacetime tended to one or theother of two excesses. On the one hand, Stein admonishes that

If the distinction between inertial frames of reference and thosewhich are not inertial is a distinction that has a real applicationto the world; . . . and if this structure can legitimately be regardedas an explication of Newton’s “absolute space and time”; then thequestion whether, in addition to characterizing the world in justthe indicated sense, this structure of space-time also “really exists,”surely seems to be supererogatory.31

On the other hand, we find Earman and Friedman claiming that in arguing(against an array of conventionalists) that the real content of Newton’s laws isan empirical claim concerning the geometrical structure of spacetime they arethereby supporting a ‘realist interpretation of space-time theories’ ([1973], pp.

329 and 358). Their paper has long been read as an argument in favor of what isnow called substantivalism (i.e., anti-relationalism) about spacetime—although,in fact, it only makes the case that there are many good reasons to postulate aneo-Newtonian geometry for spacetime.32

What is missed in either case is the possibility that we can agree about geo-metrical structure, even while disagreeing about the ontology which instantiatesit.33 Indeed, I would suggest that it is best to think of the early modern debateas realizing just this possibility. Newton and Leibniz certainly agreed that spaceis three dimensional and Euclidean. They even, I maintain, agreed that spaceis—in some sense—a real thing.34 What they disagreed about was the natureof the instantiation of this structure in the world. Newton claimed that thereexisted a physical three dimensional Euclidean continuum, whose parts main-tained their identities and respective geometrical relations over time. Leibnizpostulated a permanently existing network of possibilities for the placement of bodies, but explicitly denied (L.V.47) that this ontology provided a means of identifying parts of space over time independently of the disposition of matter.

I will try to exploit this reading of the classic debate, according to whichrelationalism and its denial (substantivalism) are theses concerning the onto-

31Stein ([1967], p. 193). Malament ([1976], §III) and DiSalle ([1994]) further develop Stein’stheme.

32Earman and Friedman mention realism only in their abstract and conclusion. See Sklar([1976], p. 9) for an early, savvy, example of the standard reading. See Butterfield ([1989], p.2) for a more recent example of an author who sees a tight connection between realism andsubstantivalism.

33This point is made very elegantly in Sklar ([1990]), and something like it drives his earliercriticisms of hasty inferences to substantivalism from inertial effects ([1974], §III.F, [1976]).See also Horwich ([1978], §6).

34This is, of course, less clear in Leibniz’s case than in Newton’s. Here it may suffice to notethat around the time of his correspondence with Clarke, Leibniz wrote to Conti as follows:‘Space is something [quelque chose ]; but, like time . . . [it] is a general order of things. Space isthe order of co-existents, and time is the order of successive existents. They are true things,but ideal, like numbers.’ (Dutens [1768], p. 446.) Alexander’s ([1956], p. 185) translationomits the first sentence of this passage.

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logical instantiation of a given physical geometry.35 I contend that under thisconstrual the debate about the nature of geometry has clear content, non-trivial

connections with questions about the nature of motion and relativity principles,an undeniable relation to the early modern debate, and a pleasing neutralitybetween the spatial and spatiotemporal cases. The first subsection below eluci-dates and defends this approach to the substantival-relational debate; the nexttwo evaluate the debate’s status for the space and spacetime of classical me-chanics; the final subsection concerns classical field theories, which have oftenbeen taken to present an especial difficulty for relationalists.

The most influential view—developed by Friedman ([1983]) and Earman([1989]) and endorsed, I believe, by the majority of commentators—accordssubstantivalism a healthy advantage over relationalism. My account of thesubstantival-relational debate is influenced by, and draws upon, the relationalistresponses to this consensus, due to Barbour ([1982]), Manders ([1982]), Mundy([1983]), Saunders ([2000]), and Teller ([1991]).36

I will not discuss attempts to open up a third way between relationalismand substantivalism. The most promising is probably Earman’s suggestion,inspired by work of Geroch ([1972]) and others, that the formulation of ourtheories in terms of algebras of observables rather than in terms of tensorsdefined on manifolds might clarify interpretative issues ([1977], [1989], §9.9).37

This is, in the first instance, a proposal to replace one sort of formulation of our theories with a mathematically equivalent alternative. As such it is, asobserved by Earman ([1977], pp. 110–111) and as emphasized by Rynasiewicz([1992]), a strictly technical proposal which cannot resolve any interpretativedifficulties whatsoever. The problem of specifying the interpretation of thetheory suggested by this particular reformulation remains, I think, open. Itmight be worth noting, however, that the strategy of doing geometry in terms

of functions rather than points has in fact proven tremendously fruitful withinpure mathematics and mathematical physics in recent years.

5.1 Counting Geometrical Possibilities

Let us begin with the case of three dimensional Euclidean space. Suppose thatwe specify the relative distances between the four vertices of some irregulartetrahedron. How many ways are there to embed these four points in Euclidean

35See Earman ([1989], p. 12) for a related characterization of substantivalism. This ap-proach is only distantly related to other ways of distinguishing between substantivalism andrelationalism: Friedman ([1983], §VI.1) and Field ([1985], p. 335) take a Quinean approach,under which substantivalists quantify over the larger set of variables; Maudlin ([1993], p. 184)and Norton ([1992], §§5.12.1 and 5.12.2) characterize relationalists as unwilling to countenancethe existence of geometrical degrees of freedom in addition to material degrees of freedom.

36I would classify some other authors—such as Hoefer ([1996]) and, to a lesser extent,DiSalle ([1994])—as crypto-relationalists.

37Two other ideas often come up in this regard: Sklar’s ([1974], §III.F) proposal thatrelationalists help themselves to a primitive notion of absolute acceleration, which I count asa variety of relationalism; and Teller’s ([1987]) suggestion that we think of spacetime pointsas properties rather than particulars, which Teller himself has abandoned under pressure fromthe hole argument ([1991], p. 395).

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space? I make two stipulations, neither of which is uncontroversial. On theone hand, I require substantivalists to maintain that there are a large number

of such embeddings: place the four points as you like; you generate a distinctpossible embedding by acting upon this first one by any Euclidean isometry. Onthe other hand, I require relationalists to maintain that there is a single suchembedding.38 These stipulations imply that relationalists will side with Leibniz,while substantivalists will side with Clarke, on the question of whether it wouldhave been exactly the same if things had all been shifted somewhat. Let me,then, call these two ways of counting possibilities Clarke’s rule and Leibniz’s (other) rule .

Relationalists and substantivalists who adhere to these rules find their dis-agreement about the instantiation of a given geometrical structure reflected ina disagreement concerning the set of physical possibilities. We will see in §§5.2and 5.3 below that this latter disagreement has interesting implications for theform and content of dynamics.

Unfortunately, many supposed substantivalists reject Clarke’s rule.39 I thinkit fair to say that most of them are reacting to the hole argument, which is sup-posed to show that adopting an analog of Clarke’s rule in the context of generalrelativity leads to a (weak) form of indeterminism.40 So perhaps one has goodreason to move away from this way of counting possibilities in the general rel-ativistic case, and perhaps consistency forces one to back away in the nonrela-tivistic case as well.41 One way of accomplishing this is to endorse Leibniz’s rulewhile calling oneself a substantivalist (and assigning relationalism some other,typically quite barren, demesne). I prefer to maintain the traditional links be-tween ontological stances towards geometry, ways of counting possibilities, anddynamical considerations. Those who ob ject to my stipulation will read thisdiscussion as an adjudication of a dispute between two sorts of substantivalist.

The association between relationalism and Leibniz’s rule is also controver-sial. I have been proceeding as if relationalists were entitled to speak of pointsstanding determinate geometric relations which fix, up to a rigid motion, theembedding of the points in Euclidean space. But standard axiomatizations of Euclidean geometry take betweenness and congruence as their only non-logicalprimitives. So it might be thought that the expressive resources of relationalistsare restricted to just these two relations. It would follow that relationalists whowould like to be able to speak of three points which determine some particular

38Perhaps there is conceptual space for another kind of relationalist, who identifies con-figurations only if they are related by orientation preserving isometries; such relationalistswould recognize exactly two possible embeddings for every (nonplanar) set of relative dis-tances. There is no technical obstacle to developing this view—although one loses the link torelationalism about motion developed in §5.2 below. On the much discussed question of theconceptual cogency of this brand of relationalism, see Earman ([1989], Chapter 7), Brighouse

([1999]), and Hoefer ([2000]).39Here is a partial list: Bartels ([1996]), Brighouse ([1994]), Butterfield ([1989]), Field

([1985]), Healey ([1995]), Hoefer ([1996]), Mundy ([1992]). DiSalle ([1994]) argues that Newtonhimself would have rejected Clarke’s rule.

40I believe that Field ([1985], fn. 15) was the only substantivalist to reject Clarke’s rulebefore Earman and Norton’s ([1987]) resurrection of the hole argument.

41Maudlin ([1990]) attempts to pry the two cases apart.

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scalene triangle were in fact only entitled to speak of three non-collinear points,no two pairs of which determine congruent segments—leaving them very far

indeed from being able to determine embeddings up to isometry.42

Relationalists will turn to modality to meet this objection. But, as Field([1985], §§7–10) points out, merely appealing to a modal analysis like ‘AB :BC :: 2 : 1 if it is geometrically possible that (holding fixed the actual rela-tions between A, B, and C ) there should be a D between A and B such thatAD is congruent to DB and BC ’ will not in itself solve the problem—sinceholding fixed the betweenness and congruence relations between A, B, and C

evidently radically underdetermines the ratios. Thus, relationalists must positsome primitive structure in addition to the occurrent betweenness and congru-ence relations. One way to do this is to accept metric facts—along the lines of AB : BC :: 2 : 1—as primitive. Even if this is done, problems remain. Supposethat we fix the distance relations between some finite—perhaps null —set of

material points. How can relationalists hope to capture facts about geometricstructure outside of the finite—perhaps vanishing —region occupied by thesepoints?43 Is space finite or infinite? Does it have the same structure everywhere?Only by accepting a large number of modal facts which do not supervene onoccurrent facts can relationalists give content to such questions.

Thus it appears that any tenable relationalism must incorporate a powerful,primitive notion of geometric modality. Manders ([1982]) provides the mostpromising route to such a relationalism.44 The core of his paper is the followingtechnical result. Suppose that we have a standard axiomatization of Euclideangeometry in terms of betweenness and congruence in some language quantifyingover geometrical points. Manders shows that this formulation can be replacedby an equivalent one which quantifies over finite configurations of points andmentions only the following relations: betweenness and congruence restricted

to members of finite configurations; membership of points in configurations;containment of one configuration in another; a relation c1x1 ∼ c2x2 which holdswhen point x1 occupies the ‘same place’ in configuration c1 as x2 does in c2.Manders further observes that the existential quantifiers in the second theorycan be given a modal interpretation—that there exists y midway between x andz, can be understood to mean that it is geometrically possible for there to besuch a y (whether or not there actually is one).

The equivalence of the two theories shows that all facts about Euclideangeometry can be captured by talking about geometrical relations among possi-ble finite configurations of point particles, without reference to the totality of geometrical points. In particular, given three non-collinear points arranged in a

42Clarke appears to have urged just this objection on Leibniz: ‘the situation or order may

be the same, when the quantity of time or space intervening is very different’ (C.V.54).43See Sklar ([1974], §III.B.2) and Earman ([1989], §§6.10, 6.12, and 8.6) for versions of thisobjection.

44See Mundy ([1983]) for a related approach. Mundy purports to offer a relationalist accountof Euclidean and Minkowskian geometry which does not rely upon modality. But he takes asgiven the structure of the inner product, which would appear encode a large number of modalfacts.

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scalene triangle, one can prove—just as one can in the standard approach—thatthere must be determinate real numbers measuring the ratios of the sides. The

upshot is that we have a formulation of Euclidean geometry which invites amodal relationalist reading. If we ask substantivalists to consider two copies of R3 each containing a copy of our scalene triangle differently placed, they will

grant that we treat these models as representing the same triangle in geometryclass, but will maintain that they ought to be conceived of as (at least beingcapable of) representing distinct possible physical situations. The advantage of Manders’ reformulation of the geometrical theory is that it makes this lattermove less tempting—the individuals dealt with in geometry are the vertices of the triangles, not the points of R3 underlying a model of the theory.45

Much of this discussion carries over to other geometries: Manders’ schemeapplies to any geometry that can be formulated in terms of a finite numberof nonlogical relations; this class includes arbitrary Riemannian spatial geome-

tries and the geometries of Newtonian, neo-Newtonian, Minkowski, and generalrelativistic spacetimes.46 Thus one can be a modal relationalist about any phys-ically interesting geometry.47 One is able to talk not only about the geometricalrelations instantiated by physical particles and events, but also about whichfurther relations are geometrically possible. And this means that modal rela-tionalists about any geometry are entitled to facts about, e.g., the curvature of a short finite segment on which no betweenness relations are satisfied, in thesame way that relationalists about Euclidean space are entitled to determinateratios of segment lengths.

As in the special case of Euclidean geometry, fixing the structure of a (spatialor spatiotemporal) geometry imposes constraints on possible relations betweenmaterial bodies or events. Given a geometry which admits symmetries and ageometrically possible configuration of matter, substantivalists and relationalists

will disagree about how many ways it can be physically realized. If, for instance,the geometry is homogeneous—i.e., if there is a group of isomorphisms whichcan map any given point to any other given point—then substantivalists, butnot relationalists, will recognize a notion of absolute position.

5.2 Relationalism about Space and Relationalism about

Motion

The substantival-relational debate about Euclidean space has quite direct im-plications for dynamics. Substantivalists count each possible embedding of a set

45See Wilson ([1993], §§XV and XVI) for some related points about standards of individu-ating representations in mathematics and physics.

46All of these except the first (which requires a further relation of genidentity for points of

absolute space) can be axiomatized in terms of betweenness and congruence alone (see Field[1980] and Mundy [1992]).

47Horwich ([1978]) proposes a variety of non-modal relationalism about Newtonian space-time, which is explored and generalized by Mundy ([1986]) and Maudlin ([1993], pp. 193–4),who observe that non-modal relationalists about Minkowski spacetime and neo-Newtonianspacetime are unable to fix embeddings of non-colinear sets of points up to isomorphism. Thepresent approach owes much to Teller ([1991]).

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of N particles into R3 as (being capable of) representing a distinct possibility—

which is just to say that they will work with the standard 3N dimensional

configuration space and the standard 6N dimensional phase space when con-structing mechanical theories. Relationalists about space will deny that em-beddings related by rigid motions can represent distinct possibilities; so theywill identify points in the standard configuration space so related; thus theywill employ the 3N − 6 dimensional configuration space (parameterized by therelative distances) and the 6N − 12 dimensional phase space (parameterized bythe relative distances and velocities) of the relational theory of §4.48 Thus rela-tionalism about space implies relationalism about motion. Conversely, it wouldseem that anyone who adopts the relational phase space ought to be committedto Leibniz’s rule. So in this context, relationalism about space is equivalent torelationalism about motion.

5.3 Dynamics and Relationalism about Spacetime

Just as relationalists about space accuse their substantivalist counterparts of miscounting possible dispositions of particles in space, so relationalists aboutspacetime take their opposite number to miscount possible four dimensionalconfigurations of events. We can represent a possible history of a system of particles by a collection of world lines embedded in a manifold with the structureof neo-Newtonian spacetime. If we act upon one of these representations byshifting the worldlines by a Euclidean isometry or a boost, a substantivalistwill affirm, while a relationalist will deny, that the new representation can beconstrued as representing a distinct physically possible history. Substantivalistssee each solution to the equations of motion as (being capable of) representing adistinct possibility; relationalists believe it necessary to identify solutions related

by symmetries before a bijective correspondence with physical possibilities canbe established.

Thus far, we have thought of the standard phase space for our system of particles, its space of dynamical possibilities, as the being the space of possibleinitial data for a theory motion, the space of possible instantaneous dynamicalstates. But in the context of a deterministic theory, the space of initial dataand the space of solutions are isomorphic—with each choice of a time at whichto pose the initial data determining an isomorphism. And, as we can viewisometries and boosts as acting on the space of solutions, we can also view themas acting on the phase space, once we have chosen an isomorphism. 49 Thus,while substantivalists about spacetime, like substantivalists about space, holdthat the standard phase space is isomorphic to the set of physical possibilities,relationalists contend that it is the quotient of the standard phase space by the

Galilei group which is isomorphic to the space of possibilities.

48This is a viable course of action only because the universe has no appreciable angularmomentum and the relevant interparticle forces depend only on the relative distances.

49I set aside the time translation symmetry of classical mechanics, the elimination of whichcan cause technical problems. (For ergodic dynamics, time translation is an improper groupaction; quotienting by this action results in a non-Hausdorff space.)

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We can proceed as follows. Beginning with the standard theory with its 6N

dimensional phase space, we identify points which are related by translations

and/or boosts, killing off the six variables describing the position and velocityof the center of mass of the system.50 If we go on to identify points related byrotations and reflections, we end up with a 6N − 9 dimensional reduced phase space , which carries a geometrical structure and Hamiltonian inherited from theoriginal phase space.51 The dynamics of the resulting reduced theory capturesthat part of the Galilei invariant content of the dynamics of the original theory.

The reduced phase space can be thought of as the result of adjoining threevariables to the relative phase space of §4 (it is an R

3 bundle over this space). Itcan also be viewed as the union of subspaces constructed in the following man-ner: begin with the standard phase space; identify points related by translationsand boosts; fix a value, R, for the magnitude of the angular momentum of thesystem; identify points in the resulting space which are related by rotations(including improper rotations). If R = 0, this construction leads to relativephase space; otherwise, the result is a phase space of dimension 6N − 10, whichincludes two variables parameterizing the surface of a sphere of radius R inaddition to the variables of the relative phase space (i.e., each such space is asphere bundle over the relative phase space). Each of the spaces so constructeddescribes the invariant content of the sector of the standard theory in which themagnitude of the angular momentum is R, with the sphere variables encodingthe body angular momentum (i.e., the direction of the angular momentum vectoraccording to a frame rotating with the system).52 These manifolds fit togetherin a natural way to form the reduced phase space.53

Relationalists about neo-Newtonian spacetime will consider this reducedphase space to be the most perspicuous setting for dynamics, since it is theresult of identifying points in the standard phase space that are related by the

isomorphisms of neo-Newtonian spacetime, boosts and rigid motions. It hasthe advantage over the reduced phase (preferred by relationalists about spaceand motion) that it captures the full invariant content of the Newtonian theory,

50The result is the same if we fix an arbitrary value for the total linear momentum andthen identify points related by translation (Souriau [1997], Theorem 13.15). The action of thegroup of boosts has a curious feature—it fixes the set of dynamical trajectories even thoughit does not leave the Hamiltonian invariant. Indeed, that there are can b e no Galilei invariantHamiltonian is an algebraic fact which has interesting consequences for the classification of classical elementary particles (Ibid., §§11 and 14).

51The Poisson structure of the algebra of Galilei invariant functions of the original theorydrops down to define a Poisson structure on the reduced phase space. Note, however, thatthe resulting Poisson space is not a symplectic (stratified) manifold.

52See Iwai ([1987]). These are symplectic (stratified) manifolds. The Coriolis and cen-trifugal forces are enforced by extra terms which appear in the symplectic structure and theHamiltonian (both of which depend upon R). The resulting theory models the motion of a

particle carrying an so(3) Yang-Mills charge moving on the curved background geometry of the relative configuration space.

53See Marsden ([1992]). The reduced phase space (like any Poisson manifold) is foliatedby symplectic submanifolds—in our case, the sphere bundles. When one factors out a groupof symmetries, conservation laws of the original theory are reflected in the geometry of thereduced theory—in our case, each leaf corresponds to a distinct possible state of rotation, andno Hamiltonian can carry the system from one leaf to another.

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countenancing even rotating universes; it has the advantage over the standardphase space (preferred by substantivalists about space and spacetime) that it

does without the dynamically inert variables (position and velocity of center of mass, orientation about the center of mass) associated with the symmetries of the theory.

Specifying initial values in the reduced phase space determines a set of initialvalues in the standard phase space up to a rigid motion or boost. Thus the threevariables of the reduced theory beyond the relative distances and velocities areprecisely what is necessary in addition to the relational initial data in order tofix the state of rotation of the system, and to determine the history of the systemup to an isomorphism of neo-Newtonian spacetime. Substantivalists who preferthe standard approach may want to claim an advantage here—they can see thatthe further non-relational variables correspond to the magnitude of the angularmomentum of the system and to the body angular momentum. Relationalistsabout spacetime will retort that this is merely a paraphrase of their claim thatthese are the variables which fix the state of rotation of the system, once therelative distances and velocities have been specified.

I do not see that either relationalism or substantivalism enjoys much of an advantage over the other. Some, like Saunders ([2000]), will be drawn torelationalism because it allows one to avoid the unnecessary postulation of in-discernible possibilities. Others will feel that the increase in complexity of thetheory, with its complicated phase space and Hamiltonian, is too high a priceto pay for its ontological parsimony. One can always hope that some furtherphysical consideration will settle the issue.54

5.4 Fields

Let me bring this rather long discussion of classical mechanics to a close with afew comments about classical field theories, whose bearing on the substantival-relational debate has been the subject of some disagreement. Friedman, takingrelationalism to imply that the set of physical events is a proper subset of theset of existents recognized by substantivalists, worries that the substantival-relational debate would collapse if relationalists were to recognize the existenceof space-filling fields ([1983], pp. 222–3). But, as Earman ([1989] §6.6) observes,this seems to be too quick: Leibniz was a plenist who thought he had a realdifference of opinion with substantivalists; surely we ought to look for a construalof the debate which keeps this possibility open.

On the other hand, Field claims that the inclusion of fields more or less settlesthe debate in favor of substantivalism ([1980], p. 35, [1985], §3; see also Earman[1989], §§8.1–8.3). According to one influential school of thought, fields are

assignments of properties—perhaps dispositional ones—to parts of space (see,e.g., Hesse [1961], p. 192; Stein [1970] pp. 266 ff.). If fields are so understood, itbecomes difficult to see how relationalists can countenance them—is it possible

54Quantization and reduction of symmetry do not always commute, but there is everyindication that they do in our case.

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to attribute properties to parts of space even while denying the existence of suchparts? Field believes that the only viable option for relationalists is to replace

standard field theories by action-at-a-distance alternatives:

Instead of predicting and explaining the behavior of matter in termsof fields, i.e., properties of. . . regions of space-time, a relationalistphysical theory would have to predict and explain the behavior of matter in terms only of that matter and other matter. (Field [1985],p. 40.)

Anyone accepting this obligation faces what appears to be an insuperable diffi-culty. A standard theory describing the interaction of a finite number of particleswith a field posits infinitely many degrees of freedom, associating all but a finitenumber of these with the field. An action-at-a-distance theory, if it remainswithin the framework of classical mechanics, would have only the finitely many

particle degrees of freedom to work with. It is perhaps not quite certain that theconstruction of such alternatives to field theories is an impossible task: perhapsone can find empirically adequate truncations of standard field theories whichcan be mimicked by finite dimensional theories; but historical attempts to carryout this program have not met with much success (see, e.g., Arntzenius [1994],§2 for a discussion of the Einstein-Ritz dispute).

Fortunately, there is an second reading of classical field theories, which doesnot support the link that Field seeks to establish between relationalism andaction-at-a-distance theories. The central idea is that the same considerationswhich lead one to posit infinitely many degrees of freedom for the field—thefact that its parts store energy and momentum, which propagate even betweeninteractions with matter—also militate in favor of thinking of the field itself as an infinitely extended object whose parts have properties (Malament [1982],fn. 11; Teller [1991], p. 382). Relationalists—and others—can treat fieldsas they would, say, rigid bodies—as extended objects whose parts stand indeterminate spatial relations to one another, and to which differing propertiescan be attributed.55

It is sometimes objected that this is a cheat which trivializes the substantival-relational debate: if relationalists adopt this construal of field theory, they arein effect helping themselves to an ontology as extensive as that of those substan-tivalists who attribute properties to parts of spacetime; and the debate collapsesinto an uninteresting quibble over the correct name for the resulting position(Field [1985], pp. 41–2; Rynasiewicz [1996], §V).

55There is some question concerning the extent of the field-body so-construed—does itexist in regions where the field intensities vanish? See Friedman ([1983], pp. 222–3), Maudlin

([1990], pp. 200–2), and Teller ([1991], p. 382). Here the most sensible attitude appears tobe that of Weyl ([1949], p. 172): ‘It should be noted here that the validity of the equationE = 0 in some portion of space does not mean that the electrical field E is interrupted in thatportion, but merely that it is in the “state of rest” there which fits continuously into all otherpossible states.’

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This conclusion does not follow if we (unlike Field) require substantival-ists and relationalists to adhere to Clarke’s and Leibniz’s rules. In the case of

Maxwell’s theory, substantivalists about space can take as their configurationspace the possible states of the magnetic field in R

3, while relationalists aboutspace can construct their configuration space by identifying magnetic fields re-lated by rigid motions. As in §4, relationalists will construct a theory which isadequate only to the rotation-free sector of the standard theory (see Barbourand Bertotti [1982], p. 303). Correspondingly, relationalists about spacetimecan insist on identifying points in the standard phase space which are related byboosts and rigid motions—resulting in a convoluted reduced phase space whichcaptures the invariant content of the standard theory. Far from causing thedebate to collapse, allowing relationalists to construe fields as extended objectspermits a smooth extension of the debate to new contexts.

6 General Relativity and General Covariance

In this final section, I would like to say a bit about the general covariance of general relativity. The story of this principle’s role in the discovery and in-terpretation of general relativity is a long and complicated one (see Norton[1993]). Here I will touch briefly on the themes that have been the focus of recent philosophical discussion—the relationship of general covariance to rel-ativity principles and to ontological questions—and conclude by preaching ashort sermon.

General covariance played a prominent role in Einstein’s canonical formula-tion of general relativity:

The general laws of nature are to be expressed by equations which

hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant).

It is clear that a physical theory which satisfies this postulatewill also be suitable for the general postulate of relativity. For thesum of all substitutions in any case includes those which correspondto all relative motions of three-dimensional systems of co-ordinates.([1916], p. 117.)

A description of a Newtonian system in rotating coordinates will include‘fictional force’ terms which do not appear in inertial descriptions of the samesystem. So we have a link between the principle of Galileian relativity and theform of the equations: the equations assume the same, simple, form in inertial

frames; such frames are, in a sense, equivalent. Einstein hoped to generalize theprinciple of Galileian relativity, as it had been realized in the special theory of relativity, to a principle according to which all frames were equivalent to oneanother. It seemed clear to him that any theory in which the equations assumedthe same form in every system of coordinates would satisfy such a principle (seeNorton [1989a] for a careful reconstruction).

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The fatal flaw was pointed out by Kretschmann in 1917 (see Rynasiewicz[1999] for an account of the Einstein-Kretschmann dispute). Any theory which

can be formulated in the language of the tensor calculus can be given a coordinate-free formulation, in which it satisfies Einstein’s principle of general covariance.At the time, of course, general relativity was the only theory to have been soformulated; but, Kretschmann observed, it ought to require only mathematicalingenuity (provided in the event by Cartan only a few years later) to recastNewtonian mechanics, with its privileged inertial observers, into such a form.Einstein granted the point, but maintained that the general covariance of gen-eral relativity did nonetheless have important physical content—content whichwould not accrue to a generally covariant formulation of Newtonian mechanics.Einstein’s claim continues to enjoy a certain popularity among physicists.

The idea that the general covariance of the new theory implied a gener-alization of the principle of Galileian relativity found its way—unalloyed byKretschmann’s reservations—into the philosophical reports on the new physicswhich appeared around 1920 (Cassirer [1953], p. 374; Reichenbach [1960], pp.7–9; Schlick [1963], p. 44); and for some time afterwards, this idea played aconsiderable role in the non-specialist philosopher’s stereotype of the theory.

What is now the received view (among philosophers) of the relationshipbetween relativity principles and principles of covariance emerged in papers of Friedman ([1973]) and Earman ([1974]) which took up and developed ideas of Anderson ([1967]); this account is given its canonical formulation in Friedman’sbook ([1983], §§II.2, IV.5, V.4, and V.5). In this scheme, a physical theory isdetermined by a set of equations governing the behavior of geometric and matterfields, Oi, on some fixed manifold, M. We want to isolate three subgroups of the group of diffeomorphisms from M to itself.

• The covariance group. This is the set of diffeomorphisms which leave in-variant the form of the equations. Using a coordinate-independent versionof the tensor calculus, we can give any theory a generally covariant for-mulation. Of course, many theories also possess formulations with smallercovariance groups: in the case of classical mechanics, we can specify thelaw of inertia and the geometry of Euclidean space in Cartesian coordi-nates associated with an inertial frame; characterizing the dynamics andgeometry via these same specifications in any other kind of coordinateswould lead to physical nonsense (free bodies do not follow straight paths

in the (r, φ)-space of polar coordinates).

• The symmetry group. The set of absolute objects of the theory is the set{Oi} such that whenever we are given two models of the theory, there is

a diffeomorphism between them which carries the absolute objects of thefirst model on to those of the second. The absolute objects are ‘the same’in each model of the theory. The group of symmetries of the theory is theset of automorphisms of the models which preserve the absolute objects.56

56This definition is not perfect (Friedman [1983], p. 59 fn. 9).

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The symmetries of a theory set in Newtonian spacetime include rigid mo-tions and time translations (preserving the temporal and spatial metric

and the relation of genidentity for points of space); the symmetry groupof a theory in neo-Newtonian spacetime is larger, including also boosts(which preserve the affine structure, but not genidentity); the Lorentzgroup (which preserves the spacetime metric) will be the symmetry groupfor a theory in Minkowski spacetime.

• The indistinguishability group. This is best handled informally. The ideais that inertial frames in Newtonian spacetime—or in neo-Newtonian orMinkowski spacetime—are indistinguishable from one another in the sensethat no experiment can distinguish between a system in a given frame, andthe boosted analog of the system in a boosted frame. It is, however, pos-sible to detect the difference as soon as one moves to a larger class of frames—things behave differently in rotating frames than in inertial ones.

Call the group which preserves the class of privileged frames the indistin-guishability group. This feature of the theory depends upon the dynamicsand on the notion of observability in play, as well as on the geometry of the spacetime involved: for theories set in neo-Newtonian and Minkowskispacetime, the indistinguishability group typically coincides with the sym-metry group; not so for standard theories in Newtonian spacetime, forwhich the indistinguishability group includes boosts, although these arenot part of the symmetry group.57

If we rely upon formulations in which the geometry and dynamics are de-scribed relative to inertial frames, then the three groups will coincide for anyreasonable theories set in neo-Newtonian or Minkowski spacetime. In this con-text, it is easy to conflate the concept of indistinguishability—most closely as-

sociated with the principle of relativity—with the concept of symmetry. Buteven here, we can see that adopting a generally covariant formulation of one of these theories has no bearing on questions of indistinguishability or relativity.

When it comes to the theory of general relativity, the symmetry and thecovariance groups will include arbitrary diffeomorphisms of spacetime—the the-ory is generally covariant, and no geometric or matter fields are invariant frommodel to model. What about the indistinguishability group? Only if this, too,includes arbitrary diffeomorphisms will general relativity embody the principleof general relativity. Following Friedman ([1983], §§V.4 and V.5), it is natural totake the class of local inertial frames (i.e., those frames adapted to freely fallingobservers) as our class of privileged frames. And, quite clearly, an arbitrarydiffeomorphism can be expected to carry us out of this class.58

57

That this shows that neo-Newtonian spacetime provides the proper setting for nonrela-tivistic physics is one of the characteristic theses of contemporary philosophy of space andtime.

58There is, however, some difficulty in applying Friedman’s notion to the case of generalrelativity. If φ is in the indistinguishability group for a model of the theory, then φ maps localinertial observers to local inertial observers. But this means that φ maps the set of timelikegeodesics through an arbitrary point, x, onto the set of timelike geodesics through φ(x). It

25



This neat and convincing account makes it clear that general relativity doesnot, after all, generalize the principle of Galileian relativity—while rendering

its one time plausibility comprehensible. It should not be surprising, then, thatmost philosophers now take this account of the content of the principles of general covariance and general relativity for granted. Philosophical interest ingeneral covariance has shifted to a new question: Is there some tension betweensubstantivalism about the spacetime of general relativity and the judgment thatthe theory is deterministic?

This question, too, has its roots in Einstein’s early struggles with generalrelativity. For a time, Einstein had himself convinced that a generally covarianttheory could not be deterministic, since such a theory could determine evolutionof initial data only up to diffeomorphism and not uniquely.59 He eventually cameto believe, however, that this consideration—now known as the hole argument —was easily overcome. In his fundamental paper on general relativity he reasonedthat ‘the results of our measurings are nothing but verifications of . . . meetingsof the material points of our measuring instruments with other material points,coincidences between the hands of a clock and points on the clock dial, andobserved point-events happening at the same place at the same time’ ([1916],p.117); but diffeomorphisms preserve such coincidences; so distinct evolutionsof the same initial data have the same physical content.

Differential equations which fail to uniquely determine the evolution of ini-tial data raise a prima facie threat of indeterminism. Einstein observed that theequations of motion of generally covariant theories fall into this category; thenpointed out that in this special case there is a reasonable criterion of observ-ability under which there can be no observable difference between the variouspossible evolutions of a given set of initial data. Leaving aside the historicalquestion of Einstein’s view (on this question, see Howard [1999]), there would

appear to be three ways of helping ourselves to his insight. (1) We can take averificationalist line, and maintain that the question of indeterminism simplycannot arise since the divergent evolutions of the initial data are observation-ally indistinguishable. (2) We can allow that there is a genuine indeterminismhere—each evolution of our set of initial data really does correspond to a distinctphysical possibility—but maintain that this is harmless, since the indeterminismin question has no observable repercussions. (3) We can assert that, properlyunderstood, the entire raft of evolutions of a given set of initial data correspondsto a single possible future, so that the theory is deterministic after all.

Few people these days are attracted to (1). Earman and Norton’s resurrec-

follows that φ is a symmetry of our model (Weyl [1952], pp. 228–9). So for a generic (henceasymmetric) model of the theory, the indistinguishability group is the trivial group consistingof the identity. An observation due to Earman suffices to show that a similar conclusion

follows for the Newton-Cartan theory (Earman [1977], p. 109).But these theories do obey a relativity principle—the principle of equivalence, which is

a direct descendent of Newton’s sixth corollary to the laws of motion (see fn. 5 above).This seems to provide some support for Brown and Sypel’s ([1995]) contention that group-mongering is not the key to understanding relativity principles.

59See Norton ([1984]), Renn and Sauer ([1999]), Stachel [1989]), and Torretti ([1996], Chap-ter 5) for complementary accounts of this episode.

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tion of the hole argument has two components: the first is an argument thatsubstantivalists about general relativity are committed to (2); the second is an

attempt to awaken a fitting sense of shame in such people (Earman and Norton[1987]; Norton [1989b]; Earman [1989], Chapter 9). The most common responsehas been an attempt to show that substantivalism is in fact consistent with (3).60

Some advocates of this claim go on to complain that the substantival-relationaldebate is running low on content and excitement.61 No wonder! I believe thatin adopting (3), substantivalists are helping themselves to a position most nat-urally associated with relationalism. In the Hamiltonian formulation of generalrelativity, the group of diffeomorphisms plays a role which is in many respectsanalogous to the role of symmetry groups, like the Galilei group in classical me-chanics and the Lorentz group in special relativity.62 For this reason it would,I think, be preferable to maintain the traditional distinction between the twodoctrines by associating substantivalism with (2) and relationalism with (3), allthe while hoping to find some physical grounds for choosing between the two(see Belot and Earman [1999]).

In any case, I think that most everyone agrees that the hole argument liter-ature, in sharp contrast to earlier discussions of relativity principles, has failedto substantially deepen our understanding of general relativity and general co-variance. This is a disappointment. Many contemporary physicists believe thatthere is a great mystery wrapped up in the general covariance of general rela-tivity, the revelation of which is a prerequisite for an understanding of the deepquantum nature of our world. And among these physicists, questions closelyrelated to the ones raised by the hole argument—questions about which quanti-ties can be physically real in a generally covariant theory—are taken to lie veryclose to the heart of the matter. Yet extensive philosophical discussion of theseissues has led to little more than a cottage industry in analytic metaphysics.

I would suggest that part of the problem is the excessively rigid conceptionwhich philosophers have of the principle of general covariance itself. We allknow by heart the Anderson-Friedman-Earman construal of general covariance,and the story of its drastic under-achievement as a deep physical principle. Andthat is where philosophical thought about the formulation and content of theprinciple of general covariance seems to have ended.

Mathematicians and physicists appear to take a very different attitude.There has been intense study of the action of the diffeomorphism group onthe phase space of theories like general relativity, and active discussion of thephysical content of this action.63 There seems to be a widespread sense that

60See fn. 39 above. See also Leeds ([1995]), Liu ([1996]), and Mundy ([1992]). I will not herediscuss the reasons offered by neo-substantivalists for their heresy; Earman ([1989]), Chapter9 remains the best overview.

61

See Belot and Earman ([1999], §1) for a survey, and the contrast with the views of someinfluential gravitational physicists.62The analogy is less than perfect in the popular 3 + 1 canonical formalism (Kuchar [1986]);

it is much stronger in the covariant approach (see Crnkovic and Witten [1987]; Ashtekar,Bombelli, and Reula [1991]; and, especially, Gotay, Isenberg, and Marsden [1997/99]).

63Let me here mention three essays which I believe deserve the same sort of attentionfrom philosophers that Anderson’s work received: Guillemin and Sternberg ([1978], see also

27



we have yet to grasp the full physical content of the principle of general co-variance; and a corresponding idea that it is only reasonable to expect a long

struggle here, when one reflects upon the history of the other great principlesof mechanics, such as the principle of least action.

I think that we philosophers would do well to be as ambitious and as open-minded on this question.

Acknowledgments

I would like to thank Frank Arntzenius, Laura Ruetsche, and Steve Weinsteinfor helpful comments on an earlier draft.

Department of Philosophy New York University New York, NY 10003 USA

e-mail: [email protected]

their [1984], pp. 146–50 and 304–13); Kuchar ([1988]); and Gotay, Isenberg, and Marsden([1997/99]).

28



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