Zaret, D. a Limited Conventionalist Critique of Newtonian Space-Time

A Limited Conventionalist Critique of Newtonian Space-TimeAuthor(s): David ZaretReviewed work(s):Source: Philosophy of Science, Vol. 47, No. 3 (Sep., 1980), pp. 474-494Published by: The University of Chicago Press on behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/186956 .Accessed: 13/10/2012 15:07

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A LIMITED CONVENTIONALIST CRITIQUE OF NEWTONIAN SPACE-TIME*

DAVID ZARET

Cornell University

In this paper, I examine a number of alternative global structures for Newtonian space-time, and corresponding Newtonian theories of mechanics and gravitation. I argue that since these theories differ only with respect to questions concerning the relative distribution of inertial and gravitational forces, the choice between them is a matter of convention. Therefore, the global structure of Newtonian space-time is also a matter of convention. Since this result is based on a consideration of the nature of inertial and gravitational forces, rather than on general reductionist principles, it is a "limited conventionalist" result.

I

A number of writers have employed a four-dimensional approach in order to help clarify the conceptual foundations of Newtonian theory. These writers have proceeded by developing alternative global structures for Newtonian space-time, and corresponding Newtonian theories-namely, theories whose postulates are appropriate mathe- matical variants of the postulates of Newtonian mechanics and gravitation, and which posit the given structure as the structure for space-time. In this paper, I will focus on the relative status of these alternative Newtonian theories. I will argue that they do not represent genuine alternatives at all, but represent instead different versions of the same theory. In other words, I will argue that the choice between them is a matter of convention. It will follow that the global structure of Newtonian space-time is also a matter of convention.

In general, I view it as a necessary condition for a conventionalist result of this kind that the theories in question be shown to be equivalent with respect to all possible observational evidence. I believe that this necessary condition is satisfied by the Newtonian theories just mentioned. However, this condition is by no means sufficient. For how can the conventionalist justify an argument which proceeds from the presumed fact that the theories with which he is concerned are empirically underdetermined, to the claim that there is no fact of

*Received September 1978; revised December 1979. 'See, for example, Havas (1964), Stein (1967), and Earman and Friedman (1973).

Philosophy of Science, 47 (1980) pp. 474-494. Copyright ? 1980 by the Philosophy of Science Association.

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the matter as to which of these theories (if any) is true? The conventionalist could attempt to justify this argument by appealing to general reductionist principles, according to which the factual content of a theory is exhausted by what the theory has to say about phenomena which can be directly and locally observed. However, such general principles are themselves rather dubious. Furthermore, when he appeals to such general reductionist principles, the conventionalist can no longer regard his argument as pertaining exclusively to the status of the theories which he is examining. For by restricting the factual content of a theory in the way just mentioned, we seem to render conventional all theories which have empirically equivalent but logically incompatible counterparts.

An alternative to this general reductionist approach is what I will call a "limited conventionalist" approach. The limited conventionalist rejects any kind of general reductionist principle. Instead, he proceeds by appealing to features which are unique to the theories with which he is dealing. My aim in this paper will be to construct such a limited conventionalist argument in the case of Newtonian theory, by focusing on the status of inertial and gravitational forces.

II

In this section, I will outline two different ways of formulating the global structure of Newtonian space-time. First, the structure of "flat Newtonian space-time," F, is as follows.2 F is a four-dimensional differentiable manifold homeomorphic to R4. We can define a function t on F which represents "absolute time." The subspaces t = constant are three-dimensional manifolds, homeomorphic to R3; and through each point of F there passes exactly one such subspace. These subspaces are to be interpreted as the planes of absolute simultaneity, or "instantaneous spaces" of F. F is also endowed with a contravariant tensor g"', of signature 0+++, which satisfies ga t = 0 (where ta = df at/az").3 This last condition implies that

gt cannot be interpreted as a non-singular metric tensor for F; instead, it is to be interpreted as the metric tensor for the instantaneous spaces of F. Finally, F possesses a flat symmetric affine connection 7f, with components Fr,, which satisfies:

1) Vf t = 0. This ensures that t is an affine parameter of Vf. 2) V-g"g = 0. This ensures that the length of spatial vectors is

preserved under parallel transport by Vf.

2In my presentation in this section, I have followed Havas (1964), and Trautman (1965), (1967). See also Earman and Friedman (1973).

3Greek indices range and sum from 0 to 3. Local coordinates are (z?, z', z2, z3).

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The equations for the geodesics of Vf are given by:

3) d2za/dt2 + ra (dzP/dt)(dz/dt) = 0.

Since Vf is flat, we can find a coordinate system in which its components vanish. In such an inertial coordinate system, the geodesic equations (3) reduce to:

4) d2z /dt2 = 0.

Now to say that (3) and (4) are geodesic equations is to say that they are equations of motion for a particle which is not acted upon by any external forces. Hence (4) implies that a particle which is not acted upon by any force performs uniform straight-line motion with respect to an inertial system.

According to a second formulation, the structure of "curved New- tonian space-time," C, is derived from the structure of F as follows. First, the four-accelerations in F are defined by:

5) a = dfd2zP/dt2 + r, (dz7/dt)(dz?/dt).

Thus Newton's second law reads:

6) maP = FP.

In particular, if the forces are derivable from a scalar potential U, (6) becomes:

7) maP = -g U,

In what follows, I will be concerned exclusively with the case in which U is the gravitational potential. Thus we can use (5), (6), and (7) to write the equations of motion as:

8) d2za/dt2 + (rF, + t ptga U)(dzP/dt)(dzY/dt) = 0.

Following Havas, we can use (8) to define a new affine connection V", with components:

9) [2,r = Frp + t,ptga Us.

In order to construct C, we take Vc rather than Vf to be the affine connection for space-time. Thus (8) is seen to be in the form of a standard geodesic law for C. In particular, (8) is to be interpreted as the law of motion for a particle under the influence of inertial forces only; gravitational and inertial forces have been identified in C, just as in general relativity. C does retain the temporal and spatial metric structures of F, and it also retains the "stratification" of space-time into instantaneous spaces. But since Vc rather than Vf is the affine connection for C, C and F differ with respect to global

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structure. Specifically, when gravitational sources are present, there is no global coordinate system in which all of the components 2 Or vanish. That is, C, unlike F, is a "curved" space-time.

Let 'F*' and 'C*' denote Newtonian mechanics and gravitation formulated in the context of the space-times F and C, respectively. Such four-dimensional formulations have been presented by Havas (1964) and by Earman and Friedman (1973). It is important to observe that, from the standpoint of C*, no particular flat connection Vf

enjoys any special status. For let Vf be a flat connection whose components Fr satisfy (9), and let i be any scalar field which satisfies

t[aVCp] a//dx ax= 0. Then the connection 7f' with components:

10) rF = r - tot 8g fsa is also flat (Trautman 1967, pp. 417-419). It follows from (9) and (10) that:

11) f2n = Fr + tftg, 8( g + U ,).

Hence there is no unique decomposition of VC; for whenever one decomposition is given by (9), another decomposition can be defined as in (11).4

It follows from these considerations that there are different versions of F*, with each version corresponding to a different decomposition of V7. Thus we can talk about the theories F(Vf)*, F(7f')*, F(Vf")*, ..., where F(Vf)* posits F as the structure, and Vf as the affine connection for space-time. What is the difference between F(Vf)* and F(Vf')*? According to both of these theories, the affine geodesics of space-time are just the trajectories of particles which are not subject to the action of any forces. Hence the fact that the geodesics of V7f do not coincide with the geodesics of Vf', implies that F(Vf)* and F(Vf')* differ as to the amount and the kind of force which is present in a given situation. In particular, these theories posit different forms for the gravitational potential.

Now it turns out that, given the exact proportionality of inertial and gravitational mass, we have epistemic access only to the "sum" of inertial and gravitational effects which is represented by V7. In other words, while it is theoretically possible to determine V' by observing the trajectories of freely falling particles (i.e., particles which are not subject to any non-gravitational interactions), it is not possible to determine the relative contribution to the motion of a particle by inertial and gravitational forces. Hence F(7f)* and F(Vf')* are empirically indistinguishable.

4For more details, see Trautman (1965), pp. 110-115.

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III

Why should we suppose that the choice between C * and the different versions of F* is a matter of convention? Earman and Friedman respond to this question as follows.

What of conventionality? We saw that there were universes in which no observational test will decide between the orthodox formulation of [the law of inertia, as in F] and the new formulation employing [the space-time C]. Doesn't this mean that in such worlds the choice between them can only be decided on grounds of descriptive simplicity and that, therefore, the existence of gravitational forces is a matter of convention? No, not by our lights. For in the first place, not only is the observational-nonobservational distinction vague, but it is constantly shifting-more and more things get counted as observational as time goes on. Secondly, and more importantly, these theories do ... make different theoretical commitments; and in enlarged contexts these commitments can give rise to different observational consequences. (Earman and Friedman 1973, p. 348)

The only "enlarged context" which Earman and Friedman consider is that of electromagnetic theory. However, as Lawrence Sklar has pointed out (1976, pp. 18-19), it is simply not the case that taking electromagnetic phenomena into account enables us to distinguish experimentally between F* and C*. Instead, the conventionalist can deal with the case of electromagnetism by modifying the details, but not the basic strategy of his argument. Earman and Friedman also seek to undermine the conventionalist's position by arguing that the observational-nonobservational distinction on which that position is based is "vague, and constantly shifting." Now I do agree that there are serious difficulties involved in any attempt to state precisely just what the difference is between an observational entity and a theoretical entity. Hence it is difficult, if not impossible, to demarcate precisely and finally the boundary between the observational and the theoretical. These difficulties may pose a problem for the adherent of a general reductionist approach to science. However, it is not so clear that these difficulties pose any problem for the conventionalist who is concerned specifically with the status of F* and C*.

To see why these difficulties do not pose any problem for the limited conventionalist, consider the sense in which C* and any particular version of F* might be said to be "interderivable." First, as indicated in (8) and (9), the structure of C can be derived from that of F by "packing" the gravitational potential into the connection

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V7. Second, suppose we are given the axioms of Newtonian mechanics and gravitation as formulated in F*. We then stipulate that freely falling particles follow geodesic paths in space-time. From the given axioms, together with this stipulation, it is a straightforward matter to write down the axioms for C*.5 The converse of this procedure is similarly straightforward. For while there is no unique decomposition of V', we can certainly "unpack" Vf and the gravitational potential from Vc when, as in the present context, 7f and the gravitational potential have already been specified.

These results indicate that the actual motions of objects are the same in C* and in (any version of) F*. What differs, in the two theories, is what we say about these motions. In the one case, we say that a freely falling particle has been "deflected" from its geodesic path by gravitational forces; in the other case, we say that the same particle is following a geodesic path, and is not subject to the action of any external forces. But this is not an observable difference. In particular, it seems very unlikely that any change in what counts as observable which is brought about by theoretical or technological advances could persuade us to regard gravitational forces (as distinct from gravitational force effects) or the property of "being geodetic" as observable.

Thus the two arguments which Earman and Friedman present in opposition to the conventionalist's position are not very convincing. Note, however, Earman and Friedman's remark that F* and C* "make different theoretical commitments." One would think, given Earman and Friedman's strongly realistic predilections, that they would focus on this point in developing their anti-conventionalist position. For according to the realist, to say that two theories are logically incompatible'is just to say that they are different theories; whether or not they are observationally equivalent should not matter at all. Hence by basing their response to the conventionalist on the claim that F* and C* are not observationally equivalent, Earman and Friedman do not really do justice to the realist position. Instead of responding directly to the conventionalist's claim that the purported observational equivalence of F* and C* implies that they are different versions of the same theory, they avoid dealing with that claim by arguing that F* and C* are not observationally equivalent in the first place. And this kind of response seems to leave open the possibility that if there were no contexts in which F * and C* give rise to different observational consequences, then the realist interpretation of these

5One version of this procedure is outlined by Misner, Thorne, and Wheeler (1973 pp. 298-302).

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theories would have to be modified. As we have seen, a good case can be made in support of the claim that there are no such contexts. But the fact that there are no such contexts should not, by itself, be enough to weaken the realist position. Instead, a more cogent realist response to the conventionalist position would include the following challenge: Given that F* and C* are observationally equivalent, why should it follow that they represent different versions of the same theory? I will attempt to deal with this question in the remainder of this paper.

IV

I will begin my discussion of this question by examining the relationship between the different versions of F* in more detail. First, using (5) and (6), we can rewrite Newton's second law as follows in the context of any version of F*:

12) F" = ma" = m(d2z/dt2 + Fr (dz3/dt)(dzy/dt)).

As before, F, are the components of the integrable connection Vf. If we think in terms of the usual formulation F" = md2z/dt2 of Newtonian mechanics, where the F" represent external forces, we can proceed by regarding the quantities F`,(dzO/dt)(dz /dt) as representing "fictitious" forces. We can then transform to a coordinate system in which Fr, = 0, and in which, therefore, these fictitious forces do not appear.

Now suppose that we take as our starting point some specific version ofF *; say, F( f) *. In the context of this theory, the geodesic equation for F can be written:

13) d2za/dt2 + r, (dzP/dt)(dz/dt) = 0.

Using (10), we can rewrite this geodesic equation as:

14) d2z"/dt2 + (F'r + t,at,g ",8)(dzO/dt)(dzT/dt) = 0;

15) d2z"/dt2 + rt (dz3/dt)(dz/dt) = - gr8 .

The equation in F(7V)* for a freely falling particle is:

16) d2z"/dt2 + Fr, (dzO/dt)(dz/dt) = - g8 U .

Again, we can use (10) to rewrite this equation as:

17) d2z"/dt2 + F'r(dz/ddt)(dz/dt) = - g"8(q^ + U,,).

As before, we can find a coordinate system {z" in which the components Fr vanish. In such a coordinate system, (13) becomes d2z/dt2 = 0. In a general coordinate system, the non-vanishing

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r, are viewed as representing inertial forces. Of course, it follows from (10) that when the Fr do not vanish, neither do the terms Fr t - t,g tg",. Hence in a general coordinate system, the terms Fr, and -g

` ^^, are interpreted as representing separate contributions

to the total inertial force. On the other hand, let us examine this situation from the standpoint

of someone who has adopted the theory F( f' )*, and who, therefore, views 7f' as "the" connection for space-time. From this standpoint, the geodesic equation is:

18) d2za/dt2 + F` (dzP/dt)(dz' dt) = 0.

In a general coordinate system, the nonvanishing F are regarded as representing inertial forces. In a coordinate system (za } in which the Fr vanish, (18) becomes d2 za/dt2 = 0, and (17) becomes

19) d2z"/ dt2 = - ga8 ( + U, ).

Then -g " ' f' ,, is interpreted as a contribution to the total gravitational

force affecting a body in free fall. Hence while the term containing i represents inertial forces from the standpoint of someone who regards Vf as the connection for F, it represents external, gravitational forces from the standpoint of someone who regards Vfs as the connection for F. Similarly, it follows from (10) that the vanishing of Fr in a coordinate system (z") does not entail the vanishing of lF' in {za); and the vanishing of Fr in (z" } does not entail the vanishing of Fry in (z"'}. So a frame of reference which is inertial from the standpoint of F(V7)* may not be inertial from the standpoint of F(7 V')*; and conversely.

Now there is nothing surprising about these results. However, from the Newtonian point of view, we would like to say that at most one of {z") and (z" } is really inertial, and that at most one of (13) and (18) represents the correct geodesic equation. For according to this point of view, there is a privileged class of reference frames, namely, the class of Galilean or inertial frames. This class includes the rest frame of absolute space, as well as all frames which are in uniform rectilinear motion with respect to the rest frame of absolute space. For example, if Vf really is the connection for space-time, then the equation d2z"/dt2 = 0, expressed in the system {z"), represents the correct law of motion for a particle free of impressed forces. Of course, it follows from (18) and the fact that {z" } is inertial for 7f' that d2z' /dt2 = 0. But the Newtonian would argue that this equation is correct in form only. He would establish this point by observing that, by using (10), we can rewrite (13) in terms of {z" } as follows:

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20) d2za /dt2 + (t , t ,ga'' f.,^)(dzP /dt)(dz' /dt) = 0.

Since the ty t, g"a' a,8' do not vanish, the Newtonian would conclude that these terms represent inertial forces which are generated by the acceleration of {z) } relative to the inertial system (z"). The fact that inertial forces do arise in {z" } shows that this sytem is non- Galilean.

However, given that gravitational mass is exactly proportional to inertial mass, it follows that we cannot distinguish locally between the effects of gravitational forces, and the effects of properly chosen inertial forces. For example, suppose that we interpret equations (15) and (17) as follows: freely falling particles are deflected from their natural motion along the geodesics determined by the connection 7f', by gravitational forces which are generated in accordance with the potential f. On the other hand, suppose that we reinterpret this situation by arguing that it is correctly portrayed in terms of (14). That is, we argue that the real connection for space-time is not 7f' but 7f, and if represents inertial forces. What the equivalence of inertial and gravitational mass implies is that there is no way of experimentally distinguishing between these interpretations.

To say that we cannot distinguish between these interpretations is just to say that we cannot determine experimentally whether F( Vf)* or F(7f')* (or neither) is "correct." Again, the reason for this underdetermination is that the only question over which these theories differ is the question of how precisely to evaluate the relative contributions in space-time of inertial and gravitational forces. For example, where the proponent of F(Vf)* views the entire term F' + t,tgt, g8 as representing inertial forces (see (10), (13), and (14)), the proponent of F(Vf')* views Fr" as representing inertial forces (equation (18)), and views to tg iS ̂ 3 as representing gravitational forces (equations (17) and (19)). On the other hand, these theories agree with respect to the question of how a particle is affected by gravitational and inertial forces combined. That is, they agree as to which trajectories represent the paths of particles which are not subject to any non- gravitational interactions. But again, given the equivalence of inertial and gravitational mass, it is only this combination of inertial and gravitational force effects to which we have epistemic access.

V

It is at this point, I feel, that the limited conventionalist has his best hope of developing a cogent argument. What he must establish is that in the case of the different versions of F*, and in contrast to the general case in which pairs of theories are observationally

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equivalent, empirical underdetermination does imply that there is no fact of the matter as to which version is correct. For example, he must establish that when, as in (14) and (15), we move the terms containing qi from one side of the equation to the other, we are just manipulating the formalism. Thus in (14), we designate the entire expression Fr, + t tyg"8 ^ as 'inertial'; while in (15) we reserve the term 'inertial' for Fr, and designate cq as 'gravitational'. But the limited conventionalist would argue that this shift represents only a change in language, rather than a change in the physical content of the theory F*.

Of course, this is what any conventionalist seeks to establish about any pair of observationally equivalent theories with which he is concerned. However, in the case of the different versions of F*, the limited conventionalist can base his argument on the status of forces. More precisely, he can base his argument on the status of different forces which have the same observable effects. This is the case which concerns the limited conventionalist in his critique of the different versions of F*, because what distinguishes one version from another is the different things they have to say about the relative contributions of inertial and gravitational forces. This analysis can also be extended to C*, because C* can be regarded as what is in some sense a "conceptual limit" of the different versions of F*. That is, where these versions attribute different "portions" of a given force effect to the action of external, gravitational forces, C* interprets all such effects as inertial force effects. The general notion of spatial impressed force is still well defined in C*; there just are not any impressed gravitational forces. Again, therefore, the relative status of C* and the different versions of F* is seen to depend on the status of different forces which have the same non-null effects. And, I feel, the case of different forces which have the same non-null effects is somewhat peculiar.

For example, imagine a two-dimensional universe whose inhabitants describe the motion of particles in terms of a particular x-y coordinate system. Let T be the theory of motion for this universe, and suppose that T says the following. First, all deflections of a particle in the positivey-direction of the coordinate system just mentioned are caused by Y-forces acting alone, or by Y'-forces acting alone. Second, deflection in the positive y-direction is the only non-null effect exhibited by Y-forces and by Y'-forces. Finally, every material object possesses a Y-factor and a Y'-factor, which determine the resistance of the

object to the action of Y-forces and Y'-forces, respectively. The greater the Y-factor (Y'-factor) of a given object, the less that object is deflected by a given Y-force (Y'-force). Now according to T, it

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should be possible to distinguish the action of Y-forces from the action of Y'-forces. For suppose that A has the same Y-factor as B, but that A has a greater Y'-factor than B. Then whenever A and B receive the same deflection in the positive y-direction, T tells us that they are subject only to the action of Y-forces. But when B receives a greater deflection than A, T tells us that they are subject to the action of Y'-forces.

Now suppose it were discovered that for all objects A and B, the ratio of the y-deflection of A to the y-deflection of B always has the same value. That is, for all objects A and B, there is a number c, such that whenever A and B are a small x-distance apart and suffer a deflection in the positive y-direction, the ratio of the y-deflection of A to the y-deflection of B is equal to c. According to T, this situation must be interpreted as follows: the Y-factor of any object is exactly proportional to its Y'-factor. More precisely, there is a number d such that for any object A, the ratio of the Y-factor of A to the Y'-factor of A is equal to d.

In this situation, it may no longer be tenable to maintain that Y and Y' are different forces. For since Y and Y' have the same effect on the motion of material objects, it seems to follow that we can avoid collapsing them into a single force only if we can differentiate the physically possible conditions under which they have their (non- null) effects. Thus we might choose a particular well defined set of conditions, and then stipulate: if these conditions hold, any y-deflection is a Y-effect; otherwise, any y-deflection is a Y'-effect. However, the limited conventionalist will claim that our choice of such a set of conditions is completely arbitrary. That is, he will claim that no one set is more correct than any other set. He will attempt to support this position by observing that we can always make a stipulation of the kind just described, with respect to any force. Thus we might stipulate that there is no such thing as electrical force; instead, there is electrical, force and electrical2 force. The former arises whenever what we ordinarily call electrical force arises in odd-numbered years; and the latter arises whenever what we ordinarily call electrical force arises in even-numbered years. Of course, such a stipulation is at least gratuitous and arbitrary, if not devoid of physical meaning. But, according to the limited conventionalist, any stipulation of this kind concerning Y and Y' is similarly gratuitous and arbitrary. Now I do feel that the limited conventionalist has a strong case here with respect to the status of Y and Y'. For no stipulation concerning the conditions under which these forces arise is built into the theory T. Instead, what differentiates Y and Y' in the context of T is their different effects. Since Y and Y' have the same effects in the universe

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being considered, any stipulation about conditions does seem to be completely arbitrary.

The limited conventionalist would like to argue that the case of inertial and gravitational forces is analogous to that of Y and Y'. For like Yand Y', inertial and gravitational forces have been discovered to have the same effects on the motion of particles, despite their dissimilar conceptual origins. Of course, this analogy could not be an exact one, because inertial and gravitational forces can affect an object simultaneously, while Y and Y' cannot. However, the limited conventionalist would argue that this added complexity in the case of inertial and gravitational forces does not affect the main lines of his argument. Instead, he would argue, this added complexity indicates only that, when we attempt to differentiate inertial from gravitational effects, we have a greater range for conventional choice than we had in the case of Y and Y'. Specifically, what is conventional is not only the choice of whether a given effect is caused by inertial forces or by gravitational forces, but also the question of what is the relative contribution of inertial and gravitational forces to any given effect. By arguing in this manner, the limited conventionalist would hope to establish that if two ostensibly incompatible theories differ only with respect to the relative contribution of inertial and gravitational forces in any given situation, then these theories have the same physical content. In particular, he would hope to establish that C* and the different versions of F* have the same physical content.

However, there is a second point at which the analogy between Y and Y' on the one hand, and gravitational and inertial forces on the other, breaks down. Thus we have seen that no stipulation concerning the conditions under which Y and Y' arise is built into the theory T. But of course, such a stipulation is basic to any of the versions of F*. More precisely, there are field laws for the gravitational field. And since the field laws relate the sources of the gravitational field to the field they create, we can presumably infer from these laws that, at least in some circumstances, a given force effect must be an inertial effect. For example, to consider a situation which both Trautman and Sklar discuss, suppose our field laws tell us that at a large distance from an isolated system of bodies the gravitational field tends to zero. Then we can construct inertial frames by performing experiments with freely falling particles in a part of the universe which is sufficiently far from the source to ensure that the gravitational field is negligibly weak. More precisely, we can stipulate that the trajectories of such freely falling particles are the geodesics of space-time, and in this way we can define a flat

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connection Vf. We can then extend the inertial frames which are determined by Vf into global inertial frames. Let us suppose that the connection which is defined in this manner, and the corresponding gravitational field, are represented by the Vf and U of equation (9). According to the realist, (9) is true as a matter of physical fact. That is, Vf really is the connection for F, and the gravitational field really is generated by U. On the other hand, the conventionalist will claim that we can just as well maintain that the real situation is exemplified by equation (11). That is, we can maintain that the connection for F is Vf', and that the gravitational field is generated by f + U. Since U vanishes at a large distance from an isolated system of bodies, the equation of motion for a freely falling particle in this situation is given by (15). In particular, in a frame which is inertial for Vf' (so that F" vanishes), (15) reduces to:

21) d2z"/dt2 =-g ", .

So the conventionalist will explain the acceleration of particles with respect to a frame which is inertial for 7f' by saying that this acceleration is caused by the gravitational forces represented by fr. The realist, however, will argue that there cannot be any gravitational forces in this situation, because there is nothing to generate them. He will conclude that what is really accelerating in this situation are the frames which are inertial for 7V'.

Now the arguments which establish the observational equivalence of F(7f)* and F(Vf')* are just as applicable in the circumstances

just described as they are anywhere else. So the usual reductionist argument is unchanged in these circumstances. But what of the limited conventionalist's argument? Is it tenable for him to maintain that one set of laws which relate the sources of the gravitational field to the field they create cannot be true, to the exclusion of all other such laws, as a matter of physical fact? If he cannot establish this result, then it is difficult to see how he can maintain the conventionality of the theories in question without resorting to general reductionist principles.

The limited conventionalist might respond to this challenge as follows. He might observe that while the availability of field laws

may imply that we have epistemic access both to the effects and to the causes of a force, we still do not have any direct epistemic access to the force itself. It is not as if, when in the presence of a source, we observe the force which it is supposed to cause. Instead, when in the presence of a source, we observe the effects of the force it is supposed to cause. In particular, then, what the gravitational field laws provide is a systematization of the relationship between

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the presence of matter, and the occurrence of gravitational force effects. That is, they provide a systematization of the relationship between the presence of matter, and deviations from geodesic motion. However, C* systematizes this same relationship. The difference is that where F* ascribes changes in the motion of particles to the action of gravitational forces which are generated in the presence of matter, C* ascribes these changes to modifications in the geometry of space-time which are brought about by the presence of matter. The observable relationship between the presence of matter and changes in the motion of particles holds as a matter of empirical fact. What cannot be answered by recourse to observable phenomena is the question of whether it is the action of forces or modifications in geometry which are responsible for these changes.

Hence even if we admit that, in the "island universe" case, there is a preferred decomposition of V7, the limited conventionalist can proceed as before simply by shifting his focus from the different versions of F*, to the relative status of C* and a particular version of F*. Thus the limited conventionalist can argue that since these two theories are observationally equivalent and differ only with respect to the relative distribution of inertial and gravitational forces, the choice between them is a matter of convention.

It should be emphasized here that when the limited conventionalist refers to modifications in the geometry of C which are brought about by the presence of matter, he is not tacitly advocating a realistic interpretation of C*. Thus according to the argument just presented, modifications in the geometry of C are both the effects of material causes and the causes of the motion of other material things. And to say that the space-time C does interact in this way with material objects is apparently to invite a realistic interpretation of C, according to which C enjoys the same ontological status as the material objects with which it interacts. However, it is important to remember that when the conventionalist analyzes the ontological implications of C*, he does not restrict his analysis to C*. Instead, he evaluates the status of C* by comparing it to F*. That is, he bases his analysis on the following points.

22) According to C*, modifications in the geometry of space-time are both the effects of material causes and the causes of the motion of other material things.

23) According to F*, the geometry of space-time never changes. Instead, gravitational forces are generated by the presence of matter, and are in turn the causes of motion of other material things.

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24) C* and F* are empirically equivalent.

The limited conventionalist will argue on the basis of (24), together with considerations about the nature of inertial and gravitational forces, that C* and (any version of) F* represent different versions of the same theory. Therefore, he will argue, (22) and (23) describe the same physical situation in different words. While it may sometimes be convenient to use C* for the purpose of making predictions, our use of C* does not commit us to the existence of a space-time C which interacts with material objects. For we could have obtained the same predictions by using (any version of) F*, and by interpreting the pertinent events in terms of forces rather than in terms of geometry.

Now the fact that the limited conventionalist can argue in this manner indicates that, even in the "island universe" case, he may not have to modify his argument concerning the relative status of the different versions of F*. For if the choice between C* and the "preferred" version of F* is, indeed, a matter of convention, it follows that the field laws posited by this version of F* enjoy only a conventional status. But then it would seem that what makes this version of F* preferred in the context being considered is not the fact that it is the unique "true" theory, but rather the fact that it is more "convenient" than other versions of F*.

Thus the limited conventionalist will argue that the realist cannot, without begging the question, establish that gravitational forces exist as a matter of objective fact by citing the factual nature of a particular version of the field laws. Instead, he will argue, the ontological status of the field laws can be seen to depend on the ontological status of gravitational forces. In particular, if the precise contribution which gravitational forces make to the overall force effect in any given situation is determinate as a matter of physical fact, then the field laws which yield the "correct" intensity for the gravitational field are true as a matter of physical fact. But if the exact contribution which gravitational forces make is a matter of convention then so, it would seem, are the field laws.

Furthermore, the limited conventionalist will note that inertial and gravitational forces act on all bodies in the same way. And he will argue that we can establish, by reference to examples such as that of Y and Y', that the assignment of the relative contributions of different forces which act on all bodies in the same way can only be made by convention. Therefore, the assignment of the relative contributions of inertial and gravitational forces can only be made by convention. The limited conventionalist will conclude that the choice between C* and the different versions of F* is, indeed, a matter

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of convention. In other words, he will conclude that these theories collapse into a single theory, in which the global structure of space-time is a matter of convention.

VI

Clark Glymour discusses the relative status of C* and (various formulations of) F* in his paper, "The Epistemology of Geometry" (1977). He agrees that these theories are empirically equivalent, but rejects a conventionalist interpretation. In the final section of this paper, I will try to show that Glymour's arguments do not undermine the kind of limited conventionalist analysis I have presented.

Glymour argues, first, that C* and any particular version of F* are not "intertranslatable," and hence do not satisfy a necessary condition for synonymy. In the case of geometric theories T1 and T2 which are formulated as covariant equations, Glymour states this necessary condition as follows: The geometrical objects which consti- tute a solution to the equations of T1 must be covariantly definable from those of T2, and vice versa. Thus in the case of C* and a particular version of F *, we have seen (equation (9)) that Vc is definable from F* in terms of Vf together with the gravitational potential U. In particular, then, if we add (9) to the equations of F*, the equations of C* follow. But, as we have also seen, this process cannot be reversed; we cannot define Vf and U from V7, or from any other geometrical object of C*. Hence Glymour concludes that C* and F* "do not say the same thing."

Glymour argues further that, although C* and F* are empirically equivalent, they are not equally well confirmed by their common observational consequences. For, he argues, F* has more untested hypotheses than C*. Thus in a Newtonian world we do have epistemic access to V', via the trajectories of freely falling particles. But none of the phenomena which serve as tests for Newtonian theories of gravitation-congruences of rigid rods, time intervals, the trajectories of freely falling bodies-could enable us to determine the global inertial frames and the unique gravitational potential posited by F *. Therefore, the observational consequences which C* and F* have in common provide evidence only for V7, or for the "sum" of Vf and U; they do not provide any evidence for the additional hypothesis that Vf and U are distinct and unique.

Glymour presents these arguments in the context of a general discussion of the purported underdetermination of geometry. He seeks to show, using the kind of considerations just mentioned, that the nonstandard theories usually considered in accounts of the epistemol-

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ogy of geometry (theories which invoke "universal forces") are neither synonymous with nor as well confirmed as the theories they are supposed to prove to be underdetermined.6 Thus he suggests that the relationship between F* and C* is analogous to the relationship between any of these nonstandard theories and its standard counterpart.

Looking back at the relation between the version of Newtonian theory with a fixed affine connection [i.e., F*] and the second version, that with a dynamical connection and no potential [i.e., C*], it is clear that the former is just the latter with a universal

force. The universal force in that case is [-g8 U; 8], i.e., just the gravitational force that enters the equation of motion of the first version (Glymour 1977, p. 245).

However, I question the exactness of this analogy. For I believe that there are some grounds for viewing the case of Newtonian space-time as different, in a philosophically interesting way, from the other cases which Glymour discusses. An examination of this point will underscore the limited nature of the conventionalist result discussed in this paper.

In order to see why the case of Newtonian space-time may be different, it will be helpful to contrast it with the case of relativistic space-time. Glymour regards as analogous the transitions from C* to F*, and from general relativity to special relativistic theories of gravitation. Both transitions are acceptable formally; but in both cases, the presence of gravitation makes it impossible to determine, mechani- cally, a unique integrable affine connection. But it is clear, at least, that the details involved in the two transitions are somewhat different. Thus the spatial metric g " for Newtonian space-time is singular. As a consequence, the requirement that the affine connection be compatible with the metric (i.e., V,g"a = 0) does not determine the connection uniquely. Instead, Vc and (the various) Vf satisfy this requirement. On the other hand, let VR be the connection for the space-time of general relativity. Then VR is uniquely determined by the requirement that it be symmetric and compatible with the space-time metric. Therefore, the compatibility requirement implies that when we split VR into a flat part V? and a gravitational remainder, we will have to adopt the metric which is appropriate to flat Minkowski space-time as the "real" metric for space-time. By making this move, we preserve the compatibility of connection and metric. Of course, as Sklar points out (1976, pp. 20-22), the "gravitational remainder"

6For more on synonymy, see Glymour (1971); for more on confirmation, see Glymour (1975).

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involved in this splitting must be constructed in such a way as to have electromagnetic and metric effects as well as dynamic effects. For in general relativity, the effects of the gravitational field are "built into" the connection VR. And, by the equivalence principle, the gravitational field acts locally like an accelerated reference frame with respect to electromagnetism and metric measurements. Since these various "gravitational" effects are not expressed by V?, they must be "built into" the gravitational field. But when the gravitational field is constructed in this manner, the splitting of VR into V? and gravitational remainder is just as legitimate, formally, as any splitting of 7'.

The uniqueness of VR indicates that, in general relativity, the metric tensor is the only independent geometric structure. That is, the affine connection, curvature tensor, and other geometric structures are uniquely determined once the metric tensor has been specified. As a result, when we do make the transition to a flat space-time version of general relativity, the flat "background" space corresponding to V? is "unobservable." In other words, the results of ordinary physical measurements will yield the metric tensor and affine connection of general relativity, and the failure of such measurements to yield the "real" geometric structures will be attributed to the "universal" effects of the gravitational field.

In contrast, there is a variety of independent geometric structures in Newtonian space-time; this leaves open the possibility that structures other than Vc can play a role in describing physical phenomena. For example, as will be discussed below, structures other than V7 play a role in the formulation of a theory of electromagnetism in Newtonian space-time. Similarly, as we have seen, the connection and metric are independent of one another in Newtonian space-time; this is evidenced by the fact that the space-times F and C share the same spatial and temporal metric structures. Hence when we make the transition from C* to F*, we do not have to postulate "universal" forces which affect all physical processes, and which, in particular, "distort" measurements made with clocks and rigid rods. Instead, we need reinterpret only the affine structure of space- time. I feel that this contrast between the two cases of geometrization is more than just one of detail. For I think it shows that Newtonian gravity is not a universal force. To be sure, Newtonian gravity does have "universal" effects on particle trajectories. But in contrast to the universal forces discussed by Poincare and Reichenbach, and in contrast to the relativistic gravitational field, Newtonian gravity does not affect the length of transported rods, the rates of transported clocks, or the paths of light rays.

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Consider now how one might construct a theory of electromagnetism in Newtonian space-time. Neither F* nor C* is adequate for this purpose. Instead, as discussed by Trautman (1967), one must introduce an "ether," or a "rigging" of the hypersurfaces of simultaneity. A rigging associates with every point of a hypersurface a direction tangent to the world line of an observer at rest in the ether. Now suppose that the rest frame of the ether coincides with an inertial frame of (an appropriate version of) F*. Let ua be the vector field which is tangent to the directions of rigging, and which satisfies Vfau = 0 and uata = 1. Then the Maxwell equations for the vacuum can be written:

25) V[7 a =0 ; VfFao= 0;

Fa3 =dl ay _ a y - C2)(g8

_ U,P3/C2) F,P=df, (g,"_ u u?/c2)(g - uPu/c~)F?,

where Fa is the "Maxwell tensor." Thus we can incorporate electromagnetism into F* by introducing

Fa together with the vector field u". But it is a bit more complicated to incorporate electromagnetism into C*. For equations (25) involve covariant differentiation with respect to 7f; and so we would, apparently, have to include F,,, ua and Vf in a version of C* which has been expanded to include electromagnetism. Alternatively, we could take the covariant derivatives in (25) with respect to Vc. But in this case, in order to retain the same observational predictions, we would have to add correction terms to the right side of these

equations. Specifically, since the gravitational potential U is "packed into" V7, these correction terms would have to include U. Of course, once we are given both V7 and U, we can define Vf; so it would seem most efficient to construct the expanded version of C* by including (25) as written.

In a Newtonian world it is possible, in principle, to use optical experiments to discover which inertial frame is the rest frame of the ether. And, since the inertial frames of Vf are just the ether frame, together with all frames moving with a uniform velocity relative to the ether, such optical experiments would also enable us to determine Vf. Hence in the expanded version of F*, Vf and U are uniquely determined by the appropriate optical experiments together with the

trajectories of freely falling particles. I believe, therefore, that when we do take electromagnetism into account in the way just described, the enlarged version of F* can be seen to be just as well tested as the enlarged version of C*.

Of course, there is no ether in our world. Hence it is only in certain possible worlds that Vf can be determined. Commenting on

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this situation, Glymour writes that "what theoretical magnitudes we can determine depends on what lawlike hypotheses are available to us, and that, in turn, depends on what kinds of things there are" (1975, p. 423). But if we confine our attention to what there is (in our world), then the absolute time, metric structures, and affine connection of C* cannot be determined, either. It is true that, in many situations, the geodesics of Vc will coincide with those of VR; but what this shows, I think, is that general relativity is better tested than F*.

What Glymour objects to, in his discussion of geometry, is the claim that geometric theories are underdetermined in principle. Reichenbach and others have sought to establish this claim by developing alternative but empirically equivalent geometric theories in certain hypothetical universes; and then suggesting that similar alternatives can be developed in our own universe. But Glymour argues that, even in these hypothetical contexts, there is no underdetermination. For, he argues, the nonstandard theories are not as well tested as their standard counterparts. And this is because their positing of universal forces makes them untestable. Hence what bothers Glymour about the nonstandard geometric theories presented by Reichenbach and others is not that these theories are not, in fact, well tested (no one ever suggested that they were); what bothers him, rather, is that they are not well tested even in the hypothetical situations in which they are empirically adequate.

However, in the context in which F* and C* are empirically adequate, the fact that gravitation is not a universal force allows Vf to be determined by non-mechanical phenomena. Such a context is provided by the hypothetical case of our world, if classical mechanics and electromagnetism had turned out to be correct. And, since what seems to be at issue here is not whether certain theories have been confirmed, but whether they can be confirmed, a consideration of such contexts does seem to be pertinent. In particular, a consideration of such contexts indicates that it is not tenable to uphold the sharp contrast which Glymour seeks to draw between the confirmability of F* and C*.

But what of synonymy, and Glymour's claim that F* and C* "do not say the same thing?" Glymour characterizes his necessary condition for synonymy as "natural," but does not justify it in any detail (1977, pp. 229-230; see also 1971, pp. 279-280). However, I believe that, at least in the context of my discussion, Glymour's interdefinability criterion does require more justification than he has provided. For what is at issue in this context is whether there is any real distinction between the geometrical objects (e.g., the affine connection) postulated

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by C*, and the dynamical objects (e.g., potentials or forces) postulated by F*. According to the arguments I have presented, there is no real distinction between these objects, despite the fact that they are not interdefinable and hence apparently genuinely distinct. Hence my arguments suggest that, in this context, the interdefinability condition simply does not apply. Therefore, it would not be legitimate to reply to my arguments by maintaining that there must be a distinction because these objects are not interdefinable. Or at least, this would not be legitimate without some kind of justification for applying the interdefinability criterion to the case of Newtonian space-time; and Glymour has not provided such a justification.

Glymour's discussion of synonymy presupposes that different theories can be empirically equivalent. As indicated in the first section of this paper, I agree that empirical equivalence is not a sufficient condition for synonymy. But I fail to see why our rejecting the view that empirical equivalence is a sufficient condition for synonymy should lead us to accept Glymour's interdefinability criterion as a necessary condition. Instead, I feel that an adequate set of criteria for deciding when two theories "say the same thing" should enable us to say that, in certain cases, theories which do not meet Glymour's criterion and which, therefore, are apparently incompatible actually represent different versions of a single theory.

REFERENCES

Earman, J. and Friedman, M. (1973), "The Meaning and Status of Newton's Law of Inertia," Philosophy of Science 40: 329-359.

Glymour, C. (1971), "Theoretical Realism and Theoretical Equivalence," in R. Buck, (ed.), Boston Studies in the Philosophy of Science, Vol. 8. Dordrecht: D. Reidel.

Glymour, C. (1975), "Relevant Evidence," Journal of Philosophy, 72: 403-426. Glymour, C. (1977), "The Epistemology of Geometry," Nous, 11: 227-251. Havas, P. (1964), "Four-Dimensional Formulations of Newtonian Mechanics and Their

Relation to the Special and the General Theory of Relativity," Reviews of Modern Physics 36: 938-965.

Misner, D. S., Thorne, K. S., and Wheeler, J. A. (1973), Gravitation. San Francisco: W. H. Freeman.

Sklar, L. (1976), "Inertia, Gravitation and Metaphysics," Philosophy of Science 43: 1-23.

Stein, H. (1967), "Newtonian Space-Time," Texas Quarterly 10: 174-199. Trautman, A. (1965), "Foundations and Current Problems of General Relativity,"

in S. Deser and K. W. Ford, (eds.), Lectures on General Relativity. Englewood Cliffs, New Jersey: Prentice-Hall.

Trautman, A. (1967), "Comparison of Newtonian and Relativistic Theories of Space- Time," in B. Hoffman, (ed.), Perspectives in Geometry and Relativity. Bloomington: Indiana University Press.

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Zaret, D. a Limited Conventionalist Critique of Newtonian Space-Time