The presentation of §1.1 showed that the most common sorts of spacetime theory represent spacetime by means of a 'bare' differentiable manifold M over which certain structures, called geometric-object fields, are defined. The types of geometric-object fields split into two distinct categories: the absolute objects (or what, following contemporary physicists' parlance, I have been calling "background structures") Bi and dynamical objects . The interpretation usually given to these objects (e.g. [Earman, 1989], p. 45) is that the background structures characterize the fixed structure of spacetime and the dynamical structures characterize the physical contents of spacetime—a common metaphor is that the background structures form the stage on which the actors (the dynamical fields) perform. A model of a spacetime theory is then given by M = (M, Bl, ) (where the left-to-right ordering of the model represents the fact that M can be viewed as 'prior' to Bi, and Bl as 'prior' to Di). What structures are placed on the manifold, what symmetries they admit, what laws they obey, and whether they are background or dynamical is what distinguishes the various types of spacetime.
A natural way to view these structures is as an assignment of geometrical properties to the points of M. The background structures are fixed across the models of the theory, so that the points retain any properties determined by these structures independently of whatever processes are going on there or elsewhere. Thus, one can consider a model with twice as much energy as another, and yet the background structure remains insensitive to this. The dynamical structure, on the other hand, is allowed to vary across models, so that the points (or their counterparts in other models) can possess different properties as determined by such structures depending upon what processes are going on there or at other points.52 Background independence is more than this though, it tells us that the values of the dynamical fields, but not the (absolute) background fields, have to be solved for using the theory's field equations.
This interpretation clearly suits the substantivalist, for the points of the manifold become bearers' of properties, and there is a sense in which these points exist independently of dynamical processes played out by material objects (as represented by one or more dynamical objects). However, the relationalist only deals
52 This already involves some questionable modal assumptions concerning the 'sameness' of points across models. It also involves a questionable assumption concerning identity and individuality since we are supposing that the points are the proper subjects of predication. These assumptions will become increasingly important, and will come under closer scrutiny as we progress.
in material objects and their relations, so, for her, both the background structures and the manifold (if it is ineliminable) have to be reconstructed from these raw materials alone.
So grounded, the modern debate between substantivalists and relationalists concerns the ontological status of the elements of M, interpreted as space(time) points, as structured by B1—these points are the most natural counterparts of Newton's parts of space in the context of modern spacetime theories. The sub-stantivalist will be committed to the points along with the structure they inherit from B1, the relationalist will want to claim that they are a fiction (albeit a useful one), and that all there really are are material objects and their properties entering into various relations that define the observed structure of space and time.
Newton is generally taken to have held a substantivalist position with regard to absolute space and time.53 The reason he believed in a substantivally conceived absolute space and time was because of the work that such an interpretation could do: it could provide a physical basis for inertial effects; this was something that was supposed to be impossible for the relationalist.54 I turn to this issue at the end of this subsection. First I consider the question of what structure this space is taken to possess. In the context of Newtonian mechanics, space is represented by a three dimensional manifold of points equipped with a fixed Euclidean metric determining their distance relations—with this metric functioning as a background structure—so that the structure of space is isomorphic to E3. Time is represented by a one dimensional manifold, and it too is equipped with a Euclidean metric, so that the structure of time is isomorphic to E1. Newtonian spacetime has the simple product (topological) structure M = E3 xE1, understood as describing an enduring 3-space according to which the points of space persist through time.55 We might then reasonably expect that the substantivalists attributions of robust existence concerns the points of this space and their properties and relations (cf. [Earman, 1989] §1.1, for such an account drawn from the Scholium). Relationalism will then involve, at the very least, an outright denial of the existence of space points, but may involve an agreement about the Euclidean structure of space—the structure will just be seen to be implemented by relations between material objects rather than between substantival points.
Ontological commitment to the points of this space, including the properties they inherit from the metric, is what underwrites the substantivalist's supposed commitment to inflation. Here is why: The points of the space are indiscernible; space is homogeneous and isotropic. The symmetries of the metric on space allow for a notion of invariance under global, rigid spatial translations and rotations. This means that if a system of matter is globally and rigidly translated some distance in space, so that the parts of the system have different absolute locations
53 Note that substantivalism does not imply absolutism. Absolutism, in this case, implies 'sameness across models', whereas substantivalism implies a denial of the material dependency thesis. Neither implies the other, so that the substan-tivalist might deny that space is absolute and yet still deny the material dependency thesis. This will become important in subsequent chapters.
54 However, Berkeley defended a plausible Machian-type line according to which matter, in the form of the fixed stars, was responsible for the inertial forces. See [Barbour, 2001] for an excellent account of this controversy, and others relating to the debate between absolute and relational theories of motion.
55 In fact, this is an oversimplification. Strictly speaking, a Newtonian spacetime has a well-defined notion of timeseparation, but not spatial separation (unless the time separation vanishes).
(i.e. occupy different spatial points), the observable properties of the system are unaffected. This is simply a statement of the invariance of the laws of Newtonian mechanics under spatial translations—we can run a similar line concerning rotations, and time-translations. Now let the system of matter be the total material content of the universe. Once again, the observable properties of the two systems are identical, only this time 'the system' comprises everything material that is contained in the universe! The only difference between the translated and untranslated system concerns the roles played by the points of space: e.g. in one system the center of mass is at the point x in the other it is at the point y = gx (where g is some symmetry transformation, a spatial translation say). Inflation arises if we say that the two cases generated by the symmetry represent physically distinct states of affairs. The substantivalist is supposedly bound to say this since (1) the points of space are real (and independent of matter), and (2) the system bears different relations to these points of space in the translated and untranslated scenarios. The relationalist can clearly deflate the number of worlds represented, since any worlds agreeing on their material objects and the relational structure these objects exemplify are identified, for this is all there is to a relationalist's world.56
The generation by a symmetry of qualitatively indistinguishable worlds that differ only with respect to which objects (here, points) play which role (here, 'location grounders') is what underlies the Leibniz-shift argument.57 It arises in the context of Newtonian mechanics because that theory is developed against the background of Euclidean space, and this space possesses a lot of symmetry. We can act on systems of matter with this symmetry to produce isomorphic states. Onto-logical commitment to the points of this space (and the matter) is supposed to lead the substantivalist into the jaws of inflationism for the reasons given above. The relationalist in not being ontologically committed to the points of space is supposedly thereby rendered immune from inflation, for the worlds match up on those properties that are relevant to the relationalist's conception of space and time.
The translation argument that I presented above is, of course, just a slim-line version of Leibniz's shift argument against Newton's brand of absolutist substan-tivalism.58 Leibniz accepts the above implication between substantivalism and inflation (as does Clarke), and argues that since inflation violates the principle of sufficient reason [PSR: nothing (contingent) happens without a reason why it is so rather than otherwise], substantivalism should be rejected. I repeat, then, that in
56 I will argue in §2.4, as has been argued by so-called 'sophisticated' substantivalists, that, in fact, a similar deflationary option is available to the substantivalists as well. I will also argue that relationalists are not necessarily committed to defla-tionism either: §2.4. What leads to these two possibilities concerns the modal semantics one combines one's ontological stance on spacetime with. I argue that nothing internal to the debate on spacetime ontology can decide which theory of modality to use.
57 Of course, not all symmetries lead to indistinguishable worlds. In general symmetries map states to physically distinct ones, leading to distinguishable worlds—i.e. they are symmetries of the theory rather than symmetries of worlds (cf. Ismael and van Fraassen, 2003]). The issue in those cases were we do have indistinguishable worlds is whether they are nonetheless physically distinct or not, despite their being qualitatively identical. If we say that the multiple states represent one and the same physical state (so that the symmetry does not map physical states to physical states) then we have an example of a gauge symmetry which signals a redundancy in the theory (i.e. in its description of configurations).
58 I present the real argument in detail in the next section. In the last section of this chapter, I present the mathematical guts of the argument and show how it connects to the general account of symmetry outlined in the previous chapter.
flation is the real target of the shift argument (and, we will see, the hole argument), and I will argue that inflation is not an integral part of substantivalism.59
One might be left wondering why anyone would want to be a substantivalist given the trouble it is supposed to generate. Why not avoid the trouble and be a relationalist? Newton had his reasons of course, and they were reasons of physics not philosophy or theology, as was often believed to be the case (cf. Reichenbach  for the canonical development of this 'anti-Newtonian' line). It wasn't until Stein  that the soundness of Newton's methodology became apparent. What Newton provided was essentially an 'inference to the best explanation' argument for the existence of a substantival space, what Teller calls "the argument from in-ertial effects" (, p. 369). Newton believed that an aggregate of genidentical points (or parts) of space was necessary to explain absolute motion, and that a notion of absolute motion was necessary to explain absolute acceleration; the latter is, moreover, an objective physical effect with observable consequences. For example, when an airplane accelerates to take off, the passengers will feel their seats being forced into their backs. Relationalism comes unstuck on such effects, since the loved ones of the passengers standing stationary in the airport will also be accelerating relative to the airplane: there is a symmetry between those on the airplane and those in the airport. Yet only those on the airplane feel the effects; there is also an asymmetry here that the relationalist apparently cannot accommodate.60 The substantivalist can, for according to her there is a fact of the matter about who is accelerating relative to absolute space: motion is, then, held to be relative to the 'fixed' points of space—from this, one gets absolute rest (location at the same point as time elapses), velocity (location at distinct points as time elapses), and acceleration (any motion distinct from the two previous forms). The connection to physics, then, is clear: Newton believed that substantivalism about absolute space and time was a necessary part of an empirically adequate dynamics. The inflationary aspects brought about by absolute position and velocity were an unfortunate, but unavoidable, consequence: the price to pay for absolute acceleration. As Belot explains, "Newton's laws demand a notion of absolute motion while at the same time implying that there exist states of absolute motion [positions and velocities] which are indistinguishable one from another" (, p. 564).
The shift to (non-dynamical) spacetimes that abolish absolute space and time alters the nature of the debate somewhat—but not so much that inflation no longer
59 I am certainly not the first to notice that substantivalism per se is not the real victim of Leibniz's shift argument and the hole argument. However, opinions differ as to exactly what the victim is. Hoefer  argues that assuming primitive identities for points causes the problem; Maidens  takes a similar line, though she simply stipulates her way out of the bind to inflation. Butterfield  and Maudlin  think that the particular brand of modal semantics of the arguments is the target and offer counterpart theory and metrical essentialism respectively as alternatives. Pooley [in press] and French  suggest that it is haecceitism that is the real target; Melia  too appears to adopt a similar line, though his concern is with the relevance of haecceitistic differences to questions of determinism. I examine these options in Chapter 5 when I consider the hole argument (the real context of these interpretive options)—these issues will be a running theme from hereon in.
60 Sklar (, pp. 230-1) suggests that relationalists might overcome this problem by appealing to the notion that absolute acceleration is a primitive property that certain relatively accelerating objects possess. This is, indeed, possible and consistent, but clearly ad hoc; one might question whether the increase in ontological economy brought about by the avoidance of substantival points of space and time is really worth the attendant decrease in ideological economy. As Teller says, "as food to satisfy one's appetite for a theoretical account, this move satisfies about as well as bread made from sawdust" (, p. 370). Another alternative is to argue along Machian lines that the acceleration is to be judged 'relative to the fixed stars'.
occurs. The most important difference is that the notion of spatial points retaining their identity over time no longer makes sense—there are simply spacetime points whose existence is ' fleeting': different spatial points at different times. Thus, we do away with the view attributed to Newton that spacetime is built out of a stack of instantaneous spaces of Euclidean structure, such that there is a definite way to re-identify spatial points over time.
In the case of neo-Newtonian' spacetime, the notion of absolute space is done away with; though absolute time (simultaneity) is retained. Hand in hand with the eradication of absolute space is the eradication of the observationally inert concepts of absolute position and velocity. Absolute acceleration is preserved by effectively encoding it in the structure of the spacetime. We can still consider 'translations' in neo-Newtonian spacetime, but rather than supposing that the instantaneous contents of space are translated, we must suppose that worldlines are translated. We simply view the history as taking place over different spacetime points. Special relativity simply extends the damage done by neo-Newtonian spacetime to cover absolute time as well. Once again, absolute acceleration is preserved through the use of inertial frames. And once again, we can consider worlds that differ only as to which points of spacetime are occupied.
With the eradication of background structures, like absolute space, there will be an attendant increase in the symmetry group of the theory, resulting above in the abolishment of absolute positions and velocities, but with absolute acceleration remaining. In both of the above cases, the metric still appears as a background structure. The translation argument can easily be reapplied in such contexts: simply consider translations of events on spacetime. The symmetries will always allow one to generate such possibilities, and will, therefore, allow for inflationary scenarios.
However, the situation changes markedly in general relativity, where the landscape of the substantivalism/relationalism debate alters significantly. There are two aspects that are especially problematic from the point of view of this debate. The first concerns the definitions of 'spacetime' and 'matter' in such a context.61 The second, related to the first, concerns the fact that the only background structure in general relativity is the 'bare' manifold itself: there are no background fields. The symmetry group of general relativity is, accordingly, larger still—the largest possible in fact. However, crucial for our purposes is the fact that the symmetries of general relativity nonetheless allow for the generation of new possibilities, though this time concerning the points of spacetime themselves, or the distribution of fields over these points—inflation arises from permutations of these points. We walk the new terrain in Chapter 4; for now let us stick to the relatively terra firma of the Leibniz-Clarke correspondence.
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