Space. Naturalists believe that the world is just what is in space and time, and having seen that we should, if possible, believe that substances endure through time, the next question is what we should believe about the nature of space.

Possible explanations. There are basically three alternatives: spatiomaterialism (the belief that space and matter are both substances), materialism (and the belief that space is just spatial relations), and spacetime substantivalism (the belief that the substance that exists in addition to matter is not space, but spacetime). Though the following argument would have to be reconsidered, of course, if a fourth alternative turns up that is simpler than all of these, that does not seem likely. After describing each of these alternatives, I will consider which offers the best ontological explanation of the natural world.

Spatiomaterialism. By "spatiomaterialism," I mean the belief that the substances constituting the world include space as well as matter. It postulates matter, because it assumes that there are substances in space that obey the basic laws of physics. But it also postulates space as a substance. ("Substantivalism" is the traditional name for the view that space is a substance, though as we shall see, substantivalism about space should be distinguished from substantivalism about spacetime, the kind of substantivalism that is taken seriously today.) Finally, spatiomaterialism assumes that the bits of matter are all contained by space in the sense that each of them coincides with some part(s) of space or other. That is how these two substances exist together as a world, and thus, it is the basic relationship that spatiomaterialism assumes as the other part of every ontological cause.

Spatiomaterialism assumes that space is a substance by our definition, for it assumes that each part of space has both the essential and the existential aspects of the nature of substance as substance. The parts of space are all the locations in a single, three dimensional space.

Each part of space has both aspects of the existential nature of a substance, temporality and particularity, because in addition to never coming into existence and never going out of existence, each location in space has an existence that is distinct from from all the other locations in space (not to mention from any bits of matter that may coincide with it). Though spatiomaterialism is compatible with both theories about the nature of time, we shall take it to include the endurance theory, for as we have just seen, the endurance theory is the best ontological explanation of the temporal aspect of substances. (Only the endurance theory is compatible with the present being different from the past and the future, and the perdurance theory even denies that change involves properties coming into existence and going out of existence.)

Though parts of space have the same kind of existential nature as bits of matter, parts of space have the opposite essential nature. Whereas bits of matter exist independently of one another in the sense that each could still exist, even if the other bits of matter did not exist, parts of space depend on one another in the sense that no part of space can exist without all the other parts of space. The essential nature of each part of space includes having geometrically coherent relations to every other part of space. That is, the essential nature of each part of space is defined by its location relative to all the other parts. Thus, parts of space exist only if space exists as a whole. (Indeed, it is the wholeness of space and what it contributes to the natural world that is the key to almost all the new necessary truths based on spatiomaterialism.)

In other words, space has an opposite essential nature from matter because its parts are not prior to the whole. Since each bit of matter is capable of existing independently of all the other bits of matter in the world, there is a sense in which the parts of matter are prior to the totality. But that is not true of space, because no part of space can exist without all the other parts of space. That does not mean, however, that, in the case of space, the whole is prior to the parts, because the whole of space cannot exist without all its parts. Space is whole in a unique way, as we shall see, and one indication of its uniqueness is the way that the parts of space and the whole are equally basic.

Spatiomaterialism assumes that space has a three dimensional Euclidean structure. Though non-Euclidean geometries can be described coherently, they are not as simple as Euclidean geometry. Euclidean geometry is assumed here, because, as it turns out, there is no reason to doubt that the simplest alternative is true.

To be sure Einstein's general theory of relativity implies that spacetime can be curved and can, therefore, be represented only by a non-Euclidean geometry. But what it implies is curved is not just space, but spacetime, and as we shall see (in Change: General theory of relativity), curved spacetime can be explained as an aspect of space as a substance with a Euclidean geometrical structure (basically by variations across space in the velocity of light relative to space).

The version of spatiomaterialism considered here will also assume that space is infinite.

The infinity of space will be assumed, because that is the simplest nature space can have. Even though the parts of space cannot exist without one another, they are distinct substances, and the essential nature of each particular spatial substance is necessarily unique in the sense that it involves a unique relationship to every other particular spatial substance. But the simplest assumption is that all the parts of space have the same kind of essential nature, that is, the same kind of relation to other parts of space as every other part of space. However, if the parts of space all have the same kind of essential nature, a Euclidean spatial substance must be infinite as a whole. For if there were an end to space, no two parts of space could have the same kind of essential nature. Each part would have a different relation to the edge of space (if makes sense at all to talk about an end to space). Thus, the simplest form of spatiomaterialism would hold that space is infinite.

To be sure, most astronomers and astrophysicists currently assume that space is finite, because its finitude is entailed by the use of the general theory of relativity to represent the big bang and the subsequent expansion of the universe. And it would be possible, if necessary, to formulate a version of spatiomaterialism in which space is finite. But it would be a more complex ontological cause than is assumed here, for its parts would have to have systematically different kinds of essential natures. And it may not turn out that the big bang theory is true, as argued in Change: Cosmology.

Notice, however, that the infinity of space is twofold. There are finite distances in space, for that is entailed by its having a geometrical structure, and there are opposite way ways in which it is possible to generate an infinite series. It is possible, in one way, to keep doubling its size, step by step, forever in the same direction, and it is also possible to keep dividing it in half, step by step, for ever. The former means that space is infinite in extent, whereas the latter means that space is continuous (or that parts of space are connected continuously). Both kinds of infinity are assumed here as simply part of the essential nature of space as a substance.

The continuousness of space means that the number of points on a finite line is greater than the number of whole numbers, which is already infinite. Thus, the points on a line are said to be nondenumerable. (It can be shown, furthermore, that there are just as many points on a finite line as there are points on a finite plane and as there are points in a finite volume.) This can seem puzzling, if it is assumed that lines are made up of points, because more than an infinite number of points would be required to make up a line. This may be problematic for mathematics, but not for ontological philosophy, because we do not assume that space is made up by combining points at all. What is essential about points in space is their distances (and directions) from one another (or the metric of their geometrical relations), not how many points there are in any finite distance. In other words, space is not made up of points in the way that ordinary material objects are made up of simpler bits of matter, that is, by assembling them alongside one another; indeed, according to spatiomaterialism, that way of putting bits of matter together as a whole is something that is possible only because the bits of matter all coincide with parts of space. Rather space is made up of points in the sense that the points all have as their essential natures determinate distances from one another in all three dimensions as parts of a single whole space. Indeed, points can be picked out at all only because the parts of space have such spatial relations to one another. That is, once again, the wholeness of space.

Time has a twofold infinite nature like space. Given any finite period of time, it may be doubled forever or divided forever. Thus, not only does time go on eternally, but it also flows continuously (that is, the moments in time are continuous with one another). And that is likewise simply the nature of time. (Two moments in time are related by the amount of time that passes between them, not the number of moments between them; the temporal metric is what makes it possible to pick out moments in time.) The direction of time, however, introduces an asymmetry that is not found in space, separating the issue about whether time extends infinitely toward the past from the issue about whether it extends infinitely toward the future. Though the big bang theory denies the former, it leaves open the possibility that the future may be infinite. We will, however, proceed on the assumption that time is infinite in both directions, postponing discussion of the big bang until Change: Cosmology

The belief that space is a substance may have been what the ancient atomists, Leucippus and Democritus, meant by insisting that the arche, or "first principle," includes two elements, both atoms and the void. In other words, the usual interpretation of atomism, mentioned in Ontology, may be mistaken. What they meant by the void may not have been merely a kind of stuff between atoms that is so subtle that, unlike other atoms, atoms can move through it without resistance. They may have meant that the void is something that exists not only in between atoms, but also underlies each and every atom. This way of thinking about the nature of space may have been obscured by the lack of any better way than "the void" to refer to what they meant. That is, arguably, one of interpreting the ancient atomists, which would make them the discoverers of the view that is assumed here.

Notice that it is not inconsistent to hold that space is contained by space or exists in space, though it holds for a different reason from matter. It is not inconsistent, because the parts of space are substances and each part of space is, indeed, is located in space or is contained by space in the sense that it has a location relative to all the other parts of space. Indeed, that is part of the essential nature of each part of space. Bits of matter are also contained by space or in space in the sense of having a location relative to other bits of matter. But according to spatiomaterialism, that is not part of the essential nature of matter. Nor is it simply how bits of matter exist together as a world. It is, rather, a result of each bit of matter coinciding with some part(s) of space. It is space that gives bits of matter their spatial relations to one another. Given that space and matter are both ontological causes, the ontological cause of bits of matter all having spatial relations to one another is the basic relationship by which the two opposite substances exist together as a world. It is because the parts of space are contained by space that the bits of matter coinciding with any part are contained by space.

Spatial relationism. By "spatial relationism," I mean the belief that matter is the only kind of basic substance in the world and that space is nothing but the relations that hold among bits of matter. Matter is assumed to have the essential nature described by the basic laws of physics, and spatial relations can be explained in one way or another as how bits of matter exist together as a world. And we will take spatial relationism to include the endurance theory of time (as we did spatiomaterialism), since that is the best explanation and spatial relationism is compatible with it.

Spatial relationism is basically a negative thesis. It is the denial that space is a substance distinct from the material substances in the world. It denies that space exists independently of matter by holding that spatial relations have the same status as properties of matter.

It is not necessary to postulate any substances in addition to matter in order to account for spatial relations, any more than it is necessary to postulate additional substances to account for the properties of material substance. They can be understood as nothing but aspects of the material substances postulated. Just as (monadic) properties are aspects of the substances themselves taken separately, the relations among them are aspects of their existence together as a world.

There are subtly different versions of spatial relationism, as mentioned in Ontology: Nature of relations, depending on this ontology explains the possibility change in spatial relations. How the substances postulated exist together as a world is the most basic relationship that an ontology must assume in addition to the substances to explain how they exist together as a world. That basic relationship determines how substances can be combined and recombined as time passes in order to explain ontologically the diversity and change in nature. Thus, if spatial relations are nothing but how material substances exist together as a world, their basic relationship involves change. That is possible, because the basic relationship among the material substances can be simply having spatial relations of some kind or other, not having any particular spatial relations. That basic relationship does not change even when the particular spatial relations among material substances are changing. But since particular spatial relations do change, there must be some way to explain the possibility of such change. (And since spatial relations change in regular ways, it must also be able to account for those regularities.)

It is possible to hold, on the one hand, that the basic relationship by which material objects exist together as a world has a temporally complex nature. "Having spatial relations" would be defined by describing the regularities in how the spatial relations between material objects change, for example, that they change only continuously over time, that is, by motion. (Material objects do not flit about discontinuously form one place to another.) This would be to define the essential nature of the basic relationship among material substances dispositionally, much as the essential natures of material substances are defined dispositionally when they are assumed to obey the basic laws of physics, that is, in terms of the regularities in how they move and interact.

On the other hand, it may be possible to hold that the basic relationship is temporally simple. "Having spatial relations" would be understood merely as a kind of condition that holds among material objects at each moment as it is present, and the change that occurs in spatial relations would be a result of the essential natures of the material substances. That is, the essential natures of the material substances would be temporally complex, as when they are defined in terms of the basic laws of physics, and the ways in which spatial relations change over time would simply be a consequence of how the basic laws of physics work out.

There are problems with this view, however. One is that the laws of contemporary physics include quantum mechanics, and since they do not describe a fully deterministic process, spatial relations cannot be fully determined by the basic laws of physics (unless there is a so-called hidden variable involved).

But even if the laws of physics were deterministic, there is another problem, for the laws of physics can determine the spatial relations that hold of bits of matter at any one moment only if their spatial relations at some other moment is given. Since the past determines the future, the particular spatial relations may have to be fixed for some earlier point in the history of the universe, presumably at the beginning of the world, such as the Big Bang or when God created it. In any case, the basic relationship would not be temporally simple after all, for having spatial relations would not be a condition that holds only at the present moment, but must include all the particular spatial relations that hold at some other moment in the history of the world.

In either case, however, spatial relationism holds that space is nothing but spatial relations, where those relations are, ontologically, just the basic relationship by which material objects exist together as a world, that is, ultimately, an aspect of the material substances themselves.

Leibniz denied that space is a substance. (And he debated the issue with the Newtonian, Samuel Clarke. See Earman.) But spatial relationism as defined here should be distinguished from the kind of spatial relationism entailed by Leibniz's ontology. Leibniz did not take spatial relations to be how the basic substances exist together as a natural world. The substances Leibniz postulated to explain the natural world were monads, or minds of various kinds, and he explained spatial relations as how monads appear to one another, that is, as ideas in those minds. The way that monads existed together as a natural world was by each being created by God, though Leibniz did hold that the appearances of spatial relations in all the different monads fit together coherently as a natural world.

In any case, defenders of Newton were never able to refute spatial spatial relationism, because they assumed that the only way to prove that space is a substance is by empirical science, that is, by confirming some crucial prediction. Even Newton's theory did not provide any way to measure absolute rest or motion, and all the same phenomena (including the famous rotating bucket; see Earman, pp. 61-90) could also be explained on the assumption that space is nothing but spatial relations among material substances (by taking into account spatial relations to distant stars).

According to Ryansiewicz, the classical debate between spatial relationism and substantivalism about space is no longer meaningful in the context of contemporary physics. But that position is compellingly refuted by Hoefer98.

Spatiotemporalism. Spatiotemporalism agrees with spatiomaterialism that the spatial relations among bits of matter depend on another substance, in addition to matter. What makes it different from spatiomaterialism is that it takes that substance to be spacetime, rather than space. In fact, that makes it fundamentally different from both other theories. Spatial relationism (or materialism) and spatiomaterialism can both accept the endurance theory of time (and both do, as we they have been defined here). But the belief in spacetime as an ontological explanation of the world entails the perdurance theory of time. That is what it means, when speaking ontologically, to deny that space and time are absolute. Though this view is usually called "substantivalism about spacetime," I will call it "spatiotemporalism" in order to bring out the contrast with spatiomaterialism and spatial relationism.

This preference for spatiotemporalism over spatiomaterialism is justified by Einstein’s relativity theories. Minkowski introduced the notion of spacetime in 1908 as a way of summing up what Einstein’s 1905 special theory of relativity implied about the world, and he predicted that it was the beginning of the recognition that space and time are not independent of one another. Einstein then took spacetime as basic in constructing his general theory of relativity, that is, in his explanation of gravitation as a result of a curvature imposed on spacetime by large accumulations of matter in it. And since spacetime must be a substance in order for it to interact with matter in that way, it is now common for philosophers of spacetime to hold that spacetime is the opposite kind of substance that exists in addition to matter and explains why bits of matter have spatial relations. (See Friedman and Earman, for example.)

The basic nature of spacetime is determined by Einstein's special theory of relativity. Einstein called it a theory of "relativity," because it holds that the places and times at which events occur depend on the inertial frame of reference from which they are observed. Different inertial frames have different velocities, and according to Einstein's special theory, they assign different spatial and temporal coordinates to events in the universe. For example, observers on two different inertial frames that happen to be located at the same point at the same time will have different views about which events in the histories of distant objects are occurring at the same time their paths cross. Einstein's special theory holds that their views are equally true, and that implies that the distant objects actually exist now at both moments in their histories. (With additional inertial observers, this means that the distant objects must exist equally at all the moments in their histories that can be said to be simultaneous with different inertial observers here and now). This loss of agreement about the simultaneity of events at a distance might seem to be just a theoretical problem about the nature of objects at a distance, but since Einstein's special theory holds that all possible inertial frames are equivalent, it has the same implications for inertial observers here and now. That is, observers on different inertial frames observing us from a distance would similarly disagree about which moment in our history is simultaneous with their present moment, and thus, in order for all their views to be true, we must actually exist equally at different moments in our history, indeed, equally at all of them. This is to deny presentism, because it forces us to believe that our past and our future exist in the same way as the present.

It is the loss of agreement about simultaneity at a distance that makes the belief in spacetime so problematic. To be sure, no problems arise for physics, because it is always possible to predict from one inertial frame what observers on all the others will say. But when spacetime is understood ontologically, that is, as describing the basic nature of space and time, the denial of simultaneity at a distance entails the perdurance theory of time. What really exists are not substances with spatial relations enduring through time, but a kind of eternal, unchanging four-dimensional world whose parts are spread out not only in the spatial dimension, but also in the temporal dimension. Since the world is constituted by all its parts, different moments in the history of each permanent substance are different parts of the world in exactly the same sense that different permanent substances (including different locations in space) are different parts of the world.

The substantival nature of spacetime is entailed, at least for scientific realists, by the interaction between its curvature and matter that explains gravitation according to Einstein's general theory of relativity. Spacetime could not be what causes material objects to exhibit gravitational acceleration unless it were something that exists in addition to those material objects. This ontological interpretation of spacetime, or substantivalism about spacetime, is what I mean by "spatiotemporalism." It holds that time is ontologically on a par with space (by contrast to spatiomaterialism, which holds that matter is ontologically on a par with space). That is the perdurance theory of time. To hold that time is just another dimension relating parts of substances geometrically is to hold that just as different places in space all exist in the same way, so different moments in time all exist in the same way. This implies that ordinary, permanent substances (that is, substances with a temporal aspect to their existential aspect as substances) do not endure through time, but perdure over time.

The best ontological explanation of space. If we follow the empirical method, we will believe the theory about the nature of space that provides the best ontological explanation of the natural world. That is clearly spatiomaterialism, if it is possible (in particular, not falsified by the any phenomena covered by contemporary physics), because it is better than spatial relationism and better than spatiotemporalism. And since, as I will show, physics does not make it impossible, naturalists who follow the empirical method in deciding which ontology to believe will accept spatiomaterialism.

Spatiomaterialism is better than spatial relationism. It may seem at first that spatial relationism is a better explanation of space than spatiomaterialism, because it postulates only one kind of basic substance, rather than two. Spatial relationism is basically just a kind of materialism that has made its beliefs about space explicit, whereas spatiomaterialism holds that space is a substance existing independently of matter which contains all the bits of matter in the world. On grounds of simplicity, therefore, it might seem that we should prefer spatial relationism to spatiomaterialism.

Simplicity is not, however, the only empirical grounds for preferring one theory over another. There is also the criterion of scope, and by it, spatiomaterialism is clearly superior. Thus, if spatial relationism were simpler than spatiomaterialism, there would be a trade-off between them which keeps the empirical method from choosing between them.

But that standoff is not the final word, because when we look closer at the criterion of simplicity, we will find that spatiomaterialism is at least as simple as spatial relationism, if not simpler. Simplicity is not a simple criterion, for there are two ways in which ontological explanations can be simpler (not only by the number of ontological causes, but also their natures), and by one of them, spatiomaterialism is clearly simpler than spatial relationism. There is, therefore, a standoff on grounds of simplicity, and that makes the criterion of greater scope decisive. Empirical ontologists should prefer spatiomaterialism.

But a decision in favor of spatiomaterialism is even more clear-cut than this may make it seem, for the way in which spatiomaterialism is simpler also reveals another way in which it has a greater scope. In the end, there is no doubt that spatiomaterialism explains more with less. The empirical method will require ontologists, therefore, to prefer spatiomaterialism over spatial relationism. Let us begin by considering the issue about the scope of explanation.

Scope. Spatiomaterialism explains more about the natural world than spatial relationism, because it explains why bits of matter have spatial relations, whereas spatial relationism merely assumes that they do.

Space is a substance with an opposite essential nature from matter, and so if it contains all the bits of matter in the sense that each bit of matter coincides with some part of space or other, the bits of matter acquire their spatial relations from the spatial relations that already hold among the parts of space with which they coincide. That is, space and matter work together as ontological causes to produce spatial relations. Having spatial relations is not just an ontological assumption about bits of matter, since three different ontological assumptions are needed to explain it.

By contrast, spatial relationism does simply assume that bits of matter have spatial relations. To be sure, materialism can "account for" spatial relations; the fact that bits of matter have spatial relations does not show that materialism is false (as presentism and the fact of real change falsify the perdurance theory of time). But that is not to explain why bits of matter have spatial relations. Thus, since we are seeking the best explanation, we must prefer the theory which explains more.

Materialists may demur by insisting that postulating space to explain spatial relations is ad hoc and, thus, not an explanation at all. Though it may appear to be an explanation, they will hold that substantivalism about space is equivalent to assuming that bits of matter have spatial relations. Indeed, it is the same assumption that spatial relationists make, except for being disguised as a substance. Substantivalism about space merely reifies spatial relations.

This objection will not stand, however, because spatial relations are not all that substantivalism about space explains ontologically. It also explains, together with matter, the possibility of change (as well as certain ontologically necessary truths about how bits of matter change, as we shall see later).

The assumption that all the bits of matter are contained by space as a substance implies only that each bit of matter coincides with some part of space or other. But that leaves open which place it is. Moreover, parts of space are connected with one another continuously, so that there are no gaps, so to speak. That is just the essential nature that spatiomaterialism takes space to have. Thus, as space and matter endure through time, it is possible for spatial relations among bits of matter to change, because bits of matter can move across space over time without giving up any of spatiomaterialism's assumptions. The continuousness of time works together with the continuousness of space to make motion possible. Neither space nor matter changes their essential natures, and the bits of matter are always contained by space, always deriving their spatial relations from space. (This way of explaining motion ontologically also implies that motion is the only way that bits of matter can change their spatial relations as time passes. See in Local Regularities under Change.)

Since substantivalism about space explains something more than just why bits of matter have spatial relations, it is not ad hoc, but genuinely explanatory. Spatial relations are only one of several basic phenomena explained by spatiomaterialism.

This is to explain the possibility of change by motion, but it is also possible for spatiomaterialism to explain the possibility of change in another way: by interaction. That is how motion changes, as we shall see. Being in space, bits of matter can move to the same location, and when they do, they are in a position to act on one another, because they not separated from one another by space. (The impossibility of action at a distance -- that is, with spatial substances separating the bits of matter -- is also shown in Local Regularities under Change.)

Thus, since spatiomaterialism can explain the possibility of both motion and interaction, that is, both kinds of change described by the laws of physics, its explanation of spatial relations is not ad hoc. In other words, the greater scope of spatiomaterialism is shown by its ontological explanation of at least three basic facts about the natural world that are just assumptions of spatial relationism -- that bits of matter have spatial relations, that they can change by motion, and that their motion (and other properties) can change by interaction.

This may not be an original argument for substantivalism about space. The way in which space makes motion and interaction possible may have been what Leucippus and Democritus were getting at when they insisted on postulating the void as an element along with the atoms, though that is still a controversial interpretation of ancient atomism.

Simplicity. Since empirically minded ontologists must prefer ontologies that explain more, they have a good reason to prefer spatiomaterialism over spatial relationism. But materialists might hope for a standoff between these two ontologies at this point. The empirical method might fail to choose between them. Although the criterion of greater scope favors spatiomaterialism, the criterion of simplicity may favor spatial relationism, because spatial relationism postulates only one kind of basic substance, not two. Simplicity is not, however, a simple criterion, and when we consider both ways in which explanations can be simple, we will see that there is also a way in which spatiomaterialism is simpler than spatial relationism.

Such a standoff on grounds of simplicity would force ontologists to prefer spatiomaterialism, because they are equal except for the greater scope of the latter. But the empirical method is even more decisive than this suggests, because the way in which spatiomaterialism is simpler is another way in which spatiomaterialism explains more than spatial relationism. Thus, it will be clear in the end that, even though spatiomaterialism postulates two basic substances, rather than one, it explains more with less.

The reason that materialism is not necessarily simpler than spatiomaterialism is that simplicity is judged not only by the number of basic ontological causes, but also by the simplicity of their natures. That is, materialism may not be simpler than spatiomaterialism, even though it postulates only one kind of basic substance, because it may require either matter or the basic relationship among them to have a more complex essential nature than spatiomaterialism. This is how it will turn out, and in order to see why, let us look more closely at the basic relationship assumed by materialist ontology.

The basic relationship among bits of matter, according to spatial relationism, is that bits of matter all have spatial relations with one another. But we are assuming that they are all geometrically coherent, that is, that their spatial relations all fit together as parts of a three dimensional world. That assumption about their basic relationship can be understood in two different ways, and together they pose a dilemma for spatial relationism, for both have consequences that make spatial relationism more complex than spatiomaterialism.

"Having coherent spatial relations" may be taken as an aspect of the spatial relations that all the particular bits of matter have at the present moment, or it may be taken as an aspect of their particular spatial relations at every moment in the history of the world. In the first case, materialism must explain why bits of matter have coherent spatial relations at every moment, and the only way of doing so undermines the way that physical explanations are ordinarily understood. In the second case, the basic relationship of materialism must have a temporally complex nature, for its essential nature must include a fact about the world that spatiomaterialism explains by ontological causes that are temporally simple. Let us consider each horn of this dilemma in turn.

Notice, by the way, that in both ontologies, the basic relationship is not the particular spatial relations that bits of matter actually have, but an aspect of those particular spatial relations. For materialism, it is the geometrical coherence of those spatial relations, whereas for spatiomaterialism, it is that those spatial relations come from bits of matter coinciding with parts of space. The particular spatial relations that bits of matter actually have are taken by both theories to be something that can be known only by experience of the world.

Temporally simple basic relationship. The basic relationship assumed by spatiomaterialism is temporally simple. It assumes that the way that space and matter exist together as a world right now is by each bit of matter coinciding with some part of space or other, and that is temporally simple, for its two basic substances can have that relationship completely at one moment in the existence of the world. And this assumption needs to be made only about the present moment, because if all the bits of matter are contained by space at the present moment, then the fact that substances endure through time, never coming into existence and never going out of existence as time passes, entails that they have the basic relationship at every other moment. There is no way for a bit of matter to escape from space altogether, for it exists now as part of the same world by coinciding with some part of space or other and space endures through time along with matter. For the same reason, it could not get into space, if the bit of matter did not already coincide with some part of space or other.

The basic relationship assumed by spatial relationism can also be temporally simple. It is the assumption that bits of matter have coherent spatial relations, and that relationship can hold completely at any moment in the history of the world. But unlike spatiomaterialism, if that basic relationship is assumed to hold at the present moment, it does not necessarily hold at all other moments in the history of the world. It is conceivable that bits of matter would move and interact in ways that would give them spatial relations that are not geometrically coherent. (Similarly, it is conceivable that the present spatial relations are a result of the motion and interaction of bits of matter from earlier states in which their spatial relations were geometrically incoherent.)

It is conceivable, for example, that when matter of a certain kind is given the shape of a cave, the cave from inside is larger than the cave from the outside. That is, when measuring rods are taken inside the cave and used to measure how large the cave is, the internal distances measured turn out to be greater than the size of the cave when it is measured from the outside.

It might be argued that such spatial relations are not geometrically incoherent, but merely show a curvature about space. They are only incoherent according to Euclidean geometry. But postulating non-Euclidean geometry will not always preserve geometrical coherence. For example, suppose that when matter of a certain kind was shaped into a cave and extended into a tunnel, another entrance cut in the distant wall of the cave would open up in some far distant location in three dimensional space.

Topology explores many such possible connections among regions of spatial relations, and we need only think of them as being the effect of shaping matter in certain ways in order to conceive how the motion and interaction of bits of matter could lead to their having incoherent spatial relations.

In fact, spatial relations do not become geometrically incoherent in such ways, and so having geometrically coherent spatial relations at every moment is a basic aspect of the world than an ontology needs to explain. Now, spatial relationists may insist that they can explain this aspect of the world by the essential nature of matter. They assume that the essential nature of matter is defined by the basic laws of physics, and so they can argue that, if spatial relations are geometrically coherent at the present moment, they will be geometrically coherent at all moments in the future (and in the past), because those future (and past) spatial relations are determined by those bits of matter moving and interacting according to the basic laws of physics. Their geometrical coherence is, in other words, a consequence of the nature of matter. The reason that spatial relations will be geometrically coherent at other times, if they are geometrically coherent now, is that it is physically necessary.

This possibility is suggested by the structure of explanations in physics. As explicated by the deductive-nomological model, the basic laws of physics together with initial and boundary conditions make it possible to predict (or retrodict) any future (or past) state. Thus, if we take momentum to be a property of the bits of matter, the particular spatial relations among bits of matter at any one moment will determine their spatial relations at any other moment. Hence, they will be coherent at every moment, if spatial relations are coherent now.

Notice that this way of explaining why spatial relations are geometrically coherent depends on complete determinism, such as was assumed in Newtonian physics. (It was Laplace who first argued that the laws of Newtonian physics made this possible.) It is not, however, compatible with quantum mechanics, if Heisenberg's principle is taken to represent an indeterminism about what happens in the world as bits of matter move and interact, for such indeterminism would leave plenty of room for bits of matter to acquire incoherent spatial relations. It is, of course, possible that Heisenberg's principle represents merely an inevitable incompleteness about physical theory. There could be a hidden variable that makes physical processes deterministic, though it cannot be measured. But most naturalists would be surprised to find that they are committed ontologically to such an interpretation of quantum mechanics by their acceptance of materialism (that is, reducing space to spatial relations).

Even if physical laws are deterministic enough to explain why bits of matter always have coherent spatial relations, there is an intolerable cost to be paid in our understanding of how physical processes take place. Physicists ordinarily think of what happens in nature as a result of how bits of matter move and interact in space, where their spatial relations are one factor that combines with their motion and the forces they exert as a different kind of factor to determine what happens to them. But that is not possible, if the regularities described by laws of physics are what make spatial relations coherent in the first place, for then there is no way to explain how spatial relations work together with motion and forces as different kinds of efficient causes. Both kinds of factors are simply contingent aspects of bits of matter, and ontologically, they have the same status.

When material objects exert forces on one another, for example, when a planet exerts a gravitational force on a material object near its surface, we naturally think of the acceleration of the object as having two kinds of efficient causes: its distance from the planet and the force exerted by the planet. If object were released farther away, the same force would give it more momentum before colliding with the planet. And if the force were greater where it was released, the object would also have more momentum before colliding. We think of spatial relations and forces as two different kinds of efficient causes determining the later state. But if the basic laws of physics are supposed to explain why bits of matter have coherent spatial relations in the future, there is no way to distinguish the effect of forces from the effect of spatial relations. Instead, laws of physics must be seen as operating on the spatial relations, velocities, and forces that exist at one time to determine new spatial relations, velocities, and forces at a later (or earlier) time. Though what happens is predictable, the cause is not the planet's gravitation force acting on the object across space, because there is no way to distinguish the effect of the space from the effect of the force. Both depend on the regularities described by the laws of physics in the same way.

Likewise for motion. When a material object has a certain velocity, we think of its future spatial relations to stationary material objects as a result of its motion through a space that is already there. Its momentum is something that the object has, and we ordinarily see its future locations as being determined by its momentum together with its location in a space that is somehow independent of it. But that way of conceiving physical causes must be given up, if the conservation of momentum helps cause bits of matter to have coherent spatial relations. The future spatial relations are not caused by moving through space. They are caused by its motion and its past spatial relations, changing according to a basic law of physics which is taken as defining the nature of matter.

This form of spatial relationism makes almost everything that happens in the world depend on the nature of matter, rather than on what it assumes about the nature of spatial relations (namely, that they are geometrically coherent at the present moment). . This can also be seen how materialism explains other aspects of motion and interaction that spatiomaterialism explains by substantival space. Whether or not it is ontologically necessary, it is true that bits of matter do not change spatial relations by flitting about from place to place discontinuously, but only by moving across space as time passes. Spatial relationists would deny that this depends in any way on the nature of spatial relations. They would explain this regularity as just another aspect of the regularities described by the basic laws of physics, which define the nature of matter. The same holds for the materialists' explanation of why bits of matter do not act one one another at a distance.

In other words, to assume that having coherent spatial relations is a basic relationship that holds only at the present moment is to assume, in effect, that matter has an essential nature that is more complex temporally than the matter assumed by spatiomaterialism. Materialists have to attribute more aspects of what happens in the world to the nature of material substance as an ontological cause than do spatiomaterialists. The greater complexity of the essential nature of matter is what contradicts the claim that materialism is simpler than spatiomaterialism.

Temporally complex basic relationship. Instead of making the coherence of spatial relations a consequence of the basic laws of physics, materialists can take the basic relationship by which bits of matter exist together as a world to be having geometrically coherent spatial relations at every moment. This would be to postulate a basic relationship with a temporally complex nature, for the basic relationship would have to work together with the forces described by the laws of physics as another efficient cause determining what happens. And the basic relationship would have to work together with velocity as a second efficient cause determining its future spatial relations. (This form of materialism could also use its basic relationship to explain why bits of matter change spatial relations only by motion and why they do not interact at a distance.)

Though this would allow materialists to interpret the laws of physics as descriptions of how bits of matter move and interact in space, it would be to assume that the basic relationship does for materialism what substantival space does for spatiomaterialism. The materialists' basic relationship would not be simply how bits of matter exist together at the present moment, but a way of existing together at the present moment that also constrains how they can exist together at future (or past) moments in a way that is independent of the constraints imposed by their forces and velocities. Since that is to assume that the basic relationship entails that particular spatial relations can change only in certain ways, it would be to assume that the basic relationship has a temporally complex nature.

But that makes spatial relationism more complex than spatiomaterialism. The materialists' basic relationship would be replacing both space and the basic relationship of spatiomaterialism. Though one assumption is replacing two assumptions, materialism is arguably more complex than spatiomaterialism, because its one assumption has a temporally complex nature, whereas both of the spatiomaterialists' assumptions are temporally simple. That is, aspects of regularities about change that are merely assumed by spatial relationism are explained ontologically by spatiomaterialism, including not only that bits of matter have geometrically coherent spatial relations at every moment, but that they change spatial relations only by motion and that they do not interact at a distance.

Furthermore, it might be argued that, if materialism builds these regularities about change into its basic assumption about how bits of matter exist together as a world, it is violating the spirit of ontological explanation. Ontology tries to explain basic aspects of the world by showing how they are constituted by substances that endure through time with an unchanging nature. Since the basic relationship does not endure through time on its own like a substance, but is merely how the substances exist together as a world, it cannot be a source of regularities about change in the same way as substances can. Thus, not only does spatial relationism fail to explain ontologically why bits of matter always have coherent spatial relations, it also fails to account for them in the way expected of an ontology. In short, its need to postulate a basic relationship with a temporally complex nature is itself a reason for rejecting an ontology.

Although materialism seems to be simpler than spatiomaterialism, therefore, there is, in either case, a way in which it is more complex than spatiomaterialism. Either its assumption about the essential nature of matter is more complex, or its assumption about the basic relationship by which material substances exist together as a world is more complex.

Thus, there is, at least, a standoff between spatial relationism and spatiomaterialism on grounds of simplicity. And that means that spatiomaterialism is favored by the empirical method, since spatiomaterialism has a greater scope (explaining the possibility of change by motion and by interaction).

Furthermore, the way in which spatiomaterialism is simpler than spatial relationism also a way in which it explains aspects of the world that spatial relationism can only assume. Spatiomaterialism can explain ontologically why spatial relations are always geometrically coherent (not just now, but in the future and past), whereas materialism must build that assumption either into the nature of the matter it postulates or into the nature of the basic relationship it assumes bits of matter have.

Though spatiomaterialism postulates two basic substances, rather than just one, its ontological causes are simpler than those of spatial relationism. But since both ontologies account for the same basic facts, that means that spatiomaterialism explains more with less. At the outset, we saw the greater scope of spatiomaterialism in its ontological explanation of why bits of matter have spatial relations and how change is possible (not to mention what will be show later, that they change spatial relations only by motion and do not interact at a distance). But in showing that spatiomaterialism is simpler than spatial relationism, despite initial impressions to the contrary, we have seen that its ontological causes explain another aspect of the world that materialism can only assume, namely, why bits of matter always have coherent spatial relations. In the end, therefore, it is how spatiomaterialism explain more less than makes the decision in favor of spatiomaterialism clear, at least, for naturalists who accept the empirical method.

Spatiomaterialism is better than spatiotemporalism. Substantivalism about spacetime entails, as we have seen, the perdurance theory about the temporal existential aspect of substances. But since we have already seen that the empirical method in ontology requires us to prefer the endurance theory to the perdurance theory of time, we ought to believe either spatial relationism or spatiomaterialism rather than spatiotemporalism. Both allow us to accept the endurance theory (thereby giving us an explanation of why the present is different from the past and future and allowing us to believe that change is real in the sense of properties coming into existence and going out of existence as time passes). But having established that the empirical method prefers spatiomaterialism to spatial relationism (that is, to materialism), we must conclude that spatiomaterialism is the best ontological explanation of the natural world (assuming, of course, that there is no fourth theory that is still better than spatiomaterialism).

Defenders of spatiotemporalism will, however, object to this conclusion. They believe they are forced to accept spatiotemporalism by contemporary physics. Einstein’s discovery of the special and general theories of relativity was a revolution that led to the overthrow of the Newtonian belief in absolute space and time in physics. It is clear that any ontology that holds that material substances endure through time entails that space and time are absolute. To hold that the substances constituting the world always exist at only one moment in their histories is to hold that they all exist at the same moment, for they are part of a single world and they must exist together to be parts of the same world. Thus, if there are substances with spatial relations to one another, the spatial relations they have at the present moment hold for every possible observer. This is even clearer, if space is also a substance, for in that case the spatial relations are all constituted by a substance that exists only at the present moment. Since that is to believe that space and time are absolute, to choose to believe spatiomaterialism, or for that matter spatial relationism, would be a counterrevolution in physics. Thus, it is not likely to attract many followers among physicists and philosophers of science.

What led physics to reject Newtonian absolute space and time in favor of the spacetime of Einstein’s relativity theories was the empirical method of science. Physicists were merely inferring to the best explanation of what they could observe about the world. The special and general theories both predicted quantitatively precise measurements that were not expected by classical Newtonian physics, and they have been confirmed repeatedly. Nor does anyone dispute the mathematical simplicity of Einstein’s theories. The special theory was a paragon of simplicity by comparison with the cobbling together of ad hoc constraints by which Lorentz had proposed to explain the same phenomena. The general theory of relativity was based on the special theory, and though its mathematics was novel in physics, there is no question about its elegance. These two theories were clearly the best explanation that physics offered of the space and time found in the natural world, and that caused a revolution in physics, because neither theory had any use for absolute space or absolute time. All that was required for them to be true was spacetime, that is, the ontological equality of all points in space and time.[1]

The empirical method in science is, however, different from the empirical method in ontology, and thus, what is the best scientific explanation may not be the best ontological explanation. Science tries to explain what happens, and thus, it infers to the best efficient-cause explanation of what can be observed. Its criteria of truth are prediction and control. But there is, as we have seen, a difference between efficient-cause and ontological-cause explanations. Ontology tries to explain everything in the world, not only what happens there, but also what exists there — including the properties and relations of the objects found in the world, and how it is possible for anything to happen in the first place. Such things are explained ontologically by showing how they are constituted by basic substances and relations among them. Thus, empirical ontology tries to explain the world most completely using the fewest and simplest substances with the fewest and simplest relations. Though the goal of explaining the most with the least is the same, the kinds of explanation involved are different.

Since we know on empirical grounds, that spatiomaterialism is the best ontological explanation of the world, empirically minded ontological naturalists must prefer it to spatiotemporalism, if it is possible. Thus, the only relevant question is whether it is possible that spatiomaterialism is true, given that Einstein’s special and general theories of relativity are the best efficient-cause explanations of all the relevant phenomena.

The answer is Yes. It is possible to explain all the observational predictions of what will happen that is entailed by either the special or the general theory of relativity on the assumption that space is a substance enduring through time and, thus, absolute. To be sure, spatiomaterialism must make certain additional assumptions about the nature of space and matter and how they interact, which are, in effect, new laws of nature. But it is possible. (And the fact that spatiomaterialism is able to explain the truth of Einstein's two theories is further reason for preferring it over spatial relationism, because spatial relationism cannot explain them. It can only assume them in the same way it does the geometrical coherence of spatial relations.)

The spatiomaterialist interpretation of Einstein’s special and general theories of relativity is given a detailed defense below (in Change: Special theory of relativity and Change: General theory of relativity), as one of the implications of spatiomaterialism for physics. But we can see the possibility of such an interpretation in the abstract, and since this may seem unlikely to some, let me sketch briefly just how the truth of Einsteinian relativity will be explained ontologically by spatiomaterialism.

Given the endurance theory of time, what substantivalism about space implies is that only one moment in the history of each location in absolute space exists. That is the present moment, and it is the same for all of them, since the parts of space are all parts of the same world. Thus, all that a spatiomaterialist interpretation requires is that each and every part of space (along with the bits of matter coinciding with parts of it) be in a state at the present moment that is consistent with Einstein’s two theories. What that means is that, among all the possible inertial frames, which relativity takes to be equivalent, one, and only one, is true. This is not to say that it is possible to determine by some measurement which one it is. That is clearly precluded by Einstein’s theories; if it weren't, they could not be called relativity theories. But it is equally clear that there can be an inertial frame at absolute rest, even though it is not possible to detect which one it is.

What makes such an easy accommodation possible is that empirical science and empirical ontology have different criteria of truth. Since the empirical method of science seeks the best efficient-cause explanation of what happens in the world, its criterion of truth depends on predicting and controlling what happens, and thus, given that inertial frames are all equivalent in that regard, it can take the truth to be what is the same for all of them. In ontology, however, the empirical method seeks the best ontological-cause explanation of what exists in the world. Its criterion of truth is the simplest substances and relations that will explain everything in and about the world, and thus, it must explain how all the different inertial frames could be part of the same world. That is something that science can take for granted, because one observer can always predict what coordinates will be assigned by other observers. And since the reasons for believing that there is an absolute frame of reference are ontological, the lack of any difference in the predictions made from different inertial frames is not a reason to doubt that it exists.

The detailed spatiomaterialist explanation of the two relativity theories shows, from the point of view of the inertial frame at rest in absolute space (whichever one that is), how it is possible that all the other inertial frames are observationally indistinguishable from it. Here is the gist of the explanations given in CHANGE.

The special theory of relativity implies that the various possible inertial frames (that is, the various possible unaccelerated material objects that might be used as the basis for measuring distances in space and intervals of time) are all equivalent, making it impossible ever to determine by measurement which one is at rest in absolute space. But the undetectability of absolute rest does not mean that there is no such thing. Indeed, as Lorentz began to show early in the twentieth century, it is possible to explain the observational equivalence of inertial frames which makes absolute rest undetectable on the assumption that all the material objects are located in an absolute space in which light has a constant velocity. Lorentz showed that it would not be possible to detect absolute motion by measurements of the velocity of light, if material objects with a high velocity relative to absolute space suffered several distortions (including the shrinking of their lengths in the direction of motion, the slowing down of their clocks, and increase in their mass at a certain rate). It is also possible to show that, if observers on all inertial frames accept Einstein's definition of simultaneity at a distance (and synchronize their clocks on the assumption that the velocity of light is the same both ways, back and forth, in every direction -- that is, as if they were at rest in absolute space), those same "Lorentz distortions" will make all inertial equivalent even when it comes to their measurements of one another. (Observers on both of any pair of inertial frames will see the other's clocks slowed down, the other's measuring rods as shrunken, etc.)

In other words, if we think of the effect of space on the material objects it contains as the "ether" in which Newtonian physicists thought that light had a fixed velocity (or what I will call an "inherent motion" in space), and if we assume that the motion of material objects through the ether has certain distorting effects on them and their physical processes, then all of the observational consequences of Einstein’s special theory of relativity follow. We will have explained all the phenomena without referring to spacetime. Thus, it is not necessary to give up the assumption that space is a substance enduring through time to explain what is described by Einstein's special theory of relativity.

The general theory of relativity is a theory about gravitation formulated in terms of spacetime. It holds, in effect, that matter accumulation in spacetime imposes a curvature on spacetime, and that in curved spacetime, the path of inertial motion is not straight, but curved, or in other words, accelerated. But all the predictions that follow from assuming that spacetime is curved can also be made on the assumption that the velocity of light relative to space varies from place to place in space. That is, the spatiomaterialist interpretation of Einstein’s special theory of relativity is, in effect, an ontological interpretation of what is meant by "spacetime,"and that is what makes it possible to explain the observational adequacy of Einstein’s general theory of relativity on the assumption that space is a substance enduring through time.

As suggested above, talk of "spacetime" can be replaced by talk of an ether, in which the velocity of light is equal both ways in every direction, though that is really just a way of describing how space interacts with the matter it contains. In our ontological explanation of the special theory, we assume that the ether is at rest in absolute space and we explained all the other inertial frames as observers on the one that is at absolute rest. In order to explain the general theory, we assume that the ether itself can have a velocity in space, one that varies across space according to the accumulations of matter nearby. That means that the absolute velocity of light varies from location to location in absolute space (that is, at different locations in the inertial frame at absolute rest, from which we are giving this explanation). But it also means that material objects, which interact with one another by way of electromagnetic interactions through the ether, are accelerated with the ether, and such a moving, accelerated ether is what "curved spacetime" comes down to ontologically, for as we shall see, it explains all the observational predictions of the general theory of relativity. It could all be just the effect that space has on the light and matter it contains, if the right states for space to have such effects are imposed on space by the accumulation of matter in space. Precisely the same observational predictions follow from this theory as from Einstein’s general theory of relativity, and thus, it is possible that space is a substance enduring through time, that is, absolute.

These sketches of the spatiomaterialist explanation of the truth of Einstein's two relativity theories may be too brief for most people to follow easily. But they are included here because even a suggestion of the nature of these arguments may clarify what is meant by saying that it is possible that spatiomaterialism is true, notwithstanding the Einsteinian revolution in physics. But at this point, it is still just a promise, and thus, we accept the obligation to show in detail how it is possible as we take up showing what holds necessarily, if spatiomaterialism is true. It is like taking out a mortgage in order to construct the ontological foundation for this philosophical argument. If it should turn out, as we build the edifice of ontological philosophy, that relativistic phenomena cannot be explained on the assumption that space is a substance existing in time, spatiomaterialism will have been falsified and we will not be entitled to use it as a foundation to support any conclusions about the world. We will have to concede that we do not have a new way of doing philosophy after all.

As it stands, however, spatiomaterialism is a better ontological explanation of the natural world than either spatial relationism or spatiotemporalism, because the latter two theories have opposite failings. Spatial relationism (that is, materialism) can explain why the present is different from the past and future (and, thus, can hold that change is real), but it cannot explain spatial relations. Spatiotemporalism can explain spatial relations, but it cannot explain why the present is different from the past or the future, that is, except as another kind of relation like that of space (and, thus, cannot hold that change is real). Spatiomaterialism, however, can explain both spatial relations and why the present is different from the past and the future (and, thus, can hold that change is real).