The Atlas of Reality

by Robert C. Koons,Timothy Pickavance

  17.3 Spatial Relationism

  17.1A Spatial Relationism. Places are not G-fundamental.

  Since the kind of grounding involved here is conceptual grounding (Section 3.4), Spatial Relationists reject the real existence of places. There are only located entities, material bodies of some kind.3 Spatial Relationism had some ancient defenders. Aristotle and many of his followers, for example, seem to have been Relationists. The view has also been popular in modern times, beginning with George Berkeley and Gottfried Leibniz in the seventeenth century.

  There are two schools of Relationism, and the two schools coincide roughly with the two periods, ancient and modern. The schools differ on the question of which spatial relation or property is most fundamental. Modern Relationism has it that distance is most fundamental, while Aristotelian Relationism has it that extension (that is, shape and volume) and contiguity are most fundamental.

  17.1A.1T Modern Relationism. Spatial Relationism is true, and shape and volume are properties of material bodies that consist entirely in the holding of certain distances between the pairs of proper parts of those bodies.

  17.1A.1A Aristotelian Relationism. Spatial Relationism is true, and shape and volume are metaphysically fundamental properties of material bodies, and contiguity or contact is a fundamental relation.

  There is a related distinction to consider. Are fundamental material bodies point-sized (without volume) or extended (voluminous)? The two distinctions seem to be tightly connected. If fundamental bodies are point-sized, then the fundamental spatial relation must be distance, since point-sized bodies have no shape or volume and are never contiguous. Conversely, if fundamental bodies are extended, then it seems natural to take the properties of size and shape and the relations of contiguity (touching) and non-contiguity (non-touching) as fundamental.

  17.2T Fundamentality of Distance. The fundamental spatial property is distance between point-sized things.

  17.2A Fundamentality of Shape and Contiguity. The fundamental spatial properties are shape, volume, and contiguity between extended (voluminous) things.

  We are tacitly excluding the view that distance is the fundamental spatial relation but holds between extended things rather than points. The distance between two extended things could be defined in a variety of ways on this view. It could be the distance between their closest points, the distance between their farthest points, the distance between their geometric centers, or the distance between their centers of mass. However, in each case, it seems that the distance between the extended things is derivative, and the distance between two points fundamental.

  Distance a Relation Between Spatial Points. The distance between two extended things is not fundamental but derives from the distance between pairs of points occupied by those things.

  Given the thesis Distance a Relation Between Spatial Points, Modern Relationism and the Fundamentality of Distance entail each other, as do Aristotelian Relationism and the Fundamentality of Shape and Contiguity. We leave it to the reader to discern these entailments.

  POINTS, CURVES, SURFACES, AND REGIONS Before moving on, we should clarify what we mean by points and point-sized bodies, as well as such contrasting terms as extended regions and surfaces. We'll start with the definitions of points, curves, surfaces, and extended or voluminous regions, as they would be defined for Substantivalists:

  Def D17.3.1 Point. A point is a zero-dimensional place (lacking length, width, and depth, as well as area or volume).

  Def. D17.3.2. Curve. A curve is a one-dimensional place, having length, but lacking width and depth, as well as area or volume. A curve can be either finite or infinite in length. Points can be located on curves. A curve is a line if and only if any three points located on the curve are such that one is exactly between the other two.

  Def D17.3.3 Surface. A surface is a two-dimensional place, having length, width, and area, but no depth or volume. A surface can be either finite or infinite in area. Points and curves are located on surfaces. A surface is a plane if and only if any any two points located on the surface are also located on a line on the surface.

  Def D17.3.4 Region. A region is a three-dimensional place, having length, width, depth, and volume. Regions can be either finite or infinite in volume. Points, curves, and surfaces are located in regions.

  For simplicity's sake, we are going to ignore the possibility of fractal entities, entities with dimensionality other than 0, 1, 2, or 3, such as space-filling curves with dimensionality between 1 and 2. For Spatial Substantivalists, these definitions can be applied to define point-sized, curve-sized, surface-sized, and region-sized material bodies. For example, a point-sized material body, or point-mass, is a body that exactly occupies a spatial point and nothing else.

  Spatial Relationists do not believe in places and so cannot use Defs D17.3.1 through D17.3.4. However, they can make do with very similar definitions, such as Def D17.3.1′:

  Def D17.3.1' Relationist Point. A point-sized body is a zero-dimensional body, lacking length, width, and depth, as well as area or volume. The distance between any two parts of a point-sized body is zero.

  ARGUMENTS FOR AND AGAINST SPATIAL RELATIONISM According to Spatial Relationism, there are no places, strictly speaking. There are only things (material bodies of some kind) bearing various spatial relations (like distance or contiguity) to each other.

  The main argument for Relationism is simplicity (PMeth 1). Substantivalists, realists about places, must posit two kinds of entities, namely, places and things that occupy places. Relationists believe only in place-occupiers (material objects), together with facts about the spatial relations among those objects. This difference threatens to disappear if the Substantivalists adopt the Theory of Spatial Qualities, especially if spatial relations are internal relations between such qualities. Even in that case, however, there does remain this difference: Substantivalists will want to posit the existence of unoccupied places, which for Spatial-Quality Theorists will amount to the existence of uninstantiated location-qualities. Relationists, in contrast, have only place-occupiers, with nothing corresponding to unoccupied places.

  This lack of unoccupied places poses an immediate problem for Relationism, however. Relationists seem forced to deny many, if not all, of the usual axioms of geometry. For example, given Relationism, it is false that there always exists a spatial point between any two distinct points. If two points were occupied by material objects, but nothing existed between them, then of course it would be false to assert that something (namely, a point) exists between them.

  This problem creates another problem for Relationism (as Hartry Field (1984) argues). Without the usual axioms of geometry, Relationists cannot provide the usual reductionist account of spatial quantities like distance, area, volume, and angle measure. Substantivalists can suppose that the only fundamental spatial relation is betweenness, which is a relation among three points. Given the usual axioms for geometry, expressed entirely in terms of betweenness, spatial quantities can be defined in such a way that any permissible definitions are isomorphic to one another, differing only in the arbitrary unit of measure. However, without those standard axioms, such a definition of spatial quantity is impossible. This implicit definition of quantities is supported by a representation theorem, a mathematical proof in which it is shown that any system of objects that satisfies the standard axioms of geometry can be assigned a set of real-numbered values that satisfy certain basic postulates for the quantity in question (e.g., distance), and, moreover, that any permissible set of assigned values is essentially equivalent to any other such set, in the sense that the one set can be converted into the other by multiplying by some constant number (representing the conversion from one arbitrary unit of measurement to another).

  In the absence of such a representation theorem, Relationists would have to adopt a position that Field calls “heavy-duty Platonism”. Heavy-duty Platonism requires that there is a single, fully natural unit of distance (the metaphysical meter), and that real
numbers enter directly and ineliminably into the constitution of physical facts about distance, area, volume, and so on. The distance between two points, for example, consists in those points' being jointly related to some real number. It's possible that Relationists could make do with a somewhat less heavy-duty version of Platonism. It might be enough, for example, to assume that any two pairs of points are related uniquely to some real number r, in the sense that the distance between the first pair of points is r times the distance between the second pair. This still involves some degree of heavy-dutiness, since these relations to real numbers must be among the fundamental relations in the world. Field argues that this is undesirable, since it involves treating numbers as though they were themselves physical objects, whereas in fact numbers seem utterly separate from the physical world, however useful they may be in describing that world.

  No Heavy-Duty Platonism. There are no fundamental relations between real (and complex) numbers and material things.

  One way out of this problem is for Relationists to verify the standard axioms of geometry by positing the existence of possible material objects located at each point in space or the possible and counterfactual location of actual material objects at each point in space. This approach would be quite attractive to those who adopt Concretism (14.1T.1T) with Counterpart Theory (16.1A.1). We can suppose that there exists a concrete possible world, the Plenum world, with no vacuums at all (all points in space are occupied) and with counterparts of all actual bodies, with those counterparts standing in the same spatial relations (in that world) that corresponding actual bodies stand in in our world. We could then take the points of geometry to be small, point-sized bodies in the Plenum world, with an actual body occupying one of those points just in case the point is part of the counterpart of that body in the Plenum world.

  At this point it is unclear whether there is any advantage in terms of ontological economy for Relationism. Relationists on this alternative are committed to an infinite number of possible material objects in an infinite number of concrete possible worlds, a totality at least as large in number as the real places of Substantivalism. There is perhaps some advantage to Relationists in that they can claim to make do with only one basic kind of thing, namely, material objects, whether actual or merely possible, rather than two, namely, material objects and spatial locations. (We will discuss the problem of vacuums and empty space further in Section 17.4.)

  Instead of positing concrete possible worlds without vacuums, Relationists could instead simply “bite the bullet” and embrace heavy-duty Platonism. In fact, we discussed a version of heavy-duty Platonism in our survey of theories of quantity: Composite Intensity Theory (10.2T.3). According to Composite Intensity Theory, intensities or magnitudes are simply real numbers, and a body or ordered plurality of bodies has a certain magnitude of some quantity (like distance) by jointly instantiating the determinable universal (DISTANCE) and some real number. This would be an attractive option for Relationists who are Realists about universals and Abstractionists about possibilities.

  There is a second argument for Relationism, in addition to ontological economy. Leibniz argued that Substantivalism needlessly multiplies possibilities. For example, if Substantivalism is true, then the universe could have been located six feet over, in some direction or other, from where it is in fact. If Substantivalism were true, then each of the following would represent a real possibility, distinct from the actual world:

  Everything is exactly as it in the actual world, except that everything (both near and far) has been, is, and always will be located exactly six feet from its actual location in the direction to which the earth's axis was pointing at noon Greenwich Mean Time on 1 January 2001.

  Everything is exactly as it is in the actual world, except that everything in the universe has been, is and always will be rotated in an angle of 43° to the west along the line defined by the earth's axis at noon Greenwich Mean Time on 1 January 1 2001.

  Everything is exactly as it is in the actual world, except that the entire universe is reflected, as in a mirror about the plane that coincided with the earth's equator at noon Greenwich Mean Time on 1 January 1 2001.

  Everything is exactly as it is in the actual world, except that every length and distance is doubled compared to its actual value.

  Of course, there are, for each of these four transformations, an infinite number of alternatives with different values for the magnitude of the transformation and different axes or planes of reference. For Substantivalists, each of these transformations corresponds to a distinct possibility, a different way the whole world could have been. For Relationists, in contrast, these supposed alternative possibilities would correspond to different conventions for describing a single set of spatial relations among the bodies. For example, a mirror-reversal like (3) would correspond merely to a difference in the meaning of ‘left-’ and ‘right-handedness’, not to a real difference in the relation between things and space.

  Field (1984) argues that this needless multiplication of possibilities is especially problematic for Spatial-Quality Theorists, since on that account such possibilities would be qualitatively different. The Sixth Corollary of Ockham's Razor (PMeth 1.6) enjoins us to minimize the class of possibilities. This is a corollary of Ockham's Razor because the Razor asks us to prefer the theory that posits the smallest class of fundamental properties. The fewer fundamental properties there are, the fewer the number of distinct possibilities involving different combinations and permutations of fundamental properties. Substantivalists posit both fundamental relations of occupation (between material things and locations) and of spatial distance, while the Relationists posit fundamental relations only of the second kind. Arguably, Leibniz's argument provides a tie-breaker in favor of a modal version of Relationism over Substantivalism.

  Isaac Newton offered a famous thought-experiment against Relationism in the General Scholium of his Principia (1687), the bucket experiment. Newton imagines a bucket containing water that is being swung in a circular path on a rope. As Newton hypothesized, the surface of the water will become concave, rather than flat, due to what could be described as the ‘centrifugal force’ of the rotation. This curvature of the water would distinguish the bucket from any bucket that is at rest or that is moving in a straight, unaccelerated path (i.e, an inertial path). Newton suggested that it is only the accelerated motion (in this case, circular motion) of the bucket relative to Absolute Space that can ground the difference between a bucket that does and one that does not experience these centrifugal effects.

  However, Relationists have a number of effective responses to Newton's argument. Ernst Mach suggested that centrifugal forces depend on the motion of the bucket in relation to the mass of the earth and other celestial bodies (a hypothesis known as ‘Mach's Principle’—see Mach 1960.). Still more effectively, Relationists can argue that accelerated and non-accelerated paths differ from each other intrinsically, and not due to changes in velocity relative to Absolute Space. All that is needed is a geometrical description of spatiotemporal relations that distinguishes the two kinds of paths, something that is accomplished by the so-called ‘Minkowski’ representation of spacetime, which is consistent with the non-existence of places.

  Finally, Einstein's theory of general relativity may be relevant to the Substantivalism-Relationism debate. As standardly interpreted, general relativity includes the claim that spatial relations are merely part of a comprehensive spacetime structure, which includes both temporal and spatial relations in a flexible relationship that varies to some extent from one frame of reference to another. More importantly for our purposes here, general relativity entails that this spacetime is curved or warped by the presence of massive bodies. A massive body, like a large star, warps the shape of spacetime in its vicinity, resulting in the curving of the paths of moving bodies toward the center of the mass. This curvature is the relativistic explanation of the phenomenon of gravity. Instead of thinking of gravity as a force acting on bodies in a
flat or uncurved spacetime, Einstein encouraged us to think that bodies move toward massive bodies by simply following the shortest, laziest path through local spacetime. For this reason, even mass-less entities like photons are curved toward massive stars, as if they were being attracted by a force. This curvature of light is actually observed experimentally.

  If spacetime can be curved by massive bodies, then it seems that it must be something more than a set of relations between those bodies. It seems natural, at least, to think of such spacetime as an entity in its own right, subject to influence (in respect of its pattern of local shapes or curvatures) by the action of massive bodies. In support of this picture, the equations of general relativity enable us to describe a spacetime world that is entirely empty, unoccupied by bodies of any kind.

  However, this Substantivalist tendency can be and has been resisted. First, one can refuse to believe that every possible solution of the general relativity equations represents a possible world. Perhaps an empty spacetime is metaphysically impossible, even if describable by consistent mathematics. Second, one can make a distinction between the bare, spatiotemporal relatedness of the world, the spacetime manifold, and the measurable shape or curvature of spacetime at various places, the metrical field. The spacetime manifold could be understood entirely in relational terms, and the metrical field could be interpreted as matter's effects upon those relations, and not as its effect on a substantial spacetime.

  17.4 Absences and Vacuums

  As we have seen, a critical difference between Spatial Substantivalists and Spatial Relationists concerns their treatments of empty space. For Spatial Particularists, empty regions of space are wholly unproblematic. They are ordinary, first-class entities, differing from occupied regions only in the absence of an occupying body. For Spatial-Quality Theorists, empty regions of space correspond to uninstantiated properties. For Spatial Relationists, in contrast, there are strictly speaking no empty regions of space; there are (at most) merely possible bodies that would (if they existed) be in contact with or at certain distances from certain actual bodies.