LUDOVICIAN CONSTRUCTIONS David Lewis (Lewis and Lewis 1970, Lewis 1980a, 1993, 2004a) introduced, in the second half of the twentieth century, a quite different approach to replacing problematic entities with constructions. The strategy involves Ludovician constructions. Lewis's idea is closely related to the notion of “loose and popular identity” introduced by Joseph Butler (in Flew 1964: 166–172) and further developed by Roderick Chisholm (1976: 92–113, 1989: 25–41, 124–128). A Ludovician construction of points could be one that identifies points with very small regions, rather than with sets of regions. Intuitively, a point is a very small, spherical region. How small? Small enough for whatever is one's present purpose. Which small regions count as points depends on the context. For example, when discussing geometrical results in school, we often draw “points” and “lines” with chalk. The points are not really zero-dimensional objects. They are just very thin and relatively small smears of chalk on the board. Nonetheless, they are often small enough for the purpose at hand. The key is that whatever we count as points must have negligible volume (in the relevant context).1
Of course, if we count one very small region as a point, there will be infinitely many other, equally small regions that largely overlap the region in question. In fact, there will be infinitely many tiny spheres that share exactly the same center, intuitively speaking. We don't want to say that each one of them counts as a distinct point or else we'll be forced to contradict some basic axioms of geometry, such as the one stating that two distinct lines can intersect in at most one point. David Lewis's answer was to introduce the same-X relation—in this case, the same-point relation. If two tiny spheres are concentric, then they stand in the same-point relation. Each is counted as the same point as the other. Thus, if one line (a narrow tube) intersects another line (another narrow tube), there will be just one point of intersection, even though there are many tiny spheres in the intersection, since the many spheres will together count as just “one” point.
The same-point relation, as well as the similar same-curve and same-surface relations, can do additional work for Ludovician constructors. When a material body moves, its surface, with all of its attendant points and edges, moves with it. We cannot identify the surface of the body with a region of space or spacetime, since neither of them move with the body. Think of a ball that is rotating. The surface of the ball is also rotating, but the space in which the ball is sitting is not moving at all. The surface is the two-dimensional boundary between the ball and the surrounding environment. A Ludovician construction can enable us to replace such two-dimensional surfaces with very thin, three-dimensional skins or peels. Each material body has infinitely many such skins of a given thickness or less, but all of these can be counted together as the “same” surface, since they stand in the same-surface relation. There is no problem with saying that the ball's surface is rotating with the ball, since every outer layer of the ball, no matter how thin, is rotating.
The same idea could be applied to certain immaterial bodies or surfaces, such as holes or shadows. A hole or shadow could be identified at each moment with a region of spacetime, namely, the empty space constituting the hole at a moment or the region from which light is excluded by the occluding object. Shadows and holes can move through space, while regions of spacetime cannot. We can accommodate this fact by making use of appropriate same-hole or same-shadow relations. We can say that a shadow is moving through space so long as a series of spacetime regions, each located at a different time, are connected by the same-shadow relation.
The Ludovician construction has one principal advantage over the more traditional Whitehead-Russell logical construction: it identifies points (holes, shadows, etc.) with concrete things, like parts of material objects or parts of spacetime, and not with abstract objects, like sets. This enables us to say with more confidence that we can indeed perceive these things, and it enables us to assign them some real causal role in explaining phenomena.
The main drawback of Ludovician constructions is that it leads to a number of counterintuitive results. For example, it would be literally true that some points are larger than others or that some points overlap other, distinct points, even though it would be inappropriate to say these things in any given, fixed context. Everything that one refers to in one context as a dimensionless point could be accurately described in another context as having finite volume and containing many points as parts.
18.2 Points vs. Regions
We take it as obvious that space has parts. These parts are three-dimensional regions with finite volume. It also seems reasonable to concede that points, which are zero-dimensional spatial objects, exist in some fashion if places exist at all. However, it is not at all obvious which are fundamental. Are the fundamental things volumes, points, or both?
18.1T Spatial Pointillism (Extreme Indivisibilism). Indivisible, dimensionless parts of space (points) are (of necessity) more fundamental than extended regions; extended regions can be wholly grounded in points.
18.1A Spatial Anti-Pointillism. Necessarily, finite regions or volumes are at least as fundamental as points.
If locations are properties of material things, then Spatial Pointillism means that point-location properties are more fundamental than region-location properties, while Spatial Anti-Pointillism implies that region-location properties are at least as fundamental as point-location properties. If Spatial Pointillism is true, then a material object has a class of fundamental properties, each corresponding to some point that it occupies, while if Spatial Anti-Pointillism is true, then each material object has at least one fundamental property that corresponds to the region it occupies, or perhaps some finite set of fundamental location properties, each corresponding to some part of the region it occupies. Pointillism is also known as “Extreme Indivisibilism” because points are indivisible, and Pointillism implies that indivisible points are the fundamental spatial reality.
Spatial Pointillists advocate the conceptual grounding of truths about regions by fundamental truths about points. That is, they do not suppose that regions really exist, as entities with an essence that demands that their existence and properties be grounded in that of points. Rather, they suppose that it is the essence of our concepts of region and volume that permits us to explain how propositions ostensibly about regions and volumes can really be true in a world consisting only of points (or sets or sums or pluralities of points).
We are going to assume that space has its structure of fundamental entities as a matter of metaphysical necessity, whether those entities are points, volumes, or both. It seems very unlikely that space might be contingently made up of fundamental points and nothing else, or contingently made up of regions, or whatever. If space is a certain way in any world, it is essentially and necessarily so.
One issue concerns the relationship between one's view about fundamental spatial entities and one's view about the fundamental occupiers of space, namely, material bodies. We'll begin an investigation of this issue by considering the following plausible correspondence principle:
18.2T Fundamental Entity Space-Matter Correspondence. Necessarily, the fundamental spatial entities are points (as opposed to regions) if and only if the fundamental occupiers of space are point-sized bodies (as opposed to voluminous bodies).
Could Fundamental Entity Space-Matter Correspondence be false? There are two kinds of things that would be counterexamples to it: extended material atoms occupying a fundamentally Pointillist space and point-masses without any proper location.
18.2A.1 Extended Material Atoms in a Pointillist Space. There could be extended material atoms that occupy fundamental spatial points without point-sized parts occupying those points.
18.2A.2 Material Simples without Proper Location. There could be an extended body composed of infinitely many indivisible, zero-dimensional, volume-less material bodies, each without a unique fundamental location. That is, each indivisible body occupies infinitely many spatial regions, each of which is also occup
ied by infinitely many other bodies.
The likelihood that the thesis Material Simples without Proper Location could be true is remote. We would have to imagine a body that consists of point-sized fundamental entities, point-masses, which share a common extended region as their joint location, even though each of the point-masses have no unique fundamental location. But if the mass consists of point-masses, and there is some natural relation of location, then the point-masses must each bear the location relation to a unique and unshareable location, that is, a spatial point. Every body should have a unique, unshareable or proper location.
The truth of Extended Material Atoms in a Pointillist Space is more controversial. The main argument against the possibility of Extended Material Atoms in a Pointillist Space is given by Dean Zimmerman (1996a), and it goes like this. If some material entity A fills an extended region R, then there must be a part of the entity located in each part of the region. For example, R must have both a left and a right half. It seems reasonable to think that entity A must also have a left and a right half, one occupying the left half of the region R, the other the right half. This seems especially clear if we think of A as a mass of stuff of some kind. But in order for extended material atoms to be atoms, they cannot have any parts. So extended material atoms cannot occupy Pointillist spatial regions. Neither sort of counterexample to Fundamental Entity Space-Matter Correspondence is possible; that principle would seem to be true. And we have further happened upon another correspondence principle:
Divisibility Space-Matter Correspondence. If a mass M occupies an extended region R, and R has proper parts, then the mass M has proper parts corresponding one-to-one to those spatial parts.
There is an influential argument for Divisibility Space-Matter Correspondence, namely the “supercutting” argument of John Hawthorne and Brian Weatherson (2004). If there were extended material bodies in a Pointillist space, then it seems plausible that the following infinitary supertask would be possible. Take the extended material object E, and cut it into two pieces, leaving a small spatial gap between the two pieces. Repeat the process in half the time, splitting each of the halves into half. Keep repeating the process over and over, doubling the number of pieces in progressively shorter and shorter periods of time. After a finite period of time has elapsed, an infinite number of cuttings will have taken place, resulting in an infinite number of pieces, each with zero width and each some finite distance from any other piece. The same procedure could then be undertaken along the other two spatial dimensions, resulting in an infinite collection of point-sized bodies. However, if the extended body were a metaphysical atom, with no real parts, then the procedure would be impossible. Contradiction. Space-Matter Correspondence must be true.
Consider Spatial Anti-Pointillism, in which it is extended spatial regions (in three dimensions) that are metaphysically fundamental. We can ask whether these metaphysically fundamental regions have smaller parts. That is, we can ask which of Fundamentally Gunky Space and Discrete Space is true:
18.1A.1T Fundamentally Gunky Space. All fundamental volumes of space have proper, extended parts.
18.1A.1A Discrete Space. There are extended spatial simples.
Gunk was introduced by David Lewis as the concept of something which has proper parts, but also such that every part of it also has proper parts. A gunky thing is not simple and it has no simple parts. Every part of a gunky thing is divisible into still smaller parts.
If space is fundamentally gunky, then all of the regions of space, whether fundamental or not, have still smaller regions as actual parts. In contrast, if space is discrete, then the fundamental regions of space are simple, lacking any actual parts. There are therefore two versions of Discrete Space:
Berkeley-Hume indivisibles. There are indivisible spatial minima, each with a finite volume. On the most plausible version of this theory, each finite volume of space consists of a finite grid in three dimensions of cubical or dodecagonal “pixels”.
Aristotelian discrete space with infinite divisibility. Each region of space has only finitely many actual parts, but each part is further divisible without limit. Each region is potentially divisible without limit, but some regions are not actually divided.
If we accept Fundamental Entity Space-Matter Correspondence, then we can reduce Spatial Pointillism and Spatial Anti-Pointillism to the following:
18.3T Material Pointillism. Necessarily, the only fundamental bodies are point-sized (dimensionless). Truths about extended bodies are wholly grounded conceptually in truths about point-sized bodies.
18.3A Material Anti-Pointillism. Necessarily, if there are any extended bodies (bodies with finite volume), then there are fundamental bodies with finite volume.
18.3 Arguments against Points as Fundamental
There is a certain initial plausibility to Material Pointillism. It seems to be implicit in our modern approach to geometry, which takes planes and space to be composed of parts, and in the analytic, bottom-up approach of modern physics, especially in the search for fundamental particles. There are three sorts of arguments against it and for Material Anti-Pointillism: (i) arguments from Finitism, (ii) arguments from the nature of contact, and (iii) arguments from mathematical paradoxes like the Banach-Tarski theorem. In addition, we should bear in mind the arguments for Priority Monism that we considered in Chapter 11, especially those of Bradley (Section 11.2.3) and Schaffer (Section 11.2.4).
The dispute between Material Pointillism and Anti-Pointillism would not be settled if it turned out that certain material bodies (like photons, for example) are point-sized, so long as it was necessarily the case that extended bodies are not composed of an infinite number of such fundamental, point-sized bodies. Material Anti-Pointillism does not entail that there couldn't be fundamental bodies that are point-sized; it simply entails that, if there are extended bodies, then not all fundamental bodies are point-sized.
We are going to assume that extended bodies are metaphysically possible.
The Possibility of Extended Bodies. It is possible that material bodies exist with finite volume.
Consequently, we will take Material Anti-Pointillism to entail that metaphysically fundamental extended bodies are possible. In contrast, Material Pointillism clearly entails that such fundamental extended bodies are impossible.
We will now turn to the three arguments for Anti-Pointillism.
18.3.1 Finitism
One way of rejecting Material Pointillism is to be a spatial Finitist or a Finitist about the whole concrete world.
18.4T Finitism. There are (with the possible exception of sets and numbers) only finitely many actually existing things.
18.4A Infinitism. There are infinitely many actually existing things, other than sets and numbers.
If Finitism is true, and there are actually existing extended things, then Spatial Anti-Pointillism and Material Anti-Pointillism must be true, since any spatial region contains infinitely many points, as well as infinitely many subregions.
Infinite Numbers of Points. Any finite spatial extension contains infinitely many points.
How can it be the case that there are not infinitely many spatial entities, given the obviously infinitary nature of geometry? Finitists have two options. First, they could maintain that space is discrete, consisting of a finite number of extended but indivisible places, a kind of minimum unit of spatial volume. George Berkeley and David Hume accepted such a view, as did some ancient Greek philosophers. This is Discrete Finitism. Second, as an alternative, Finitists might place special emphasis on the idea that only finitely many things exist in actuality, as opposed to potentially. The infinitely many points, lines, and planes, and the infinitely many extended regions of standard geometry must be thought of as being merely potential in nature. This was the position of Aristotle and of his many followers, so this is Aristotelian Finitism.
Although Finitism entails that Pointillism is false, it doesn't follow that every Anti-Pointillist must embrace Finitism. It is possibl
e to be an Anti-Pointillist and an Infinitist. This was in fact the position of Whitehead, who believed that there are infinitely many voluminous regions in any finite region, but also that there are no points. However, Anti-Pointillists who embrace Infinitism lose the finitary arguments against Pointillism that we will consider in this section.
The arguments for Finitism are reductios, attempts to show that granting the possibility of actually infinite entities in space leads to metaphysical absurdities. There are two kinds of infinity to consider: entities of infinite extension or volume and entities with finite extension that include an infinity of actual point-masses.
If there could exist an extended entity of infinite extent or volume, then such an entity could exist with a finite, homogeneous density. Such an infinite object would be infinite in mass. Consequently, its momentum and kinetic energy would be infinite. Moreover, if such an entity could exist, then it could move with a constant velocity. A moving entity of infinite mass would contain infinite energy and momentum.
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