Brian Skyrms draws this conclusion:
At this stage of the game, the elimination of non-measurable sets may appear a rather quixotic goal. Lebesque measure has extended measurability to a far richer domain than Zeno and Aristotle imagined possible. It meshes with an elegant and powerful theory of integration adequate to the needs of the physical sciences. Perhaps Lebesque measure should be taken as the theory of measure for physical space, and the existence of non-measurable sets should be viewed as just a mildly surprising consequence of the theory rather than as a real difficulty. This is, I believe, the dominant view among mathematicians and mathematical physicists. ]source[(Skyrms 1983: 247)
Despite Skyrms's sanguine attitude, the upshot seems to be some support for Aristotelian Finitism, according to which extended regions are not fusions of sets of points. Three-dimensional volumes may have points as parts of a sort (i.e., as parts of the region's internal and external boundaries), but they are not constituted, without remainder, of points, even when the points fill the space. If regions were fusions of sets of points, then there would be no good reason to deny that every set of points constitutes a region. Aristotelian Finitists can explain why some sets of points have measurable volume and why some do not. Those sets of points that are interior to some region can be assigned a volume, and those that are not, cannot.
18.3.3 Spatial boundaries and physical contact
Our third and final argument against Pointillism concerns the nature of contact between bodies. These arguments have their roots in medieval and late scholastic philosophy, especially the philosophy of Francisco Suárez, and were further developed in the early twentieth century by Franz Brentano (Brentano 1976/1988—these manuscripts were published after his death by Brentano's students). They can be found in recent essays written by Dean Zimmerman (1996a and 1996b).
These arguments proceed, as we shall see, from a somewhat naive and pre-scientific view of the nature of matter. They assume that an extended body occupies a volume of space that is absolutely filled with some material substance, in such a way that two such bodies can come into direct contact with one another. Modern views of matter (since the sixteenth century) have been corpuscular in that they assume that matter consists of a finite number of particles, widely separated from each other and held together by electrostatic forces. Does the unrealistic or counterfactual nature of these scholastic and Brentanian arguments render them unpersuasive?
There are at least two reasons for taking these arguments seriously. First, even if they assume facts about matter that are not realized in the actual world, we can still consider them thought experiments about a metaphysically possible world. So long as it is possible for two material bodies that fill volumes of space to come into direct contact, the argument provides support for Anti-Pointillism. Second, there is a historically important way of interpreting modern quantum theory, namely, the Copenhagen interpretation, according to which reality must be divided into two separate and incommensurable realms, namely, the quantum realm of microscopic particles and the classical realm of observable bodies. Even if matter within the quantum realm consists of isolated particles, it doesn't follow that classical bodies may not come into direct contact.
Here's the thought experiment. Suppose that we have two spheres completely filled with some homogeneous stuff. It should be possible for the two spheres to come into contact with each other. When two perfect spheres touch, there is a single point of contact between the two—call it ‘P’. This spatial point is occupied by a point-mass, M (on the assumption that every filled location corresponds to a unique physical object). This point of contact will also be on the surface of the two spheres. Now suppose that we separate the two spheres. Each sphere will now have its own point on its surface, P1 and P2, corresponding to the former point of contact, P.
Figure 18.1 Two Perfect Spheres Lose Contact.
The problem is whether points P1 and P2 filled with matter. Here are the possibilities:
Neither P1 nor P2 are occupied by matter. 1a. No points on the surface of physical objects are occupied by matter (so, the point of contact P is also unoccupied).
1b(i). Some points on the surface of physical objects are occupied, and some are not. In this case, none of P, P1, or P2 are occupied.
1b(ii). Some points on the surface of physical objects are occupied and some are not. In this particular case, P is occupied, but neither P1 nor P2 are occupied.
Exactly one of P1 and P2 is occupied by matter.
Both P1 and P2 are occupied by matter. 3a. One of these is occupied by M, the other by a new point-mass.
3b. Both are occupied by new point-masses.
Two of these options, 1b(ii) and 3a, seem absurd immediately. Option 1b(ii) requires the point-mass M that occupied P to be annihilated simply by separating the two spheres. Options 3a and 3b requires that one or two point-masses are created ex nihilo (out of nothingness) by the separation. Such annihilation or creation cannot be a necessary consequence of simply moving the two spheres apart. In other words, the following principle seems highly plausible:
PNatPhil 2 Independence of Motion and Substantial Change. Neither the movement of two extended things nor the division of an extended thing into two parts can necessitate the creation or annihilation of material parts (not even point-masses).
Two of the options, 2 and 3a, require something asymmetric to happen as a result of a purely symmetric operation. But metaphysical necessities ought to be grounded in the natures of the things involved (PMeta 3), and there is nothing to distinguish the nature of one sphere from the other in order to ground such a difference in results. This is Symmetry:
Symmetry. Wherever possible, avoid theories that require violations of symmetry (e.g., that require symmetrical operations to result in asymmetric outcomes).
It might seem that option 3b is objectionable for the same reason 3a was, since 3b involves the creation ex nihilo of two new point-masses. However, this need not be a consequence of 3b: we could suppose instead that the original point-mass M had two parts, M1 and M2, each with one-half the mass-density of M and both co-located with M at the same very point. When the spheres are pulled apart, the two halves are split. This does involve some violation of symmetry, nonetheless, since we cannot explain why one of the parts went one way and the other the other way.
But there is a second, weightier objection to 3b. The two parts M1 and M2 must each have half the mass-density of the original point-mass M (where the density of a point-mass x is equal to the average density of a region filled with point-masses intrinsically identical to x). This means that after the splitting of the two spheres, the density of each of the two spheres varies discontinuously, with a sharp discontinuity in density occurring at the location of M1 and of M2. Alternatively, the density of the two spheres-in-contact must have had a discontinuity at the location of M, which would have had to have twice the density of all of the surrounding matter. Each of these seems to be impossible. If the average density, and hence the total mass, of a body is to be well defined, the mass-density of its parts must vary continuously. Thus, option 3b is in conflict with Mass a Function of Density:
Mass a Function of Density. The mass of any extended body is a function of the density of its parts, and the mass of such a body is always well defined.
Thus, the alternatives left are 1a and 1b(i). Both require that the relevant points were never occupied. Option 1a means that the space occupied by matter is always topologically open, lacking its own external boundary. A physical sphere must fill all of those points that are less than a certain distance from the center, not the points that are located exactly that distance from the center on the dimensionless skin around the sphere's interior.
However, option 1a violates another very plausible principle:
Unlimited Divisibility of Matter. Any extended mass can be divided (separated) into parts along any two-dimensional surface passing through its interior.
Consider a single sphere A. Take a circular cr
oss-section of A that passes through its center, and call this circular surface C. C divides A into two hemispheres, A1 and A2. Suppose that sphere A is completely filled with matter. Consequently, C, A1, and A2 are all fully occupied by this matter. Now suppose that we separate A1 and A2. The matter filling the cross-section C must be annihilated, on the assumption that the two separated hemispheres must each fail to occupy their outer boundaries. So, either the mere division of the sphere necessitates the annihilation of some matter, violating the Independence of Motion and Substantial Change or the sphere is not divisible. Either result is absurd.
Therefore, we seem to be forced, by process of elimination, to adopt option 1b(i), the view that surfaces of bodies are partly closed and partly open. That is, that some of the points on the surface of bodies are occupied and others are not. This view has a numberof odd metaphysical consequences. For example, this view requires that two bodies can be pushed into contact at a point only if one of the bodies has an open surface at that point and the other has a closed one. If both surfaces are closed, then the two occupied points will exclude each other, so that the two bodies must always be some finite distance apart. This seems absurd, as can be seen by considering the plausibility of this principle:
Potentiality of Contact. Any two extended masses can be brought into mutual contact.
Potentiality of Contact might be taken to be a consequence of the Principle of Epistemology 1: that imagination is a guide to possibility. We can certainly imagine (in some loose sense) the contact of any two extended bodies. But, if both surfaces are open, then once again the two bodies cannot be brought into contact, since what would be the point of contact between them must always be either unoccupied or occupied by a point-mass that belongs to neither body. This would also violate Potentiality of Contact.
18.4 Voluminism vs. Volume-Boundary Dualism
If we conclude that spatial regions and material bodies are not composed of points, that is, if we conclude that points are not metaphysically more fundamental than three-dimensional regions and bodies, then we still have a choice concerning points and other superficial entities. We might suppose that there are no points at all or that points are merely logical constructions, whether sets of regions, pluralities of regions, or Ludovician very small regions. Alternatively, we might believe that regions have boundaries, like surfaces, edges, and point boundaries, even though they do not consist entirely of points. That is, we might consider the possibility that both regions and points are fundamental entities.
18.1A.2T Voluminism. Entities of fewer than three dimensions (like boundaries that are points, curves, and surfaces) are not fundamental entities: either they do not exist at all or they are mere logical constructions from regions and extended entities.
18.1A.2A Volume-Boundary Dualism. Entities of fewer than three dimensions (like boundaries that are points, curves, and surfaces) exist and are not mere logical constructions, although they are not more fundamental than extended things.
Voluminists either eschew belief in points altogether or replace points with logical, pluralist, or Ludovician constructions. Volume-Boundary Dualists accept both points and regions as fundamental entities, as well as material point-masses and bodies. Moderate Volume-Boundary Dualists (following Aristotle) believe only in the actual, external surfaces of material bodies. However, we will assume that typical Volume-Boundary Dualists believe in all of the points, curves, and surfaces of classical geometry, including the ones that are interior to a material body. Volume-Boundary Dualists believe that both boundaries and material bodies are G-fundamental: that is, their existence is not wholly grounded in the existence of things of the other category. However, neither boundaries nor material bodies are O-fundamental (in the sense of definition D3.6): the essences ofbodies include the existence of boundaries, and the essences of boundaries include the existence of bodies. The two categories of things are equally G-fundamental but ontologically interdependent (as we discussed in Section 3.3.2).
ADVANTAGES OF VOLUMINISM Voluminism has at least two advantages over Volume-Boundary Dualism.
Voluminists have a much smaller and more uniform ontology (PMeth 1), consisting entirely of three-dimensional regions and bodies, plus sets and other abstract objects. In contrast, Volume-Boundary Dualists posit the existence of two kinds of fundamental concrete objects, both three-dimensional regions and bodies and zero-, one-, and two-dimensional boundaries.
Volume-Boundary Dualism requires additional necessary connections between distinct entities, since material boundaries cannot exist without material bodies and bodies cannot exist without boundaries (see PMeth 1.2). In addition, we cannot have regions with points, curves, and surfaces, nor points or other surfaces without regions. Voluminism requires no such necessary connections, since points either don't exist at all, are identical with small bodies, or are mere sets or pluralities of regions.
There is one qualification to the last point. Voluminists who identify points with sets of regions do have to accept the necessary connections associated with set theory. For example, it is a necessary truth that if x exists, then the set containing x also exists and contains x as a member. This fact could neutralize the second advantage if Volume-Boundary Dualists reject set theory.
ADVANTAGES OF VOLUME-BOUNDARY DUALISM Volume-Boundary Dualism has two advantages over Voluminism, however.
Suppose that there are natural, fundamental properties or relations of a physical or geometrical character that are borne by points, curves, and surfaces. If so, then these things cannot be mere sets, since we've assumed that the only natural relation that sets enter into is the set-membership relation. Modern physics does typically assign certain fundamental properties, like density or gravitational or electromagnetic field strength, to points in space, and not to whole regions. If physics is correct in doing this, it provides a powerful argument against taking points to be logical constructions. We'll take this up in Section 18.4.1, on continuous variation.
Voluminists who use a Ludovician construction will also have problems with natural properties that are assigned to points, since physical theory would not assign determinate values of those properties, like density, to small volumes, no matter how small. A pluralist construction might work, however, taking a property of a point to be the collective property of a plurality of regions (one that converges on that point).
In our naive, pre-scientific thinking about physical objects, we need to talk about points, curves, and surfaces in order to describe various kinds of contact or fusion between those objects. For example, two perfect spheres would, if they were to come into contact, meet in a single point. It is hard to believe that such a real point of contact could be some kind of set of overlapping bodies. The Ludovician construction is more reasonable, but it involves positing a primitive, unanalyzable relation of contact, while Volume-Boundary Dualists can analyze contact in terms of sharing a common point, edge, or surface. A pluralist construction could take care of this problem, since it would enable us to define contact in terms of sharing a common point, where the point itself is identified with a plurality of regions. We'll take this up in Section 18.4.2, on coincident boundaries.
18.4.1 Continuous variation in quantity and quality
The first prima facie problem for Voluminism concerns continuous variation. A quantity or quality varies continuously when each point in some region has a value that is different from all of its neighboring points. Since Voluminism denies that points are fundamental entities, in contrast to both Pointillism and Volume-Boundary Dualism, Voluminists must either deny that continuous variation is possible or else come up with some inventive account of how it is possible, despite the absence of fundamental points to act as bearers of the quantities or intensities.
A spatial region is uniform or homogeneous when every quality and quantity that is instantiated there takes a constant value at every subregion. A region is continuously variable with respect to some quality or quantity when it contains no homogeneous subregi
on. When a quantity or quality varies continuously within a region, the value of that quantity (like density or temperature) or quality (like color) at each point in the region is unique within some finite neighborhood around that point. If the quantity does not vary continuously throughout any subregion of a region S, then S is composed of a set of homogeneous subregions. Let's say that a region is Aristotelian if it is composed entirely by homogeneous subregions.
Def. D18.1. Aristotelian Region. A region x is Aristotelian if and only if there is a set S of homogeneous regions such that x is the sum of S.
Def D18.2 Sum. x is the sum of set S if and only if every member of S is a part of x and every part of x overlaps (has a part in common with) some member of S.
18.5T Spatial Aristotelianism. Every region of space is Aristotelian.
18.5A Continuous Variation. Some quality or quantity varies continuously throughout some region of space.
To simplify our discussion, let's suppose that the property that varies continuously throughout certain regions according to Continuous Variation is density. What then would be the fundamental properties? It would be natural to suppose that the fundamental properties are specific measures of actual densities. The bearers of these fundamental properties would have to be points in space or point-masses, since there are by hypothesis no subregions of constant density.
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