An empty region of space or vacuum would seem to consist in a certain absence of matter or bodies. Thus, before we consider vacuums in particular, we should examine the question of absences in general. Do absences really exist? Can all truths about absences be conceptually grounded in truths about bodies and their locations?
17.3T Absentism. There are real absences.
17.3A Anti-Absentism. There are no real absences.
There are three sets of arguments for the existence of absences: appeals to ordinary language and common sense, appeals to perception, and appeals to negative causation. We will consider negative causation in Chapter 27. Presently, we turn to the first two sets of appeals.
ARGUMENTS FROM ORDINARY LANGUAGE AND COMMON SENSE We talk about absences all the time. Consider (1) and (2):
(1) John's absence was widely noticed.
(2) The absence of rain ruined the crop.
Words like ‘absence’ (and ‘lack’, ‘privation’, ‘omission’, and ‘deficit’) are nouns, and noun phrases are typically used to refer to things. However, the mere fact that we often use a noun phrase does not by itself indicate that we really believe that something exists to which the noun phrase refers. If we can find a paraphrase that communicates the same information without implying that an absence exists, then we can ask ourselves whether or not common sense is really committed to the existence of the absence. (1) and (2) can easily be paraphrased using negative statements, as in (3) and (4):
(3) It was widely noticed that John was not there.
(4) The crops failed because there was no rain.
Unless we accept Truthmaker Maximalism (2.1T.1), there is no reason to think that something exists corresponding to negative statements like ‘John was not there’ or ‘There was no rain.’ Moreover, common sense dictates that negative things, like shadows, holes, and privations, are not as real as positive things. One way to make sense of this principle of positivity would be to suppose that there are no absences or other negative things.
So, ordinary language doesn't provide any strong evidence for absences, and common sense seems to weigh in on the other side. Hence, a successful case for absences will have to rely on evidence from perception or causation.
ARGUMENTS FROM PERCEPTION OF ABSENCES We often see or hear absences. This isn't always a matter of seeing or hearing that something is absent (a phenomenon that Fred Dretske (1969) has labeled ‘epistemic’ seeing). Sometimes we simply see something without knowing that we are seeing it. Barwise and Perry (1983) describe scenes or situations that can be the object of such non-epistemic seeing. In English, such non-epistemic seeing of a scene is expressed by means of naked infinitive phrases, as in (5) and (6):
(5) I saw Mary run.
(6) I saw that Mary ran.
(6) reports epistemic seeing, using a complementary that-clause, while (5) reports the non-epistemic seeing of a scene (one consisting of Mary's running) by means of the naked infinitive form of the verb ‘run’. (5) can be true even if the speaker did not recognize Mary as the person in the scene, and even if the speaker did not recognize that it was running that the person was doing (as opposed to dancing or skipping).
Scenes and situations can involve absences. One can see Mary be absent, that is, one can see Mary's absence without seeing that Mary is absent. If so, it would seem that we must acknowledge that absences exist, in order that they may serve as the object of such non-epistemic perceiving.
Absences that we can see or hear include silence, darkness, the absence of dark spot on a light background, the absence of an elephant in the room, and a hole in a doughnut. If absences and other privations, like holes, are really perceived (see Sorenson 1984, 2009, Wright 2012), then they must exist, given the Reliable Perception Presumption (PEpist 4). Moreover, it seems that they cannot be mere logical constructions, since it is implausible to suppose that we perceive sets and other abstract entities.
One strategy for Anti-Absentists is to claim that we always perceive positive entities. For example, in supposing that we see holes, we are thinking in a confused way about what we are really perceiving. What we really perceive are either perforated objects or hole linings. (Both possibilities are explored in Lewis and Lewis 1970. See also Casati and Varzi 1994.) Each proposal has its difficulties. The main problem with treating perforation as a primitive property of material objects is that it leaves us unable to count holes. We would have to introduce an infinite set of properties—singly-perforated, doubly-perforated, and so on—and then we would have to add postulates allowing us to say that an n-ly-perforated object has the same number of holes as a set of n objects has members. Casati and Varzi (1994: 25–30) point out a number of problems with the Ludovician theory that holes are hole linings. When we enlarge a hole, we do not necessarily enlarge the hole-lining. In some cases, the hole-lining becomes smaller in area. In addition, the inside of a hole is not the same as the inside of a hole-lining, since the latter is inside the hole's host, not within the space we ordinarily associate with the hole. Finally, to create a hole is not to create the hole-lining. It is rather to hollow out the material contained by the lining. Similarly, to remove a hole-lining is not to remove the hole; it is in fact to make the hole larger.
The Lewises responded that each of the words and concepts involved—inside/outside, enlarge/shrink, create/remove—are systematically ambiguous, with one meaning when we think of the hole/hole-lining as an ordinary material object and another meaning when we think of the hole/hole-lining as a hole or absence. This is certainly a cost of the theory. A theory that predicts ambiguity where none seems apparent is to that extent less credible (PMeth 1.5). In addition, it is far from clear that the Ludovician account of holes can provide a systematic account of all of the ambiguities that would be needed to make the theory work.
17.5 Conclusion
We've examined in this chapter six theories of spatial location, three Substantivalist, two Relationist, and one hybrid. The Spatial-Quality version of Substantivalism combined a high degree of elegance with a good fit with common sense, despite its unorthodox postulation of unperceivable “qualities” associated with each possible location.
Relationism is even simpler than any Substantivalist account, requiring no spatial entities and no absolute distinction between motion and rest or between right- and left-handedness. However, the great stumbling block for all versions of Relationism comes in the consideration of empty space. (This may be why some of the most prominent Relationists of the past, including Aristotle, denied the very possibility of a vacuum.) As Hartry Field has argued, vacuums deprive Relationism of the straightforward verification of the standard axioms of geometry and so deprive it of the aid of representation theorems that could verify the natural emergence of distance and other quantities from the inherent structure of space. Relationists have to choose between heavy-duty Platonism with metaphysically fundamental units of distance and reliance on geometrical facts about merely possible bodies filling the empty spaces, with the consequent worries about finding actual truthmakers for the possible facts.
Notes
1. We will consistently use the word ‘place’ for a kind of thing, a part of space, and we use the word ‘location’ for the relation between a body and its place or between one body and other bodies.
2. This is equivalent to our definition of a weakly internal relation in Chapter 10, D10.2.1.1.
3. We do not mean to be ruling out the existence of immaterial beings here, only that if there are such things, that they are located in space in some direct way. So, for example, if there are Cartesian egos, then they are located in space only by being related to a body that is located in space. Which is to say: immaterial things, if there are any, are not determiners of relations of spatial distance.
18
Structure of Space: Points vs. Regions
In this chapter, we take up whether space and extended bodies are ultimately composed of points (and point-masses) or spatial regions (and voluminous bodie
s). Here there are three positions: Pointillism, according to which only points and point-sized bodies are fundamental; Voluminism, according to which the only fundamental things are regions and voluminous bodies; and Volume-Boundary Dualism, according to which both points and regions really exist and are equally fundamental. It is possible to combine any of these three positions with either Spatial Substantivalism (17.1T) or Spatial Relationism (17.1A). For Relationists, the issue concerns only bodies: are the fundamental units from which all extended bodies are composed point-sized, dimensionless bodies, or voluminous bodies? For Substantivalists, the issue concerns both spatial regions and bodies. Most Substantivalists will accept some principle of Body-Space Correspondence: the ultimate bodies are point-sized if and only if the ultimate parts of space are points.
Modern analytic geometry, especially since the work of Descartes, normally takes it for granted that points are the fundamental geometrical entities, with lines, surfaces, and spatial regions defined as certain sets or collections of points. However, this assumption, although natural and easily carried out, is by no means the only way to proceed. Other mathematicians, like Alfred Tarski and Alfred North Whitehead, have shown that it is possible to start instead with spheres or other three-dimensional objects as fundamental, defining points as sets of regions. One can also take both points and regions as fundamental, as was done by many nineteenth-century philosophers of mathematics, including Franz Brentano.
This issue is tightly bound up with the issue of the relative fundamentality of distance, on the one hand, and shape, volume, and contiguity, on the other (see 17.2T and 17.2A). If distance is the fundamental relation, then it is plausible to take the fundamental entities to be points. Conversely, if shape, volume, and contiguity are the fundamental properties and relations, then it is plausible to take extended regions as the fundamental entities.
It is possible to hold, as Alexander Pruss has suggested (in correspondence), that exact spatial distance between regions is the fundamental relation. Informally, we understand the claim that the distance between region A and region B is x to mean that x is the shortest distance between two points, one of which is from A and the other from B. Nonetheless, it could be that, metaphysically speaking, the distance relation holds between regions, and that “points” are a mere fiction, to be constructed mathematically from regions.
In Section 18.1, we look at the idea of constructing points from regions. This idea takes a little getting used to, since we are accustomed to thinking of regions as constructed from points, that is, as wholes containing points as parts. We also must more generally introduce logical constructions, a theoretical tool developed by analytic philosophers in the early twentieth century. Then we introduce, in Section 18.2, the basic terms of the debate for this chapter, which concerns whether points or regions are fundamental. In Section 18.3, we lay out three arguments against Pointillism, the thesis that points are the only fundamental entities, namely, the arguments from Finitism, from mathematical paradoxes, and from the nature of contact. Finally, we turn in Section 18.4 to a consideration of the two versions of Anti-Pointillism: Voluminism, and Volume-Boundary Dualism. Voluminism takes only voluminous regions to be fundamental, while Volume-Boundary Dualism includes both voluminous regions and points, surfaces, and other boundaries as equally fundamental.
18.1 Constructing Points from Regions
The construction of points as sets of regions was first proposed by Alfred North Whitehead (1919) using his “method of extensive abstraction”. Alfred Tarski developed an especially elegant version of it in his classic paper, “Foundations of the Geometry of Solids” (Tarski 1956). Given a domain of regions, we can define a point as a certain set of regions; intuitively, a point is the set of all of the regions containing that point. We require that any two regions belonging to a single point overlap each other, and we also require that for every region within a point, there is a second region in that point that is a proper part of the first region. This ensures that, in effect, the set of regions mutually intersect in a single, dimensionless point, even if the original domain contains no such unextended regions. Once we have constructed a domain of such points, we can then add the usual geometrical axioms as postulates.
18.1.1 Digression on logical constructions
SET-THEORETIC CONSTRUCTIONS Alfred North Whitehead and Bertrand Russell introduced into philosophy the technique of logical construction in the early twentieth century. Making use of the mathematical theory of sets, they constructed mathematical objects that could fill the role in thought that was played by various kinds of entities. Thus, they promised a dramatic simplification of our metaphysical theories of the world. For example, instead of believing that the world consists of three things, points, regions of space, and sets, Russell and Whitehead showed that we could get by with just two, either sets and points or sets and regions. We can then identify things in the missing third category with special kinds of sets. We could suppose that spatial regions are nothing but sets of points or that points are nothing but sets of extended regions. Metaphysicians have used the method of logical constructions to reduce our ontological commitments by eliminating many different kinds of things from the list of fundamental entities, including propositions, properties, relations, and events.
The method of logical construction provides a way of carrying out the program of conceptual grounding or conceptual reduction that we discussed in Section 3.4. The proponent of a logical construction can argue that there is in reality nothing that fits exactly to the conceptual profile of the entity to be replaced by the construction. Instead, we appeal to the essence of the relevant concept, arguing that that concept is such as to play a role in the formulation of propositions that are made true by the facts described in the logical construction. Thus, logical constructions provide a means for ontological deflation of the original theory, showing how that theory can be true in a world with both fewer things and fewer kinds of things. We don't have to suppose that the things replaced by logical constructions really exist, although we can affirm the truth of ordinary propositions made in terms of the replaced entities.
In addition to the style of logical construction favored by Russell and Whitehead, there is a competing strategy that uses mereology in place of set theory. Mereology, developed by Lesniewicz, Goodman, and Leonard and discussed in Chapter 23, is the calculus of parts and wholes. However, as we shall see in Chapters 22 and 23, it is not at all obvious that a whole is nothing more than its parts (taken collectively), so it is not obvious that a mereological construction enables us to reduce the number of fundamental categories of things. In addition, the mereological strategy couldn't be used to reduce points to regions, since a sum of regions is an even bigger region, not an approximation to a point.
Logical constructions in this sense—the replacement of apparently concrete things by sets within a particular theory—have both advantages and disadvantages. The main advantage to a logical construction is greater ontological economy, as demanded by Ockham's Razor (PMeth 1.4.1). By eliminating a whole category of things and replacing them with sets, we can reduce the primitive qualitative complexity of the world as we represent it.
The principal disadvantage to a logical construction is that it forces us to eliminate or reduce the apparently fundamental properties and relations that were borne by the entities being replaced by sets. If we show that entities of kind K (such as regions of space) are in reality nothing but sets, then we cannot suppose that entities of kind K enter into any natural relations other than the natural or fundamental relations that sets enter into.
What sorts of natural or fundamental relations do sets stand in relations to other things, including other sets? As we have discussed, it is plausible to suppose that sets are involved in only a single natural or fundamental relation: that of membership. If x is a member of set S, then x's membership in S is a fundamental fact about S, not reducible or explainable in terms of other relations. However, that's it. If S stands in any other r
elation to a thing, the holding of that other relation must be explained in terms of S's members, and no other facts about S. This is Membership the Only Fundamental Set-Relation (PMeta 3).
PLURALIST CONSTRUCTIONS Besides sets and mereological sums, there is a third possible approach to treating regions or points as something other than fundamental entities: we could think of regions as pluralities of points or of points as pluralities of regions. On this view, a plurality is fundamentally ‘they’ rather than an ‘it’.
Pluralist constructions could be used to eliminate regions in favor of points. To say that an object occupies a region is simply to say that there are some points that it occupies. To say that a region is connected is to say that there are some points and that any two of them can be connected by a path consisting entirely of some others of them.
Pluralism could also be used in the opposite direction, in parallel with the Whitehead-Tarski reduction of points to regions. To say that a body occupies a point is merely to say that there are some nested, concentric spherical regions and the body occupies some of them. (For more information on the plural quantification involved in the use of these plural pronouns, see Boolos 1998.)
Pluralists who reduce regions to points must embrace the view that all natural and fundamental properties and relations take points and only points as their relata. Similarly, pluralists who reduce regions to points must suppose that all natural relations take regions and only regions as their relata. Therefore, pluralists will have to suppose that there exist what are known as natural multigrade relations, relations that relate an indeterminate number of things, even infinitely many. For example, suppose that the points-only pluralist wants to say that a certain region has a volume of one cubic meter. This will have to be understood as a relation among points: there are some points, and these points stand together in the filling-one-cubic-meter relation, a relation that relates an infinity of things. If we extend the pluralist conception to extended masses, we can say that having a mass of one gram is a relation borne jointly by an infinity of point-masses.
The Atlas of Reality Page 63
