CONTINUOUS VARIATION AND VOLUME-BOUNDARY DUALISM For Volume-Boundary Dualists, the best solution to the problem of continuous variation is to adopt the idea of distributional properties, as proposed by Josh Parsons (2004). On this view, the property of having a certain density at point P is a fundamental property of some extended region or extended material object E, where P is a metaphysically dependent part of E. Points are real entities, but they cannot exist except as parts of extended things. On this view, it is hard to resist the conclusion that points are bearers of fundamental properties like density: the property of an extended object E's having density d at point P just consists in P's having density d and in P's being a part of E. This distributional properties option leaves us with a class of unexplained, brute necessities, however, which is a cost by Ockham's Razor (PMeth 1.2). Why can't points exist as independent entities?
The best response to this question would be to suppose that points have no independent spatial location. The location of a part P relative to other parts of the extended object E and to the whole of E is a fundamental property of E as a whole, not a property that P possesses on its own. In contrast, the spatial location of E is a simple quality of E, and its distance from other extended objects is a matter of the degree of similarity between the two intrinsic spatial qualities. Thus, the spatial location of an extended whole E and of a point-sized part P are radically different in character, explaining why P cannot exist on its own. Extended parts of E also derive their own location and internal organization from the internal organization of E itself, but such extended parts can potentially exist on their own, since they are the potential bearers of fundamental spatial qualities.
The metaphysical necessities involved with the distributional properties option are relatively few, and we can provide an explanation for them in terms of the essences or natures of extended objects and their parts. The only possible bearers of fundamental spatial locations are extended material objects. Internal points have only a derived location, based upon the location of the whole object and the point's role within the internal organization of the extended object, where that organization is a fundamental distributional property. Consequently, points are metaphysically dependent, since they are not the sort of thing capable of holding a spatial location on their own.
CONTINUOUS VARIATION AND VOLUMINISM Given Continuous Variation, Voluminists could suppose that the fundamental properties are properties of average density, borne by regions of space or extended material bodies. There would have to be necessary connections among these properties. The average density of matter in region S must be the volume-weighted mathematical average of the average densities of the subregions belonging to any partition of S (that is, to any set of mutually exclusive and jointly exhaustive subregions of S). Ockham's Razor (PMeth 1.2) enjoins us to minimize such brute metaphysical necessities, making such an account unattractive.
What are the alternatives? Voluminists seem to have only three further options:
Deny the possibility of continuous variation, insisting on Spatial Aristotelianism.
Claim that densities are properties of points, with points as logical constructions from regions.
Opt for fundamental properties of voluminous bodies that are simultaneously properties of both punctual (pointlike) spatial location and punctual density.
VOLUMINIST OPTION 1: SPATIAL ARISTOTELIANISM Voluminists' first option is simply to deny that continuous variation is possible. This is, scientifically speaking, still a live option, since we still do not understand how matter is organized at very small scales. However, it would be a serious defect of the view if it were inconsistent with the mere possibility of continuous variation, since such variation seems possible, and since it may be, for all we know, actual.
VOLUMINIST OPTION 2: DENSITIES AS PROPERTIES OF SET-THEORETIC OR LUDOVICIAN CONSTRUCTIONS OR OF PLURALITIES We can reject the set-theoretic version of option 2 on the basis of Membership the Only Fundamental Set-Relation (PMeta 3). Sets of regions cannot have fundamental physical properties like density.
The Ludovician version of option 2 is also problematic, since the very definition of continuity presupposes that points are dimensionless and volume-less. It doesn't make sense to talk about a distribution as approximately continuous. A distribution that is only approximately continuous is simply not continuous at all. Moreover, it will be simply false that the Ludovician points, which are in reality very small bodies, have precise density-values. The Ludovician could try to make use of average values here, but only at the cost of introducing a large number of brute necessary connections among the various average values. The various average values must satisfy the arithmetical law of averages, but the Ludovician has no metaphysical explanation of why this should invariably happen if each average value is a primitive fact.
However, a pluralist solution might be worth considering here. On this view, one treats density as a joint property of a plurality of nested regions (like Tarski's nested spheres). Pluralists can distinguish between distributive and collective predications of groups. For example, consider (1) and (2):
(1) The members of the glee club are male.
(2) The members of the glee club harmonize well.
The predication in (1) is distributive: each of the members of the glee club is a male. The predication in (2), in contrast, is collective. No member of the club harmonizes well all by himself. Proposition (2) is talking about how they harmonize with one another. Similarly, the pluralists could treat density at a point as a collective property possessed by those spheres that together make up the point, by virtue of being concentric. This has the advantage over option 1 of attributing density to something physical and concrete, namely, the spheres considered collectively, rather than to an abstract object, like a set.
Some spheres, the x's, are collectively pointlike if and only if for all spheres y and z,
(1) if y and z are among the x's, then the intersection of y and z is also among the x's,
(2) if y intersects all of the x's, then y is among the x's, and
(3) if y is among the x's, then y overlaps each of the x's.
VOLUMINIST OPTION 3: FUNDAMENTAL SPATIAL-CUM-QUANTITATIVE PROPERTIES On option 3, an extended material object has a family of properties, each corresponding to an internal point. These properties are simultaneously spatial-location qualities and densities. Any two of these primitive punctual properties bears at least two different kinds of similarity to one another: similarity with respect to spatial location and similarity with respect to density. Two properties can be very similar in the spatial-location aspect and very different in the density aspect (representing two spatially close points with very different densities) or vice versa (representing two spatially distant points with similar densities). The space of colors provides an analogy. Two color-properties can be very similar with respect to hue but dissimilar with respect to brightness (like pink and maroon) or very similar with respect to brightness and very dissimilar with respect to hue (like pink and sky blue). Both spatial distance and relative density would then be internal relations between pairs of properties.
One advantage of option 3 is that it provides a uniform account of spatial distance: spatial distance is always an internal relation between spatial/quantitative properties.
However, option 3 is incompatible with both Classical and Non-Classical Truthmaker Theory (2.1T/2.1A.1T). In particular, it violates the One Truthmaker per Fundamental Property principle (PTruth 1), since a single entity, whether a trope or an ordinary particular, is supposed to serve as the truthmaker for the instantiation of determinates of both spatial location and density.
The main disadvantage of option 3 is that it provides no basis for thinking that point-sized material objects are impossible, and so no basis for denying that point-sized material objects are fundamental entities. If extended material objects each bear some infinite set of spatial-cum-quantitative properties, then why not suppose that there are infinitely many point-si
zed masses, each possessing exactly one of those properties? We can posit that, as a matter of metaphysical necessity, any material object must possess an infinite number of spatial-cum-quantitative properties, forming a continuous spatial region, but this necessity would seem to have to be a brute, unexplainable necessity.
Another objection to option 3 comes from the thought experiment of supercutting. If option 3 were true, it seems plausible that Hawthorne-Weatherson supercutting would be possible, resulting in a collection of isolated point-masses. On option 3, each of the resulting bodies would possess a single fundamental spatial-cum-quantitative property, so there is no simple explanation for the impossibility of the procedure. If the procedure is possible, it is arguable that the point-sized bodies were there all along, since there is no reason to think that merely cutting a body in half generates new entities, since there is no change in the fundamental properties that are instantiated. Thus, option 3 seems to be forced to give up the metaphysical priority of extended wholes, eliminating it as an option for making continuous variation compatible with that thesis.
18.4.2 Coincident boundaries: Actual vs. potential boundaries
If there could be several physical surfaces, edges, or points that coincide exactly in their spatial location, in their coincident boundaries, then this would pose a serious challenge to the combination of Voluminism with Spatial Monism (17.1T.1A.1A).
18.1A.2A.1 Coincident Boundaries. There are spatially coincident points (curves, surfaces).
Brentano (1976/1988) offered strong arguments for thinking that two distinct surfaces can co-exist in exactly the same place. Suppose the face of one cube is in direct contact with the face of another. Part of the surface of the one cube is then located in exactly the same place as part of the surface of the other, unless the two surfaces somehow fuse into a single entity. However, there seems to be little, if any reason, to believe that such fusion must take place. Each surface is dependent on a different entity, so why should something come into existence that is simultaneously dependent on both cubes? Why should mere contact necessitate the annihilation of the two distinct surfaces? If the two surfaces have natural properties, like color, it would seem that each could retain that distinct property, even when they coincide in space. It is hard to see why mere coincidence should necessitate that these properties be destroyed or altered.
Aristotelian philosophers have traditionally distinguished between actual and potential boundaries. A surface exists in actuality only when it is the outer boundary of some actual, extended thing. On this view, the surfaces that intersect the interior of extended things have merely potential existence. They could exist if the extended thing were actually divided along that surface. On this view, there can be actually coincident surfaces when two really distinct bodies are in contact, but there will be no actually coincident surfaces within the interior of an undivided body.
18.5 Conclusion
We have examined two major issues in the last two chapter. First, do places exist, and second, do material bodies (the things that have location) consist fundamentally of point-sized or voluminous parts? Our examination of the first issue was inconclusive, but we did find three theories that seemed to be the simplest: the Theory of Spatial Qualities (17.1T.1T), Spatial Monism, and Relationism (17.1A). We postponed our evaluation of Spatial (or Spatiotemporal) Monism until Chapter 24, where we will take up the question of how things persist through time, leaving us the Theory of Spatial Qualities and two versions of Relationism (Aristotelian 17.1A.1A and Modern 17.1A.1T). The gravest difficulty for both theories concerns the reality of empty space. How best to handle this problem turns on other metaphysical issues, like Actualism vs. Anti-Actualism (12.1A) and Realism (7.1T) vs. Trope Theory (8.2T). On the second major issue, we found a number of arguments that support Anti-Pointillism. The problem of continuous variation suggests that we take very seriously the mixed position of Volume-Boundary Dualism, despite the fact that it requires that we posit both points and regions as fundamental entities.
In the next chapter, we examine questions concerning the structure of time, including an issue that is similar to the debate between Pointillists and Anti-Pointillists, namely, the debate between Instantists and Intervalists.
Note
1. A Ludovician reduction of regions to points would not be very plausible, since all points are equally small and equally lacking in shape or finite volume. We could, in some contexts, take a body to be some point located somewhere near the intuitive center of gravity of the body, but it would be hard to see what sense could be made of a body's having a certain shape or volume.
19
The Structure of Time
In this chapter, we will examine three issues concerning the structure of time. First, are temporal intervals (extended periods of time) metaphysically fundamental entities or are they and their attributes wholly grounded in their constituent instants? Second, if there are fundamental intervals, are instants also fundamental or are they merely derived entities (such as sets of intervals)? Finally, is it metaphysically necessary that time have a beginning or is it possible that the past be infinitely long? We will take each of these up in successive sections.
19.1 Is Time Composed of Instants or Intervals?
In the case of time, we have a set of issues that closely parallel those concerning space. In the case of space, we considered whether it is spatial simples (points), extended spatial regions, or both that are metaphysically fundamental. There is a similar issue in the case of time. Does every finite temporal interval consist in a set of durationless instants in some set of mutual relations, or are temporal intervals metaphysically fundamental, with instants being fictions, logical constructions, or metaphysically dependent boundaries of intervals?
19.1T Instantism. Temporal intervals are not G-fundamental entities: intervals are wholly grounded in dimensionless instands.
19.1A Intervalism. There are extended temporal intervals that are G-fundamental.
G-fundamental entities are entities whose existence is not wholly grounded in the existence of other entities (as we defined in D3.5). In the rest of this chapter, when we use the term ‘fundamenta’, we will mean ‘G-fundamental’.
Just as in the case of space, it is plausible to assume that there is some kind of metaphysical correspondence between time and the fillers or occupiers of time:
Time-Process Correspondence. The fundamental temporal entities are instants (as opposed to intervals) if and only if the fundamental occupiers of time are temporally unextended things.
What would it take for Time-Process Correspondence to be false? We would have to have either temporally extended processes occupying infinitely many fundamental instants without temporal parts or durationless parts of extended processes without any corresponding fundamental temporal location. The second option seems quite far-fetched. If a process consists of an infinity of durationless “time-slices”, then surely there must be an infinite set of equally fundamental temporal locations—instants—for each of the instantaneous parts. Similarly, the first option is unattractive. If a process takes place through a series of real, fundamental temporal instants, then surely the process as a whole simply consists of a set of correspondingly instantaneous parts.
If Time-Process Correspondence is true, then Instantism is equivalent to Procedural Instantism, and Intervalism to Procedural Intervalism:
19.2T Procedural Instantism. No temporally extended process is fundamental: only their indivisible, dimensionless parts (time-slices) are fundamental.
19.2A Procedural Intervalism. Some extended processes are metaphysically fundamental.
Hereafter, we will use “Instantism” to refer to the combination of 19.1T and 19.2T and “Intervalism” to refer to the combination of 19.1A and 19.2A. Intervalism comes in several varieties, depending on whether we think that all intervals (and all extended occupiers of time) have proper temporal parts (parts that are finite but still shorter in duration).
19.3T Temporal
Finitism. It is impossible for any temporal interval to have infinitely many actual temporal parts.
19.3A Temporal Infinitism. It is possible for temporal intervals to have infinitely many actual temporal parts.
Instantists must be Temporal Infinitists, since there are infinitely many instants contained in any interval of time. Intervalists can go either way. Alfred North Whitehead, for example, was both an Intervalist and a Temporal Infinitist. However, the clearest and strongest arguments for Intervalism are arguments for Temporal Finitism. Consequently, in the rest of the chapter, we will consider arguments for and against Temporal Finitism.
Temporal Discretism is the thesis that time consists of discrete time-atoms, each taking up some finite duration. Temporal Discretism is a kind of Finitism: any finitely extended interval is made up of only finitely many indivisible units of time.
19.4A Temporal Discretism. There are extended occupiers of time without proper temporally extended parts.
19.4T Temporal Anti-Discretism (Infinite Divisibility). All extended occupiers of time have proper temporally extended parts.
All Discretists are Finitists, but must all Finitists be Discretists as well? As we saw in the analogous case of space, there is some tension between Infinite Divisibility and Finitism. Anti-Discretists who are also Finitists must be Aristotelian Finitists: they must make a distinction between actual and potential temporal parts of an interval. There are intervals (and extended metaphysical processes) that are metaphysically fundamental, and these have proper temporal parts, but only in the sense that they are potentially divisible into sub-intervals. A given process might have begun earlier or ended sooner for instance, and it has potential temporal parts corresponding to these alternative scenarios, but the temporal parts are merely potential and are dependent upon the whole.
The Atlas of Reality Page 67
