Locations, bodies, and sets [electronic resource]
Keywords: Philosophy.

Mereology, as a form of philosophical or applied research, begins with the assumption that objects have parts. Cars, people, planets, and galaxies are organized part-to-whole. Indeed the entire range of concrete entities will be conceived as having a certain decompositional makeup. The study of mereology is an attempt to discern this structure and to formalize these notions in a regimented theory. The theory of nominalism has always maintained close ties with formal mereology. Nominalism is the view that abstract objects do not exist and that spatiotemporal objects exhaust the domain of existing things. Formal mereology evolved from attempts to provide a system sufficiently powerful to supplant set theory as a foundation for mathematics without abstract references of any kind. Ambitions such as these continue to the present in mereology and metaphysics more generally. Thus two important questions emerge concerning the feasibility of nominalistic mereology. Firstly, could it be, despite the assumptions of the project, that any such system unavoidably will contain references to abstract entities? And secondly, supposing we could erect such a system, how much of reality's decompositional structure could any nominalistically acceptable formal mereology capture? Ultimately, I claim that there can be no such thing as a nominalistic formal mereology in the sense envisaged. Moreover any remotely acceptable system will fail to capture the entire part-to-whole structure of concrete objects. In the dissertation, we encounter three major problems with the conception of a nominalistic mereology. The first failure concerns the status of the parthood relation itself. I argue that, although the parthood relation may be understood in some sense as an in re universal or spatiotemporally located entity, it is clearly not a particular concretum or trope of any kind. It must be multiply located and repeated wholly amid its relata. The second failure concerns the conceptions required to represent the cohesiveness of physical objects. A universe is not merely a mereological whole. For its dimensional parts are interconnected in complex ways. Either a formal topology or mereology with connectedness predicates will be required to represent the topological properties of concrete objects. And these will entangle us in commitments to set-theoretic constructions. Finally, nominalistic systems will be far too weak. To demonstrate this we take pains to select a language which is maximally acceptable. But we find that the richness of infinite spatial structures exceeds our ability to capture them in any first-order theory. Ontology. We first identify what ontological distinctions a reasonably expressive language must be able to make. An assumption of nominalism will imply that sets be rejected in favor of extensional mereological fusions. Only concrete entities must be assumed to exist. Among these are so-called locations which I define as fusions of either material or material-free substances. Locations are extensional in the mereological sense and are closed under unrestricted fusions. The existence of movements on the part of persons and motions of inanimate objects imply that subparts of reality have less dimensions than that of the entire system. Persons are observers figuring in a multitude of localized mereological arrangements. And they are capable of enduring changes in their proper parts. Although it might be thought that conceiving of reality in this way supersedes a purely nominalistic account or falls outside the pales of formal metaphysics, I claim, based on features of our relation to our bodies, that some such account must be adopted. Perhaps ironically, a view of the physical world as a comprised of situated persons provides a way to obviate explicit commitment to the topological properties and sets required to represent the interconnectedness of physical universes. Persons have intrinsically interconnected locations within a single spatiotemporally closed universe. Hence we arrive at a view that the objects postulated by nominalism are those connected via locations to our bodies. The status of the parthood relation as multiply located entails that our nominalist accept some notion of mereological state of affairs. A maximally nominalistic ontology will therefore consist of concrete individuals and mereological arrangements involving them. Some states of affairs are localized and obtain at various sub-locations of reality, but others will hold regardless of one's immediate location. I suggest that the distinction between localized and non-localized situations helps to explain issues related to time, simultaneity at a distance, and tense. Mereologic. Having provided a maximally nominalistic ontology, we can then turn our attention to defining formal notions and modeling reasoning over the selected domain. Our pilot system is a modal logic of mereology tailored precisely to the ontology. We employ a hybrid modal language. Hybrid languages are extensions of standard modal languages in which references can be made to individual objects by so-called `nominals'. The latter are atomic formulae functioning like constants in the first-order language. We adopt an extension H_m of Arthur Prior's nominal tense language with additional operators for various part and extension relations. Although expressively weak in comparison to first-order mereologies, it is shown formally that H_m is capable of denoting nominalistically acceptable states of affairs. Given the modal nature of hybrid languages, both localized and non-localized types of situation are representable. Formulas are evaluated relative to a particular location. But there are, in addition, those which "lift" the interpretation to a global perspective. Each formula of H_m is shown to represent an acceptable state of affairs relating individuals part-to-whole. In nominalistic spirit, arithmetical features and principles are thereby eliminated. In contrast to first-order systems, in H_m, counting expressions are undefinable and arithmetical facts hold only over distinguished objects. As for logics, we provide axiom systems and demonstrate the existence of various mereologics for classes of extensional mereological structures. In a novel Henkin construction, I demonstrate an axiom system analogous to Leonard and Goodman's "General Extensional Mereology" is complete with respect to the traditional classes of partial orders up to zero-deleted Boolean algebras. General completeness results for varieties of infinite atomic and atomless Boolean algebras are also demonstrated. Morphisms germane to modal logic are of equal importance in formal mereology. They allow us to gauge precisely the structural details seized by our adopted languages. Essentially, mereological reasoning is encapsulated in the notion of what I call mereobisimulation---a morphism stronger than bisimulation but weaker than a strong homomorphism. And I situate mereo-reasoning with its targeted nominalistic restrictions in relation to first-order logic: it is exactly the mereo-bisimilar fragment of the first-order logic. Can H_m detect the subtle differences between distinct parts of reality? I argue that the best way to answer this is first to identify suitable models representing the structural features we wish to preserve - in particular those that represent the decomposition of space. Then one proceeds to test how much structure the language can ``see'' of them. Two mathematical models are indistinguishable by H_m-formulas if there is a mereo-bisimulation amid them. Thus if H_m detects no differences between two models - one which has the structural features of locations and another which clearly does not - then, a fortiori, H_m will fail to capture the corresponding structural details in reality. Well-known, adequately proved results in the theory of Boolean algebras indicate that certain mathematical structures called complete Boolean algebras have the requisite features of the structure of unrestrictedly fused locations. Taking some results proven by Tarski and MacNeille in the thirties for granted, I show that any infinite n-dimensional atomless or atomic Boolean algebra expanded with a finite distinguished elements is mereo-bisimilar to its corresponding Boolean completion. In particular, I show that there is a sound and complete proof system for the class of regular open sets of R^n for finite n and the class of infinite atomic complete Boolean algebras. Our answer to the second question can then be summarized as follows. If reality contains infinitely many locations, then we will lose the ability to discriminate between uncountably many of them. Indeed if there are infinitely many locations and these decompose to a floor of atoms, single H_m-formulas will conflate reality with a finite structure. If, however, there are infinitely many locations and some of these contain no atoms (or if all are completely atomless), then up to mereo-bisimulation, portions of reality will be conflated with "pixelated" or geometrically extended, unanalyzable objects. Crystallizing on the structure of an infinite dimensional system will therefore be impossible. In conclusion, I urge that a formal language for mereology should not be restricted on nominalistic grounds. We should be inclined, despite any reluctance, to incorporate terms for sets and set-quantifiers.