Locations, Bodies, and Sets: A model theoretic investigation into nominalistic mereologies
Jeremy Meyers
Abstract:
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.
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