Mereology


Mereology is the philosophical study of part-whole relationships, also called parthood relationships. As a branch of metaphysics, mereology examines the connections between parts and their wholes, exploring how components interact within a system. This theory has roots in ancient philosophy, with significant contributions from Plato, Aristotle, and later, medieval and Renaissance thinkers like Thomas Aquinas and John Duns Scotus. Mereology was formally axiomatized in the 20th century by Polish logician Stanisław Leśniewski, who introduced it as part of a comprehensive framework for logic and mathematics, and coined the word "mereology".
Mereological ideas were influential in early, and formal mereology continues to be used by some working on the . Different axiomatizations of mereology have been applied in, used in to analyze "mass terms", used in the cognitive sciences, and developed in. Mereology has been combined with topology, for more on which see the article on mereotopology. Mereology is also used in the foundation of Whitehead's point-free geometry, on which see Tarski 1956 and Gerla 1995. Mereology is used in discussions of entities as varied as musical groups, geographical regions, and abstract concepts, demonstrating its applicability to a wide range of philosophical and scientific discourses.
In metaphysics, mereology is used to formulate the thesis of "composition as identity", the theory that individuals or objects are identical to mereological sums of their parts. A metaphysical thesis called "mereological monism" suggests that the version of mereology developed by Stanisław Leśniewski and Nelson Goodman serves as the general and exhaustive theory of parthood and composition, at least for a large and significant domain of things. This thesis is controversial, since parthood may not seem to be a transitive relation in some cases, such as the parthood between organisms and their organs. Nevertheless, GEM's assumptions are very common in mereological frameworks, due largely to Leśniewski influence as the one to first coin the word and formalize the theory: mereological theories commonly assume that everything is a part of itself, that a part of a part of a whole is itself a part of that whole, and that two distinct entities cannot each be a part of the other, so that the parthood relation is a partial order. An alternative is to assume instead that parthood is irreflexive but still transitive, in which case antisymmetry follows automatically.

History

Informal part-whole reasoning was consciously invoked in metaphysics and ontology from Plato and Aristotle onwards, and more or less unwittingly in 19th-century mathematics until the triumph of set theory around 1910. Metaphysical ideas of this era that discuss the concepts of parts and wholes include divine simplicity and the classical conception of beauty.
Ivor Grattan-Guinness explains part-whole reasoning during the 19th and early 20th centuries, and reviews how Cantor and Peano devised set theory. It appears that the first to reason consciously and at length about parts and wholes was Edmund Husserl, in 1901, in the second volume of Logical Investigations – Third Investigation: "On the Theory of Wholes and Parts". However, the word "mereology" is absent from his writings, and he employed no symbolism even though his doctorate was in mathematics.
Stanisław Leśniewski coined "mereology" in 1927, from the Greek word μέρος, to refer to a formal theory of part-whole he devised in a series of highly technical papers published between 1916 and 1931, and translated in Leśniewski. Leśniewski's student Alfred Tarski, in his Appendix E to Woodger and the paper translated as Tarski, greatly simplified Leśniewski's formalism. Other students of Leśniewski elaborated this "Polish mereology" over the course of the 20th century. For a selection of the literature on Polish mereology, see Srzednicki and Rickey. For a survey of Polish mereology, see Simons. Since 1980 or so, however, research on Polish mereology has been almost entirely historical in nature.
A. N. Whitehead planned a fourth volume of Principia Mathematica, on geometry, but never wrote it. His 1914 correspondence with Bertrand Russell reveals that his intended approach to geometry can be seen, with the benefit of hindsight, as mereological in essence. This work culminated in Whitehead and the mereological systems of Whitehead.
In 1930, Henry S. Leonard completed a Harvard PhD dissertation in philosophy, setting out a formal theory of the part-whole relation. This evolved into the "calculus of individuals" of Goodman and Leonard. Goodman revised and elaborated this calculus in the three editions of Goodman. The calculus of individuals is the starting point for the post-1970 revival of mereology among logicians, ontologists, and computer scientists, a revival surveyed in Simons, Casati and Varzi, and Cotnoir and Varzi.

Theory

A basic choice in defining a mereological system, is whether to allow things to be considered parts of themselves. In naive set theory a similar question arises: whether a set is to be considered a "member" of itself. In both cases, "yes" gives rise to paradoxes analogous to Russell's paradox: Let there be an object O such that every object that is not a proper part of itself is a proper part of O. Is O a proper part of itself? No, because no object is a proper part of itself; and yes, because it meets the specified requirement for inclusion as a proper part of O. In set theory, a set is often termed an improper subset of itself. Given such paradoxes, mereology requires an axiomatic formulation.
A mereological "system" is a first-order theory whose universe of discourse consists of wholes and their respective parts, collectively called objects. Mereology is a collection of nested and non-nested axiomatic systems, not unlike the case with modal logic.
The treatment, terminology, and hierarchical organization below follow Casati and Varzi closely. For a more recent treatment, correcting certain misconceptions, see Hovda. Lower-case letters denote variables ranging over objects. Following each symbolic axiom or definition is the number of the corresponding formula in Casati and Varzi, written in bold.

Definitions

A mereological system requires at least one primitive binary relation. The most conventional choice for such a relation is parthood, "x is a part of y", written Pxy. Nearly all systems require that parthood partially order the universe. The following defined relations, required for the axioms below, follow immediately from parthood alone:
  • An immediate defined predicate is "x is a proper part of y", written PPxy, which holds if Pxy is true and Pyx is false. Compared to parthood, ProperPart is a strict partial order.
  • Overlap: x and y overlap, written Oxy, if there exists an object z such that Pzx and Pzy both hold.
  • Underlap: x and y underlap, written Uxy, if there exists an object z such that x and y are both parts of z.
Overlap and Underlap are reflexive, symmetric, and intransitive.
Systems vary in what relations they take as primitive and as defined. For example, in extensional mereologies, parthood can be defined from Overlap as follows:

Notations

There have been many notations for mereology. The table below builds on the comparison in Peter Simons, "Parts", page 99.
Operation or termSEPSimonsLeonard–GoodmanTarskiPietruszczakOthers product of x and y
the sum of x and y
the universal object a*, Un
x is the sum of the set α,S, F
x is the product of the set αP, N
the complement of x
the sum of the x such that
the product of the x such that

Axioms

The axioms are:
  • Parthood partially orders the universe:
  • M4, Weak Supplementation: If PPxy holds, there exists a z such that Pzy holds but Ozx does not.
  • M5, Strong Supplementation: If Pyx does not hold, there exists a z such that Pzy holds but Ozx does not.
  • M5', Atomistic Supplementation: If Pxy does not hold, then there exists an atom z such that Pzx holds but Ozy does not.
  • Top: There exists a "universal object", designated W, such that PxW holds for any x.
  • Bottom: There exists an atomic "null object", designated N, such that PNx holds for any x.
  • M6, Sum: If Uxy holds, there exists a z, called the "sum" or "fusion" of x and y, such that the objects overlapping of z are just those objects that overlap either x or y.
  • M7, Product: If Oxy holds, there exists a z, called the "product" of x and y, such that the parts of z are just those objects that are parts of both x and y.
  • M8, Unrestricted Fusion: Let φ be a first-order formula in which x is a free variable. Then the fusion of all objects satisfying φ exists.
  • M8', Unique Fusion: The fusions whose existence M8 asserts are also unique. Also called "Uniqueness of Composition". P.8'
  • M9, Atomicity: All objects are either atoms or fusions of atoms.