Mereology (; from Greek μέρος 'part' (root: μερε-, mere-) and the suffix -logy, 'study, discussion, science') is the philosophical study of part-whole relationships, also called parthood relationships. As a branch of ontology, 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 § Set theory, and formal mereology continues to be used by some working on the § Foundations of mathematics. Different axiomatizations of mereology have been applied in § Ontology, used in § Linguistic semantics to analyze "mass terms", used in the cognitive sciences, and developed in § General systems theory. 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 ontology, mereology is used to formulate the thesis of "composition as identity", the theory that individuals or objects are identical to mereological sums (also called fusions) of their parts. A metaphysical thesis called "mereological monism" suggests that the version of mereology developed by Stanisław Leśniewski and Nelson Goodman (commonly called General Extensional Mereology, or GEM) 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 (as claimed by GEM) 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 (reflexivity), that a part of a part of a whole is itself a part of that whole (transitivity), and that two distinct entities cannot each be a part of the other (antisymmetry), so that the parthood relation is a partial order. An alternative is to assume instead that parthood is irreflexive (nothing is ever a part of itself) but still transitive, in which case antisymmetry follows automatically.
History Informal part-whole reasoning was consciously invoked in metaphysics and ontology from Plato (in particular, in the second half of the Parmenides) 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 (2001) 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" (Husserl 1970 is the English translation). 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 (méros, "part"), 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 (1992). Leśniewski's student Alfred Tarski, in his Appendix E to Woodger (1937) and the paper translated as Tarski (1984), greatly simplified Leśniewski's formalism. Other students (and students of 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 (1984). For a survey of Polish mereology, see Simons (1987). 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 (1916) and the mereological systems of Whitehead (1919, 1920). 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 (1940). Goodman revised and elaborated this calculus in the three editions of Goodman (1951). 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 (1987), Casati and Varzi (1999), and Cotnoir and Varzi (2021).
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