[1][2] 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.
Different axiomatizations of mereology have been applied in § Metaphysics, used in § Linguistic semantics to analyze "mass terms", used in the cognitive sciences,[1] and developed in § General systems theory.
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.
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.
Ivor Grattan-Guinness (2001) sheds much light on part-whole reasoning during the 19th and early 20th centuries, and reviews how Cantor and Peano devised set theory.
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.
In 1930, Henry S. Leonard completed a Harvard PhD dissertation in philosophy, setting out a formal theory of the part-whole relation.
A basic choice in defining a mereological system, is whether to allow things to be considered parts of themselves (reflexivity of parthood).
A mereological "system" is a first-order theory (with identity) 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.
Following each symbolic axiom or definition is the number of the corresponding formula in Casati and Varzi, written in bold.
Either axiomatization results in the system M. M2 rules out closed loops formed using Parthood, so that the part relation is well-founded.
The formula comes out true (is satisfied) whenever the name of an object that would be a member of the set (if it existed) replaces the free variable.
Leśniewski rejected Bottom, and most mereological systems follow his example (an exception is the work of Richard Milton Martin).
A system with W but not N is isomorphic to: Postulating N renders all possible products definable, but also transforms classical extensional mereology into a set-free model of Boolean algebra.
On the converse, however, if the fusion asserted by M8 is assumed unique, so that M8' replaces M8, then—as Tarski (1929) had shown—M3 and M8' suffice to axiomatize GEM, a remarkably economical result.
[6] Similarly, Ernst Schröder, in "Vorlesungen über die Algebra der Logik" (1890),[8] also used the mereological conception.
[6] It was Gottlob Frege, in a 1895 review of Schröder's work,[9] who first laid out the difference between collections and mereological sums.
[6] The fact that Ernst Zermelo adopted the collective conception when he wrote his influential 1908 axiomatization of set theory[10][11] is certainly significant for, though it does not fully explain, its current popularity.
Eberle (1970), however, showed how to construct a calculus of individuals lacking "atoms", i.e., one where every object has a "proper part", so that the universe is infinite.
[13] Philosopher David Lewis, in his 1991 work Parts of Classes,[13] axiomatized Zermelo-Fraenkel (ZFC) set theory using only classical mereology, plural quantification, and a primitive singleton-forming operator,[14] governed by axioms that resemble the axioms for "successor" in Peano arithmetic.
"[18] Potter says Lewis "could just as easily have said, gritting his teeth, that somehow, he knows not how, we do understand what it means to speak of membership, in which case there would have been no need for the rest of the book.
"[16] Forrest (2002) revised Lewis's analysis by first formulating a generalization of CEM, called "Heyting mereology", whose sole nonlogical primitive is Proper Part, assumed transitive and antireflexive.
In a series of chapters in the books he published in the last decade of his life, Richard Milton Martin set out to do what Goodman and Quine had abandoned 30 years prior.
A recurring problem with attempts to ground mathematics in mereology is how to build up the theory of relations while abstaining from set-theoretic definitions of the ordered pair.
Burgess and Rosen (1997) provide a survey of attempts to found mathematics without using set theory, such as using mereology.
A hierarchical version of Gabriel Kron's Network Tearing was published by Keith Bowden (1991), reflecting David Lewis's ideas on gunk.
See also the work of Steve Vickers on (parts of) specifications in computer science, Joseph Goguen on physical systems, and Tom Etter (1996, 1998) on link theory and quantum mechanics.
Bunt (1985), a study of the semantics of natural language, shows how mereology can help understand such phenomena as the mass–count distinction and verb aspect[example needed].