It is also widely used today as a term describing any work on system decomposition (such as that of Gabriel Kron or Maurice Jessel), or parts, wholes and boundaries. There is also great interest in such ideas in computer science and theoretical physics, often using Sheaf Theory, Topos Theory or Category Theory. In particular see the work of Steven Vickers on (parts of) specifications, Joseph Goguen on physical systems and Tom Etter on Link Theory.
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 Lesniewski elaborated this "Polish mereology" over the course of the 20th century. For a good 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 of a historical nature.
A.N. Whitehead planned but never published a fourth volume of Principia Mathematica on geometry. 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 Leonard completed a Harvard Ph.D. 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 well-surveyed in Simons (1987) and Casati and Varzi (1999).
Standard university texts on logic and mathematics are silent about mereology, which has undoubtedly contributed to its undeserved obscurity. Although mereology is an application of mathematical logic, arguably a sort of "proto-geometry," it has been wholly developed by logicians, ontologists, and computer scientists, especially those working in artificial intelligence.
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.
The treatment, terminology, and hierarchical organization below follow Casati and Varzi (1999: chpt. 3) closely. 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.
A mereological system requires at least one primitive binary relation (dyadic predicate). The most conventional choice for such a relation is Parthood (also called "inclusion"), "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:
Systems vary in what relations they take as primitive and as defined. For example, in extensional mereologies (defined below), Parthood can be defined from Overlap as follows:
The axioms are:
The systems in the table below are partially ordered by inclusion, in the sense that if all the theorems of system A are also theorems of system B, but the converse is not necessarily true, then B includes A. The resulting Hasse diagram is similar to that in Fig. 2, and Fig. 3.2 in Casati and Varzi (1999: 48).
|M1-M3||Parthood is a partial order||M||M1–M3|
|M4||Weak Supplementation||MM||M, M4|
|M5||Strong Supplementation||EM||M, M5|
|M6||General Sum Principle (Sum)|
|M8||Unrestricted Fusion||GM||M, M8|
|M8'||Unique Fusion||GEM||EM, M8'|
|M9||Atomicity||AGEM||M2, M8, M9|
|AGEM||M, M5', M8|
There are two equivalent ways of asserting that the universe is partially ordered: assume either M1–M3, or that Proper Parthood is transitive and asymmetric, hence a strict partial order. Either axiomatization results in the system M. M2 rules out closed loops formed using Parthood; so that the part relation is well-founded. Sets are well-founded if the axiom of Regularity is assumed. The literature contains occasional philosophical and common sense objections to the transitivity of Parthood.
M4 and M5 are two ways of asserting supplementation, the mereological analog of set complementation, with M5 being stronger because M4 is derivable from M5. M and M4 yield minimal mereology, MM. MM, reformulated in terms of Proper Part, is Simons's (1987) preferred minimal system.
In any system in which M5 or M5' are assumed or can be derived, then it can be proved that two objects having the same proper parts are identical. This property is known as Extensionality, a term borrowed from set theory, for which extensionality is the defining axiom. Mereological systems in which Extensionality holds are termed extensional, a fact denoted by including the letter E in their symbolic names.
M6 asserts that any two underlapping objects have a unique sum; M7 asserts that any two overlapping objects have a unique product. If the universe is finite or if Top is assumed, then the universe is closed under sum. Universal closure of Product and of supplementation relative to W requires Bottom. W and N are, evidently, the mereological analog of the universal and null sets, and Sum and Product are likewise the analogs of set-theoretical union and intersection. If M6 and M7 are either assumed or derivable, the result is a mereology with closure.
Because Sum and Product are binary operations, M6 and M7 admit the sum and product of only a finite number of objects. The fusion axiom, M8, enables taking the sum of infinitely many objects. The same holds for Product, when defined. At this point, mereology often invokes set theory, but any recourse to set theory is eliminable by replacing a formula with a quantified variable ranging over a universe of sets by a schematic formula with one free variable. 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. Hence any axiom with sets can be replaced by an axiom schema with monadic atomic subformulae. M8 and M8' are schemas of just this sort. The syntax of a first-order theory can describe only a denumerable number of sets; hence only denumerably many sets may be eliminated in this fashion, but this limitation is not binding for the sort of mathematics contemplated here.
If M8 holds, then W exists for infinite universes. Hence Top need be assumed only if the universe is infinite and M8 does not hold. Curiously, Top (postulating W) is not controversial, but Bottom (postulating N) is. Leśniewski rejected Bottom and most mereological systems follow his example (an exception is the work of Richard Milton Martin). Hence while the universe is closed under sum, the product of objects that do not overlap is typically undefined. A system with W but not N is isomorphic to:
If sets are admitted, M8 asserts the existence of the fusion of all members of any nonempty set. Any mereological system in which M8 holds is called general, and its name includes G. In any general mereology, M6 and M7 are provable. Adding M8 to an extensional mereology results in general extensional mereology, abbreviated GEM; moreover, the extensionality renders the fusion unique. Conversely, if the fusion asserted by M8 is assumed unique, so that M8' replaces M8, then Tarski (1929) showed that M3 and M8' suffice to axiomatize GEM, a remarkably economical result. Simons (1987: 38–41) lists a number of GEM theorems.
M2 and a finite universe necessarily imply Atomicity, namely that everything is either an atom or includes atoms among its proper parts. If the universe is infinite, Atomicity requires M9. Adding M9 to any mereological system X results in the atomistic variant thereof, denoted AX. Atomicity permits economies. For instance, assuming M5' implies Atomicity and extensionality, and yields an alternative axiomatization of AGEM.
Much early work on mereology was motivated by a suspicion that set theory was ontologically suspect, and that Occam's Razor requires that one minimise the number of posits in one's theory of the world and of mathematics. Mereology replaces talk of "sets" of objects with talk of "sums" of objects, objects being no more than the various things that make up wholes.
Many logicians and philosophers reject these motivations, on such grounds as:
For a survey of attempts to found mathematics without using set theory, see Burgess and Rosen (1997).
In the 1970s, thanks in part to Eberle (1970), it gradually came to be understood that one can employ mereology regardless of one's ontological stance regarding sets. This understanding is called the "ontological innocence" of mereology. This innocence stems from the mereology being formalizable in either of two equivalent ways:
In set theory, singletons are "atoms" which have no (non-empty) proper parts; many consider set theory useless or incoherent (not "well-founded") if sets cannot be built up from unit sets. The calculus of individuals was thought to require that an object either have no proper parts, in which case it is an "atom," or to be the mereological sum of atoms. Eberle (1970) showed how to construct a calculus of individuals lacking "atoms", i.e., one where every object has a "proper part" (defined below) so that the universe is infinite.
There are analogies between the axioms of mereology and those of standard Zermelo-Fraenkel set theory (ZF), if Parthood is taken as analogous to subset in set theory. On the relation of mereology and ZF, also see Bunt (1985). One of the very few contemporary set theorist to discuss mereology is Potter (2004).
Lewis (1991) went further, showing informally that mereology, augmented by a few ontological assumptions and plural quantification, and some novel reasoning about singletons, yields a system in which a given individual can be both a member and a subset of another individuals. In the resulting system, the axioms of ZFC (and of Peano arithmetic) are theorems.
Forrest (2002) revises Lewis's analysis by first formulating a generalization of CEM, called "Heyting mereology," whose sole nonlogical primitive is Proper Part, assumed transitive and antireflexive. There exists a "fictitious" null individual that is a proper part of every individual. Two schemas assert that every lattice join exists (lattices are complete) and that meet distributes over join. On this Heyting mereology Forrest erects a theory of pseudosets, adequate for all purposes to which sets have been put.
To date, the only persons well trained in mathematics to write on mereology have been Alfred Tarski and Rolf Eberle. Eberle (1970) clarified the relation between mereology and Boolean algebra, and mereology and set theory. He is one of the very few contributors to mereology to prove sound and complete each system he describes.
Topological notions of boundaries and connection can be married to mereology, resulting in mereotopology; see Casati and Varzi (1999: chpts. 4,5). Whitehead's 1929 Process and Reality contains a good deal of informal mereotopology.
Simons devotes considerable effort to elucidating historical notations. The notation of Casati and Varzi has become more or less standard for current research. Both books include excellent bibliographies.
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