Mereology (/mɪəriˈɒlədʒi/; fromGreek μέρος 'part' (root: μερε-,mere-) and the suffix-logy, 'study, discussion, science') is the philosophical study of part-whole relationships, also calledparthood relationships.[1][2] As a branch ofmetaphysics, 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 fromPlato,Aristotle, and later,medieval andRenaissance thinkers likeThomas Aquinas andJohn Duns Scotus.[3] Mereology was formally axiomatized in the 20th century byPolishlogicianStanisław Leśniewski, who introduced it as part of a comprehensive framework for logic and mathematics, and coined the word "mereology".[2]
Mereological ideas were influential in early§ Set theory, and formal mereology has continued to be used by a minority in works on the§ Foundations of mathematics. Different axiomatizations of mereology have been applied in§ Metaphysics, used in§ Linguistic semantics to analyze "mass terms", used in thecognitive sciences,[1] and developed in§ General systems theory. Mereology has been combined withtopology, for more on which see the article onmereotopology. Mereology is also used in the foundation ofWhitehead'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.[1]
In metaphysics, mereology is used to formulate the thesis of "composition as identity", the theory that individuals or objects are identical tomereological sums (also calledfusions) of their parts.[3] A metaphysical thesis called "mereological monism" suggests that the version of mereology developed by Stanisław Leśniewski andNelson Goodman (commonly calledClassical Extensional Mereology, or CEM) serves as the general and exhaustive theory of parthood and composition, at least for a large and significant domain of things.[4] This thesis is controversial, since parthood may not seem to be a transitive relation (as claimed by CEM) in some cases, such as the parthood between organisms and their organs.[5] Nevertheless, CEM'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 apartial 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.
Informal part-whole reasoning was consciously invoked inmetaphysics andontology fromPlato (in particular, in the second half of theParmenides) andAristotle onwards, and more or less unwittingly in 19th-century mathematics until the triumph ofset theory around 1910. Metaphysical ideas of this era that discuss the concepts of parts and wholes includedivine simplicity and theclassical conception of beauty.
Ivor Grattan-Guinness (2001) sheds much light on part-whole reasoning during the 19th and early 20th centuries, and reviews howCantor andPeano devisedset theory. It appears that the first to reason consciously and at length about parts and wholes[citation needed] wasEdmund Husserl, in 1901, in the second volume ofLogical 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 studentAlfred 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 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 historical in nature.
A. N. Whitehead planned a fourth volume ofPrincipia Mathematica, ongeometry, but never wrote it. His 1914 correspondence withBertrand 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" ofGoodman 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), Casati and Varzi (1999), and Cotnoir and Varzi (2021).
A basic choice in defining a mereological system, is whether to allow things to be considered parts of themselves (reflexivity of parthood). Innaive 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 toRussell's paradox: Let there be an objectO such that every object that is not a proper part of itself is a proper part ofO. IsO 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 ofO. In set theory, a set is often termed animproper subset of itself. Given such paradoxes, mereology requires anaxiomatic formulation.
A mereological "system" is afirst-order theory (withidentity) whoseuniverse of discourse consists of wholes and their respective parts, collectively calledobjects. Mereology is a collection of nested and non-nestedaxiomatic systems, not unlike the case withmodal logic.
The treatment, terminology, and hierarchical organization below follow Casati and Varzi (1999: Ch. 3) closely. For a more recent treatment, correcting certain misconceptions, see Hovda (2008). 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 primitivebinary relation (dyadicpredicate). The most conventional choice for such a relation isparthood (also called "inclusion"), "x is apart ofy", writtenPxy. Nearly all systems require that parthoodpartially order the universe. The following defined relations, required for the axioms below, follow immediately from parthood alone:
Overlap and Underlap arereflexive,symmetric, andintransitive.
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:
Simons (1987), Casati and Varzi (1999) and Hovda (2008) describe many mereological systems whose axioms are taken from the above list. We adopt the boldface nomenclature of Casati and Varzi. The best-known such system is the one calledclassical extensional mereology, hereinafter abbreviatedCEM (other abbreviations are explained below). InCEM,P.1 throughP.8' hold as axioms or are theorems. M9,Top, andBottom are optional.
The systems in the table below arepartially ordered byinclusion, in the sense that, if all the theorems of system A are also theorems of system B, but the converse is notnecessarily true, then Bincludes A. The resultingHasse diagram is similar to Fig. 3.2 in Casati and Varzi (1999: 48).
| Label | Name | System | Included Axioms |
|---|---|---|---|
| M1 | Reflexivity | ||
| M2 | Antisymmetry | ||
| M3 | Transitivity | M | M1, M2, M3 |
| M4 | Weak Supplementation | MM | M, M4 |
| M5 | Strong Supplementation | EM | M, M5 |
| M5' | Atomistic Supplementation | ||
| M6 | Sum | ||
| M7 | Product | CEM | EM, M6, M7 |
| M8 | Unrestricted Fusion | GM | M, M8 |
| GEM | EM, 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 theuniverse ispartially ordered: Assume either M1-M3, or that Proper Parthood istransitive andasymmetric, hence astrict partial order. Either axiomatization results in the systemM. M2 rules out closed loops formed using Parthood, so that the part relation iswell-founded. Sets are well-founded if theaxiom 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 setcomplementation, with M5 being stronger because M4 is derivable from M5.M and M4 yieldminimal mereology,MM. Reformulated in terms of Proper Part,MM 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 asExtensionality, a term borrowed from set theory, for whichextensionality is the defining axiom. Mereological systems in which Extensionality holds are termedextensional, a fact denoted by including the letterE 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 ifTop is assumed, then the universe is closed underSum. Universal closure ofProduct and of supplementation relative toW requiresBottom.W andN are, evidently, the mereological analog of theuniversal andempty sets, andSum andProduct are, likewise, the analogs of set-theoreticalunion andintersection. If M6 and M7 are either assumed or derivable, the result is a mereology with closure.
BecauseSum andProduct are binary operations, M6 and M7 admit the sum and product of only a finite number of objects. TheUnrestricted Fusion axiom, M8, enables taking the sum of infinitely many objects. The same holds forProduct, when defined. At this point, mereology often invokesset theory, but any recourse to set theory is eliminable by replacing a formula with aquantified variable ranging over a universe of sets by a schematic formula with onefree variable. The formula comes out true (is satisfied) whenever the name of an object that would be amember of the set (if it existed) replaces the free variable. Hence any axiom with sets can be replaced by anaxiom schema with monadic atomic subformulae. M8 and M8' are schemas of just this sort. Thesyntax of afirst-order theory can describe only adenumerable 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, thenW exists for infinite universes. Hence,Top need be assumed only if the universe is infinite and M8 does not hold.Top (postulatingW) is not controversial, butBottom (postulatingN) is. Leśniewski rejectedBottom, and most mereological systems follow his example (an exception is the work ofRichard Milton Martin). Hence, while the universe is closed under sum, the product of objects that do not overlap is typically undefined. A system withW but notN is isomorphic to:
PostulatingN renders all possible products definable, but also transforms classical extensional mereology into a set-freemodel ofBoolean algebra.
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 calledgeneral, and its name includesG. In any general mereology, M6 and M7 are provable. Adding M8 to an extensional mereology results ingeneral extensional mereology, abbreviatedGEM; moreover, the extensionality renders the fusion unique. 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 axiomatizeGEM, a remarkably economical result. Simons (1987: 38–41) lists a number ofGEM theorems.
M2 and a finite universe necessarily implyAtomicity, namely that everything either is 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, denotedAX.Atomicity permits economies, for instance, assuming that M5' impliesAtomicity and extensionality, and yields an alternative axiomatization ofAGEM.
From the beginnings of set theory, there has been a dispute between conceiving of sets "mereologically", where a set is the mereological sum of its elements, and conceiving of sets "collectively", where a set is something "over and above" its elements.[6] The latter conception is now dominant, but some of the earliest set theorists adhered to the mereological conception:Richard Dedekind, in "Was sind und was sollen die Zahlen?" (1888), avoided the empty set and used the same symbol for set membership and set inclusion,[7] which are two signs that he conceived of sets mereologically.[6] Similarly,Ernst Schröder, in "Vorlesungen über die Algebra der Logik" (1890),[8] also used the mereological conception.[6] It wasGottlob Frege, in a 1895 review of Schröder's work,[9] who first laid out the difference between collections and mereological sums.[6] The fact thatErnst 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.[6]
In set theory,singletons are "atoms" that have no (non-empty) proper parts; set theory where sets cannot be built up from unit sets is a nonstandard type of set theory, callednon-well-founded set theory. The calculus of individuals was thought[by whom?] to require that an object either have no proper parts, in which case it is an "atom", or be the mereological sum of atoms. 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 theuniverse is infinite.
A detailed comparison between mereology, set theory, and asemantic "ensemble theory" is presented in chapter 13 of Bunt (1985);[12] when David Lewis wrote his famous§ Parts of Classes, he found that "its main thesis had been anticipated in" Bunt's ensemble theory.[13]
PhilosopherDavid Lewis, in his 1991 workParts of Classes,[13] axiomatizedZermelo-Fraenkel (ZFC) set theory using only classical mereology,plural quantification, and a primitivesingleton-forming operator,[14] governed by axioms that resemble the axioms for "successor" inPeano arithmetic.[15] This contrasts with more usual axiomatizations of ZFC, which use only the primitive notion ofmembership.[16] Lewis's work is named after his thesis that a class's subclasses are mereological parts of the class (in Lewis's usage, this means that a set's subsets, not counting the empty set, are parts of the set); this thesis has been disputed.[17]
Michael Potter, a creator ofScott–Potter set theory, has criticized Lewis's work for failing to make set theory any more easily comprehensible, since Lewis says of his primitive singleton operator that, given the necessity (perceived by Lewis) of avoiding philosophically motivated mathematical revisionism, "I have to say, gritting my teeth, that somehow, I know not how, we do understand what it means to speak of singletons."[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 ofCEM, called "Heyting mereology", whose sole nonlogical primitive isProper Part, assumedtransitive andantireflexive. According to this theory, there exists a "fictitious" null individual that is a proper part of every individual; two schemas assert that everylattice join exists (lattices arecomplete) and that meetdistributes over join. On this Heyting mereology, Forrest erects a theory ofpseudosets, adequate for all purposes to which sets have been put.
Mereology was influential in early conceptions ofset theory (see§ Set theory), which is currently thought of as afoundation for all mathematical theories.[19][20] Even after the currently-dominant "collective" conception of sets became prevalent, mereology has sometimes been developed as an alternative foundation, especially by authors who werenominalists and therefore rejectedabstract objects such as sets. The advantage of mereology for nominalists is that mereological sums, unlike collective sets, are thought to be nothing "over and above" their (possibly concrete) parts.[3]
Mereology may still be valuable to non-nominalists: Eberle (1970) defended the "ontological innocence" of mereology, which is the idea that one can employ mereology regardless of one's ontological stance regarding sets. This innocence results from mereology being formalizable in either of two equivalent ways: quantified variables ranging over auniverse of sets, or schematicpredicates with a singlefree variable.
Still,Stanisław Leśniewski andNelson Goodman, who developed Classical Extensional Mereology, were nominalists,[21] and consciously developed mereology as an alternative to set theory as a foundation of mathematics.[4] Goodman[22] defended thePrinciple of Nominalism, which states that whenever two entities have the same basic constituents, they are identical.[23] Most mathematicians and philosophers have accepted set theory as a legitimate and valuable foundation for mathematics, effectively rejecting the Principle of Nominalism in favor of some other theory, such asmathematical platonism.[23] David Lewis, whose§ Parts of Classes attempted to reconstruct set theory using mereology, was also a nominalist.[24]
Richard Milton Martin, who was also a nominalist, employed a version of the calculus of individuals throughout his career, starting in 1941. Goodman andQuine (1947) tried to develop thenatural andreal numbers using the calculus of individuals, but were mostly unsuccessful; Quine did not reprint that article in hisSelected Logic Papers. 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 ofrelations while abstaining from set-theoretic definitions of theordered pair. Martin argued that Eberle's (1970) theory of relational individuals solved this problem.
Burgess and Rosen (1997) provide a survey of attempts to found mathematics without using set theory, such as using mereology.
Ingeneral systems theory,mereology refers to formal work on system decomposition and parts, wholes and boundaries (by, e.g.,Mihajlo D. Mesarovic (1970),Gabriel Kron (1963), or Maurice Jessel (see Bowden (1989, 1998)). A hierarchical version ofGabriel Kron's Network Tearing was published by Keith Bowden (1991), reflecting David Lewis's ideas ongunk. Such ideas appear in theoreticalcomputer science andphysics, often in combination withsheaf theory,topos, orcategory theory. See also the work ofSteve Vickers on (parts of) specifications in computer science,Joseph Goguen on physical systems, and Tom Etter (1996, 1998) on link theory andquantum mechanics.
Bunt (1985), a study of thesemantics of natural language, shows how mereology can help understand such phenomena as themass–count distinction andverb aspect[example needed]. But Nicolas (2008) argues that a different logical framework, calledplural logic, should be used for that purpose.Also,natural language often employs "part of" in ambiguous ways (Simons 1987 discusses this at length)[example needed]. Hence, it is unclear how, if at all, one can translate certain natural language expressions into mereological predicates. Steering clear of such difficulties may require limiting the interpretation of mereology to mathematics andnatural science. Casati and Varzi (1999), for example, limit the scope of mereology tophysical objects.
Inmetaphysics there are many troubling questions pertaining to parts and wholes. One question addresses constitution and persistence, another asks about composition.
In metaphysics, there are several puzzles concerning cases of mereological constitution, that is, what makes up a whole.[25] There is still a concern with parts and wholes, but instead of looking at what parts make up a whole, the emphasis is on what a thing is made of, such as its materials, e.g., the bronze in a bronze statue. Below are two of the main puzzles that philosophers use to discuss constitution.
Ship of Theseus: Briefly, the puzzle goes something like this. There is a ship called theShip of Theseus. Over time, the boards start to rot, so we remove the boards and place them in a pile. First question, is the ship made of the new boards the same as the ship that had all the old boards? Second, if we reconstruct a ship using all of the old planks, etc. from the Ship of Theseus, and we also have a ship that was built out of new boards (each added one-by-one over time to replace old decaying boards), which ship is the real Ship of Theseus?
Statue and Lump of Clay: Roughly, a sculptor decides to mold a statue out of a lump of clay. At time t1 the sculptor has a lump of clay. After many manipulations at time t2 there is a statue. The question asked is, is the lump of clay and the statue (numerically) identical? If so, how and why?[26]
Constitution typically has implications for views on persistence: how does an object persist over time if any of its parts (materials) change or are removed, as is the case with humans who lose cells, change height, hair color, memories, and yet we are said to be the same person today as we were when we were first born. For example, Ted Sider is the same today as he was when he was born—he just changed. But how can this be if many parts of Ted today did not exist when Ted was just born? Is it possible for things, such as organisms to persist? And if so, how? There are several views that attempt to answer this question. Some of the views are as follows (note, there are several other views):[27][28]
(a) Constitution view. This view accepts cohabitation. That is, two objects share exactly the same matter. Here, it follows, that there are no temporal parts.
(b)Mereological essentialism, which states that the only objects that exist are quantities of matter, which are things defined by their parts. The object persists if matter is removed (or the form changes); but the object ceases to exist if any matter is destroyed.
(c) Dominant Sorts. This is the view that tracing is determined by which sort is dominant; they reject cohabitation. For example, lump does not equal statue because they're different "sorts".
(d)Nihilism—which makes the claim that no objects exist, except simples, so there is no persistence problem.
(e)4-dimensionalism ortemporal parts (may also go by the namesperdurantism orexdurantism), which roughly states that aggregates of temporal parts are intimately related. For example, two roads merging, momentarily and spatially, are still one road, because they share a part.
(f) 3-dimensionalism (may also go by the nameendurantism), where the object is wholly present. That is, the persisting object retains numerical identity.
One question that is addressed by philosophers is which is more fundamental: parts, wholes, or neither?[29][30][31][32][33][34][35][36][37][38] Another pressing question is called the special composition question (SCQ): For any Xs, when is it the case that there is a Y such that the Xs compose Y?[27][39][40][41][42][43][44] This question has caused philosophers to run in three different directions: nihilism, universal composition (UC), or a moderate view (restricted composition). The first two views are considered extreme since the first denies composition, and the second allows any and all non-spatially overlapping objects to compose another object. The moderate view encompasses several theories that try to make sense of SCQ without saying 'no' to composition or 'yes' to unrestricted composition.
There are philosophers who are concerned with the question of fundamentality. That is, which is more ontologically fundamental the parts or their wholes. There are several responses to this question, though one of the default assumptions is that the parts are more fundamental. That is, the whole is grounded in its parts. This is the mainstream view. Another view, explored by Schaffer (2010) is monism, where the parts are grounded in the whole. Schaffer does not just mean that, say, the parts that make up my body are grounded in my body. Rather, Schaffer argues that the wholecosmos is more fundamental and everything else is a part of the cosmos. Then, there is the identity theory which claims that there is no hierarchy or fundamentality to parts and wholes. Instead wholesare just (or equivalent to) their parts. There can also be a two-object view which says that the wholes are not equal to the parts—they are numerically distinct from one another. Each of these theories has benefits and costs associated with them.[29][30][31][32]
Philosophers want to know when some Xs compose something Y. There are several kinds of response:
(a) Contact—Xs compose a complex Y if and only if the Xs are in contact;
(b) Fastenation—Xs compose a complex Y if and only if the Xs are fastened;
(c) Cohesion—Xs compose a complex Y if and only if the Xs cohere (cannot be pulled apart or moved in relation to each other without breaking);
(d) Fusion—Xs compose a complex Y if and only if the Xs are fused (joined together such that there is no boundary);
(e) Organicism—Xs compose a complex Y if and only if either the activities of the Xs constitute a life or there is only one of the Xs;[45] and
(f) Brutal Composition—"It's just the way things are." There is no true, nontrivial, and finitely long answer.[46]
Many more hypotheses continue to be explored. A common problem with these theories is that they are vague. It remains unclear what "fastened" or "life" mean, for example. And there are other problems with the restricted composition responses, many of them which depend on which theory is being discussed.[40]
The dictionary definition ofmereology at Wiktionary
Media related toMereology at Wikimedia Commons
