Part (mathematics)

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 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 has continued to be used by a minority in works 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'' (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 ''Classical 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. 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. 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 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) sheds much light on 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 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 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 well-surveyed in Simons (1987), Casati and Varzi (1999), and Cotnoir and Varzi (2021).

Axioms and primitive notions

A basic choice in defining a mereological system, is whether to allow things to be considered parts of themselves (reflexivity of parthood). 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 (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: 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 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:
*An immediate defined predicate is "x is a proper part of ''y''", written ''PPxy'', which holds (i.e., is satisfied, comes out true) if ''Pxy'' is true and ''Pyx'' is false. Compared to parthood (which is a partial order), ProperPart is a strict partial order.
:$PPxy \leftrightarrow (Pxy \land \lnot Pyx).$ 3.3
:An object lacking proper parts is an ''atom''. The mereological universe consists of all objects we wish to think about, and all of their proper parts:
*Overlap: ''x'' and ''y'' overlap, written ''Oxy'', if there exists an object ''z'' such that ''Pzx'' and ''Pzy'' both hold.
:Oxy \leftrightarrow \exists z zx \land Pzy 3.1
:The parts of ''z'', the "overlap" or "product" of ''x'' and ''y'', are precisely those objects that are parts of both ''x'' and ''y''.
*Underlap: ''x'' and ''y'' underlap, written ''Uxy'', if there exists an object ''z'' such that ''x'' and ''y'' are both parts of ''z''.
:Uxy \leftrightarrow \exists z xz \land Pyz 3.2
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 (defined below), ''parthood'' can be defined from Overlap as follows:
:Pxy \leftrightarrow \forall z zx \rightarrow Ozy 3.31

The axioms are:
*Parthood partially orders the universe:
:M1, Reflexive: An object is a part of itself.
:$\ Pxx.$ P.1
:M2, Antisymmetric: If ''Pxy'' and ''Pyx'' both hold, then ''x'' and ''y'' are the same object.
:$(Pxy \land Pyx) \rightarrow x = y.$ P.2
:M3, Transitive: If ''Pxy'' and ''Pyz'', then ''Pxz''.
:$(Pxy \land Pyz) \rightarrow Pxz.$ P.3
*M4, Weak Supplementation: If ''PPxy'' holds, there exists a ''z'' such that ''Pzy'' holds but ''Ozx'' does not.
:PPxy \rightarrow \exists z zy \land \lnot Ozx P.4

*M5, Strong Supplementation: If ''Pyx'' does not hold, there exists a ''z'' such that ''Pzy'' holds but ''Ozx'' does not.
:\lnot Pyx \rightarrow \exists z zy \land \lnot Ozx P.5

*M5', Atomistic Supplementation: If ''Pxy'' does not hold, then there exists an atom ''z'' such that ''Pzx'' holds but ''Ozy'' does not.
:\lnot Pxy \rightarrow \exists z zx \land \lnot Ozy \land \lnot \exists v [PPvz. P.5'

*Top: There exists a "universal object", designated ''W'', such that ''PxW'' holds for any ''x''.
:$\exists W \forall x [PxW].$ 3.20
:Top is a theorem if M8 holds.

*Bottom: There exists an atomic "null object", designated ''N'', such that ''PNx'' holds for any ''x''.
:$\exists N \forall x [PNx].$ 3.22

*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''.
:Uxy \rightarrow \exists z \forall v vz \leftrightarrow (Ovx \lor Ovy) P.6
*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''.
:Oxy \rightarrow \exists z \forall v vz \leftrightarrow (Pvx \land Pvy) P.7
:If ''Oxy'' does not hold, ''x'' and ''y'' have no parts in common, and the product of ''x'' and ''y'' is undefined.
*M8, Unrestricted Fusion: Let φ(''x'') be a first-order formula in which ''x'' is a free variable. Then the fusion of all objects satisfying φ exists.
:\exists x phi(x)\to \exists z \forall y yz \leftrightarrow \exists x[\phi (x) \land Oyx. P.8
:M8 is also called "General Sum Principle", "Unrestricted Mereological Composition", or "Universalism". M8 corresponds to the set builder notation">principle of unrestricted comprehension of naive set theory, which gives rise to Russell's paradox. There is no mereological counterpart to this paradox simply because ''parthood'', unlike set membership, is reflexive.

*M8', Unique Fusion: The fusions whose existence M8 asserts are also unique. P.8'
*M9, Atomicity: All objects are either atoms or fusions of atoms.
: \exists y yx \land \forall z[\lnot PPzy. P.10

Various systems

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 called ''classical extensional mereology'', hereinafter abbreviated CEM (other abbreviations are explained below). In CEM, P.1 through P.8' hold as axioms or are theorems. M9, ''Top'', and ''Bottom'' are optional.

The systems in the table below are partially ordered by Inclusion (set theory)">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">logical truth">necessarily true, then B ''includes'' A. The resulting Hasse diagram is similar to Fig. 3.2 in Casati and Varzi (1999: 48).

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 relation">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. 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 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 empty 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 ''Unrestricted 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. ''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:
* a Boolean algebra lacking a 0;
* a join semilattice bounded from above by 1. Binary fusion and ''W'' interpret join and 1, respectively.
Postulating ''N'' renders all possible products definable, but also transforms classical extensional mereology into a set-free model of Boolean 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 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. 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. Simons (1987: 38–41) lists a number of GEM theorems.

M2 and a finite universe necessarily imply ''Atomicity'', 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, denoted AX. ''Atomicity'' permits economies, for instance, assuming that M5' implies ''Atomicity'' and extensionality, and yields an alternative axiomatization of AGEM.

Set theory

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. 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, which are two signs that he conceived of sets mereologically. Similarly, Ernst Schröder, in "Vorlesungen über die Algebra der Logik" (1890), also used the mereological conception. It was Gottlob Frege, in a 1895 review of Schröder's work, who first laid out the difference between collections and mereological sums. The fact that Ernst Zermelo adopted the collective conception when he wrote his influential 1908 axiomatization of set theory is certainly significant for, though it does not fully explain, its current popularity.

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, called non-well-founded set theory. The calculus of individuals was thought 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 "", so that the universe is infinite.

A detailed comparison between mereology, set theory, and a semantic "ensemble theory" is presented in chapter 13 of Bunt (1985); when David Lewis wrote his famous ', he found that "its main thesis had been anticipated in" Bunt's ensemble theory.

''Parts of Classes''

Philosopher David Lewis, in his 1991 work ''Parts of Classes'', axiomatized Zermelo-Fraenkel (ZFC) set theory using only classical mereology, plural quantification, and a primitive singleton-forming operator, governed by axioms that resemble the axioms for "successor" in Peano arithmetic. This contrasts with more usual axiomatizations of ZFC, which use only the primitive notion of membership. 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.

Michael Potter, a creator of Scott–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." 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."

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. According to this theory, 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.

Foundations of mathematics

Mereology was influential in early conceptions of set theory (see ), which is currently thought of as a foundation for all mathematical theories. Even after the currently-dominant "collective" conception of sets became prevalent, mereology has sometimes been developed as an alternative foundation, especially by authors who were nominalists and therefore rejected abstract 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.

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 a universe of sets, or schematic predicates with a single free variable.

Still, Stanisław Leśniewski and Nelson Goodman, who developed Classical Extensional Mereology, were nominalists, and consciously developed mereology as an alternative to set theory as a foundation of mathematics. Goodman defended the ''Principle of Nominalism'', which states that whenever two entities have the same basic constituents, they are identical. 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 as mathematical platonism. David Lewis, whose ' attempted to reconstruct set theory using mereology, was also a nominalist.

Richard Milton Martin, who was also a nominalist, employed a version of the calculus of individuals throughout his career, starting in 1941. Goodman and Quine (1947) tried to develop the natural and real numbers using the calculus of individuals, but were mostly unsuccessful; Quine did not reprint that article in his ''Selected 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 of relations while abstaining from set-theoretic definitions of the ordered 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.

General systems theory

In general 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 of Gabriel Kron's Network Tearing was published by Keith Bowden (1991), reflecting David Lewis's ideas on gunk. Such ideas appear in theoretical computer science and physics, often in combination with sheaf theory, topos, or category theory. 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.

Linguistic semantics

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. But Nicolas (2008) argues that a different logical framework, called plural logic, should be used for that purpose.
Also, natural language often employs "part of" in ambiguous ways (Simons 1987 discusses this at length). 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 and natural science. Casati and Varzi (1999), for example, limit the scope of mereology to physical objects.

Metaphysics

In metaphysics there are many troubling questions pertaining to parts and wholes. One question addresses constitution and persistence, another asks about composition.

Mereological constitution

In metaphysics, there are several puzzles concerning cases of mereological constitution, that is, what makes up a whole. 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 the Ship 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?

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):

(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 or temporal parts (may also go by the names perdurantism or exdurantism), 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 name endurantism), where the object is wholly present. That is, the persisting object retains numerical identity.

Mereological composition

One question that is addressed by philosophers is which is more fundamental: parts, wholes, or neither? 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? 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.

Fundamentality

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 whole ''cosmos'' 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 wholes ''are 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.

Special composition question

Philosophers want to know when some Xs compose something Y. There are several kinds of response:

*One response is ''nihilism''. According to nihilism, there are no mereological complex objects (composite objects), only simples. Nihilists do not entirely reject composition because they think simples compose themselves, but this is a different point. More formally, nihilists would say: Necessarily, for any non-overlapping Xs, there is an object composed of the Xs if and only if there is only one of the Xs. This theory, though well explored, has its own problems: it seems to contradict experience and common sense, to be incompatible with atomless gunk, and to be unsupported by space-time physics.
*Another prominent response is ''universal composition'' (UC). According to UC, as long as Xs do not spatially overlap, they can compose a complex object. Universal compositionalists also support unrestricted composition. More formally: Necessarily, for any non-overlapping Xs, there is a Y such that Y is composed of the Xs. For example, someone's left thumb, the top half of another person's right shoe, and a quark in the center of their galaxy can compose a complex object. This theory also has some drawbacks, most notably that it allows for far too many objects.
*A third response (perhaps less explored than the other two) includes a range of ''restricted composition views''. There are several views, but they all share an idea: that there is a restriction on what counts as a complex object: some (but not all) Xs come together to compose a complex Y. Some of these theories include:

(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; and

(f) Brutal Composition—"It's just the way things are." There is no true, nontrivial, and finitely long answer.

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.

* A fourth response is ''deflationism''. According to deflationism, the way the term "exist" is used varies, and thus all the above answers to the SCQ can be correct when indexed to the appropriate meaning of "exist". Further, there is no privileged way in which the term "exist" must be used. There is therefore no privileged answer to the SCQ, since there are no privileged conditions for when Xs compose Y. Instead, the debate is reduced to a mere verbal dispute rather than a genuine ontological debate. In this way, the SCQ is part of a larger debate in general ontological realism and anti-realism. While deflationism successfully avoids the SCQ, it comes at the cost of ontological anti-realism, such that nature has no objective reality, for, if there is no privileged way to objectively affirm the existence of objects, nature itself must have no objectivity.

See also

* Glossary of mereology
* Gunk (mereology)
* Holism
* Holon (philosophy)
* Implicate and explicate order according to David Bohm
* ''Laws of Form'' by G. Spencer-Brown
* Mereological essentialism
* Mereological nihilism
* Mereotopology
* Meronomy
* Meronymy
* Monad (philosophy)
* Plural quantification
* Quantifier variance
* Simple (philosophy)
* Whitehead's point-free geometry
* Composition (objects)
* Emergence

References

Sources

* Bowden, Keith, 1991. ''Hierarchical Tearing: An Efficient Holographic Algorithm for System Decomposition'', Int. J. General Systems, Vol. 24(1), pp 23–38.
* Bowden, Keith, 1998. ''Huygens Principle, Physics and Computers''. Int. J. General Systems, Vol. 27(1–3), pp. 9–32.
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External links

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*Internet Encyclopedia of Philosophy:
*
Material Composition
– David Cornell
*Stanford Encyclopedia of Philosophy:
*
Mereology
– Achille Varzi
*
Boundary
– Achille Varzi

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