Mereology is a collection of axiomatic formal systems dealing with parts and their respective wholes. Mereology is an application of logic and a branch of ontology. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... In philosophy, ontology (from the Greek , genitive : of being (part. ...

History

Before the rise of set theory, part-whole reasoning was casually but unwittingly invoked all over mathematics and metaphysics, including in Aristotle. Ivor Grattan-Guinness (2001) sheds much light on this aspect of the period just before the Cantor-Peano notion of set became canonical. The first to reason consciously and at length about parts and wholes was, apparently, Edmund Husserl in his 1901 Logical Investigations., translated as Husserl (1970). However, the word "mereology" is absent from his writings, and he employed no symbolism even though his doctorate was in mathematics. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Aristotle (Ancient Greek: AristotélÄ“s 384 – March 7, 322 BCE) was an ancient Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Ivor Grattan-Guiness is a prolific contemporary historian of mathematics and logic. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Edmund Husserl Edmund Gustav Albrecht Husserl (April 8, 1859 - April 26, 1938, Freiburg) was a German philosopher, known as the father of phenomenology. ...

Stanislaw Lesniewski coined the term in 1927, from the Greek word meros (part). Between 1916 and 1931, he wrote a number of highly technical papers on the subject, translated in Lesniewski (1992). This "Polish mereology" was elaborated over the course of the 20th century by Lesniewski's students, and by students of his students. However, little work in this area dates from after 1985 or so. Stanisław Leśniewski (March 30, 1886–May 13, 1939) was a Polish mathematician, philosopher and logician. ...

Henry Leonard's 1930 Harvard Ph.D. dissertation in philosophy, often cited but never sighted, set out a formal theory of the part-whole relation which first appeared in print in Goodman and Leonard (1940), who called it "the calculus of individuals." Goodman went on to elaborate this calculus in the three editions of his Structure of Appearance, the last being Goodman (1977). Eberle (1970) clarified the relation between mereology and set theory, and 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. Properties In chemistry and physics, an atom (Greek άτομον meaning indivisible) is the smallest possible particle of a chemical element that retains its chemical properties. ... Size of the universe and observable universe Main article: Observable universe There is disagreement over whether the universe is indeed finite or infinite in spatial extent and volume. ...

For some time, philosophers and mathematicians were reluctant to explore mereology, believing that it implied a rejection of set theory, a position known as nominalism. Goodman was indeed a nominalist and his fellow nominalist, Richard Milton Martin employed a version of the calculus of individuals throughout his career, starting in 1941. The calculus of individuals began to come into its own starting only around 1970, when the "ontological innocence" of mereology began to be recognized. One can employ mereology regardless of one's ontological stance regarding set theory. Quantified variables ranging over a universe of sets, and schematic monadic predicates with a free variable, can be used interchangeably in the formal description of a mereological system. Since that recognition, formal work in ontology and metaphysics has made increasing use of mereology. Nominalism is the position in metaphysics that there exist no universals outside of the mind. ... Richard Milton Martin (1916-11. ... In philosophy, ontology (from the Greek , genitive : of being (part. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Mereology is mathematics, but has been wholly developed by logicians and computer scientists. To date, the only mathematician to write on mereology is Lesniewski's student Alfred Tarski in the 1920s and 30s (see Tarski 1984). For that matter, mereology is seldom mentioned outside of the literatures on ontology and artificial intelligence. Standard university texts on logic and mathematics are silent about mereology, which has undoubtedly contributed to its undeserved obscurity. Topological notions of boundaries and connection can be married to mereology, resulting in mereotopology. Alfred Tarski (January 14, 1901, Warsaw Poland – October 26, 1983, Berkeley California) was a logician and mathematician of considerable philosophical importance. ... In philosophy, ontology (from the Greek , genitive : of being (part. ... Hondas intelligent humanoid robot AI redirects here. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Mereotopology is a formal theory, combining mereology and topology, of the topological relationships among wholes, parts, and the boundaries between parts. ...

Mereology and set theory

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. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In philosophy, ontology (from the Greek , genitive : of being (part. ... William of Ockham Occams razor (also spelled Ockhams razor) is a principle attributed to the 14th-century English logician and Franciscan friar William of Ockham. ...

Many logicians and philosophers reject these motivations, on such grounds as:

• They deny that sets are in any way ontologically suspect;
• Occam's Razor, when applied to abstract objects like sets, is either a dubious principle or simply false;
• Mereology itself is guilty of proliferating new and ontologically suspect entities.

Nonetheless, mereology is now largely accepted as a useful tool for formal philosophy, although to date it has received much less attention than set theory.

In set theory, unit sets 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 be built up from atoms. Eberle (1970) showed how to devise mereologies such that all objects have proper parts and so can be divided at will. (Leibniz believed this to be true of matter.) Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...

Lewis (1991) argued that mereology augmented by a few ontological assumptions and some careful reasoning about unit sets, yields a formal system in which the axioms of Peano arithmetic and of Zermelo-Fraenkel set theory are theorems. David K. Lewis David Kellogg Lewis (September 28, 1941 - October 14, 2001) is considered by many to have been the leading Analytic philosopher of the latter half of the 20th century. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...

Axiomatic mereology

It is possible to formulate a "naive mereology" analogous to naive set theory. Doing so gives rise to paradoxes analogous to Russell's paradox: there is an object whose parts are all the objects that are not parts of themselves. Is it a part of itself? (However, every object is an "improper" part of itself.) Hence mereology requires an axiomatic formulation. In abstract mathematics, naive set theory1 was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... Russells paradox (also known as Russells antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Frege is contradictory. ... For the algebra software named Axiom, see Axiom computer algebra system. ...

The treatment and terminology below follow Casati and Varzi (1999: chpts. 3,4) closely. Mereology studies first-order theories, called systems, treating of wholes and their respective parts, collectively called objects. Hence mereology presupposes and builds on first-order logic with identity. Mereology is a collection of nested and nonnested axiomatic systems, not unlike the case with modal logic. A mereological theory requires at least one primitive relation e.g., dyadic Parthood, "x is a part of y," written Pxy. Parthood is nearly always assumed to partially order the universe. It has been suggested that Predicate calculus be merged into this article or section. ... It has been suggested that Predicate calculus be merged into this article or section. ... // Computer programming In object-oriented programming, object identity is a mechanism for distinguishing different objects from each other. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... A modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...

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. An object lacking proper parts is an atom. The mereological universe consists of all objects we wish to think about, plus all of their proper parts. Two other common defined predicates are: The term universe of discourse generally refers to the entire set of terms used in a specific discourse, i. ...

• Overlap: x and y overlap, written Oxy, if there exists an object z such that Pzx and Pzy both hold. 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.

A number of possible axioms follow. Lower case letters denote variables ranging over objects:

• Parthood partially orders the universe:
  • M1, Reflexive: An object is a part of itself.
  • M2, Antisymmetric: If Pxy and Pyx both hold, then x and y are the same object.
  • M3, Transitive: If Pxy and Pyz, then Pxz. The objection has been made that parthood in many natural language and philosophical contexts is intransitive.
• M4, Weak Supplementation: If PPxy holds, there exists a z such that Pzy holds but Ozx does not.
• M5, Strong Supplementation: Replace "PPxy holds" in M4 with "Pyx does not hold."
• M5', Atomistic Supplementation: If Pxy does not hold, then there exists an atom x such that Pzx holds but Pzy does not.
• Top: There exists a "universal object", designated W, such that PxW holds for any x. Top is a theorem if M8 holds.
• Bottom: There exists an atomic "null object", designated N, such that PNx holds for any x.
• M6, Sum: If Uxy holds, there exists a z, called the "sum" or "fusion" of x and y, such that the parts of z are just those objects which are parts of either x or y.
• 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 which are parts of both x and y. If Oxy does not hold, x and y have no non-empty parts in common, and the product of x and y is defined iff Bottom holds.
• M8, Unrestricted Fusion: If there exists an x satisfying the univariate formula φ, then there exists a z which overlaps precisely those y that overlap some w satisfying the formula φ. Also called "General Sum Principle," "Unrestricted Mereological Composition," or "Universalism." M8 corresponds to the axiom of replacement of axiomatic set theory.
• M8', Unique Fusion: The fusion described in M8 exists and is unique.
• M9, Atomicity: All objects are either atoms or fusions of atoms.

The above axioms all hold in classical extensional mereology. Other systems of mereology are described in Simons (1987) and Casati and Varzi (1999). There are some analogies between these axioms and those of standard Zermelo-Fraenkel set theory, if "parthood" in mereology is taken as corresponding to subset in set theory. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... Size of the universe and observable universe Main article: Observable universe There is disagreement over whether the universe is indeed finite or infinite in spatial extent and volume. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...

In the table below, strings of bold letters name mereological systems. These systems 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 Fig. 2, and Fig. 3.2 in Casati and Varzi (1999: 48). In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, inclusion is a partial order on sets. ... In the mathematical area of order theory, a Hasse diagram (pronounced HAHS uh, named after Helmut Hasse (1898–1979)) is a simple picture of a finite partially ordered set. ...

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. Either axiomatization results in the system M. M2 rules out closed loops formed using Parthood; so that the part relation is well-founded. Size of the universe and observable universe Main article: Observable universe There is disagreement over whether the universe is indeed finite or infinite in spatial extent and volume. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In grammar, a verb is transitive if it takes an object. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, a well-founded relation is an order relation R on a set X where every non-empty subset of X has an R-minimal element; that is, where for every non-empty subset S of X, there is an element m of S such that for every element...

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. In any system in which M5 or M5' are assumed or can be derived, then it can be proved that if two objects share the same proper parts, they are the same object. This property is known as Extensionality, a term borrowed from set theory, where Extensionality is a fundamental axiom. An immediate consequence of Extensionality is that no two atoms can be identical. Mereological systems in which Extensionality holds are termed extensional, a fact designated by the letter E in their symbolic names. The word complement (with an e in the second syllable, not to be confused with a different word, compliment with an i) has a number of uses. ... In mathematics, this usually refers to some form of the principle, going back to Leibniz, that two mathematical objects are equal if there is no test to distinguish them. ...

M6 [M7] asserts that any two underlapping [overlapping] objects have a unique sum [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 analogues of the universal and null sets, and sum and product are likewise the analogs of set union and intersection. If M6 and M7 are either assumed or derivable, the result is a closure system. In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ... In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ... Union generally means an organization formed to conduct an activity. ... The term intersection can mean: a road junction, where two roads intersect each other, such as a roundabout intersection; in mathematics, the set in which two or more other sets intersect each other; see intersection (set theory); a movie; see Intersection (movie). ...

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, if 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. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two... Membership can refer to: Set membership - comprising part of a set in mathematics Social group membership - in sociology, the process of socialisation aims/results in achieving membership of a social group This is a disambiguation page — a list of articles associated with the same title. ... In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ... Syntax, originating from the Greek words συν (syn, meaning co- or together) and τάξις (táxis, meaning sequence, order, arrangement), can in linguistics be described as the study of the rules, or patterned relations that govern the way the words in a sentence come together. ... In mathematics the term countable set is used to describe the size of a set, e. ...

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. Lesniewski 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: Richard Milton Martin (1916-11. ...

• Boolean algebra lacking a 0;
• 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. Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... See: JOIN, join command in SQL, a relational database keyword. ... A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ...

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 some 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.

Mereology and natural language

The interpretation of mereology is complicated by the fact that natural language often employs the words "part of" in ambiguous ways. If mereology is merely employed to add nuances to logical reasoning, this need not lead to any problems. But it is doubtful how, if at all, one can translate certain expressions in natural language into mereological predicates. The term natural language is used to distinguish languages spoken and signed (by hand signals and facial expressions) by humans for general-purpose communication from constructs such as writing, computer-programming languages or the languages used in the study of formal logic, especially mathematical logic. ...

Bunt (1985), a study of natural language semantics, shows how mereology can help understand such phenomena as the mass/count distinction and grammatical aspect. In the main, semantics (from the Greek and in greek letters σημαντικός or in latin letters semantikós, or significant meaning, derived from sema, sign) is the study of meaning, in some sense of that term. ... In English grammar, a mass noun (also uncountable noun) is a type of noun that cannot be modified by a number without specifying a unit of measurement; thus mass nouns have singular but no plural forms. ... To meet Wikipedias quality standards, this article may require cleanup. ...

Two important books

The books Simons (1987) and Casati and Varzi (1999) differ in their strengths:

• Simons (1987) sees mereology primarily as a tool for doing formal metaphysics. His strengths include: the work of Lesniewski and his descendants; the connections between mereology and a number of continental philosophers, especially Edmund Husserl; the relation between mereology and recent work on formal ontology and metaphysics; mereology and free logic and modal logic; mereology and Boolean algebra and lattice theory.
• Casati and Varzi (1999) see mereology primarily as a way of understanding the material world and how humans interact with it. Their strengths include: topology and mereotopology; boundaries and holes; the mereological implications of Whitehead's Process and Reality and work descended therefrom; mereology as a theory of events; mereology as a "proto-geometry" for physical objects; mereology and theoretical computer science.

Both books include excellent bibliographies. To meet Wikipedias quality standards, this article or section may require cleanup. ... Edmund Husserl Edmund Gustav Albrecht Husserl (April 8, 1859 - April 26, 1938, Freiburg) was a German philosopher, known as the father of phenomenology. ... In philosophy, ontology (from the Greek , genitive : of being (part. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Free logic is a logic free of existential presuppositions. ... A modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... The ordinary meaning of lattice is the basis for several technical usages A cherry lattice pastry A mathematical lattice that is a type of partially ordered set. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Mereotopology is a formal theory, combining mereology and topology, of the topological relationships among wholes, parts, and the boundaries between parts. ... Whitehead can refer to: People: Alfred North Whitehead, a British philosopher and mathematician Cortlandt Whitehead (1842-), bishop Ennis Clement Whitehead (1895-1964), Lieutenant General U.S. Air Force [1] J. H. C. Whitehead, British mathematician and nephew of A. N. Whitehead Joseph Whitehead (1867-1938), lawyer, politician, Representative Democrat Virginia... Process and Reality (1929) is Alfred North Whiteheads opus explicating the Philosophy of Organism, a philosophy of subjectivity as process itself. ...

See also

• Ontology
• Mereotopology
• Mereological Nihilism

In philosophy, ontology (from the Greek , genitive : of being (part. ... Mereotopology is a formal theory, combining mereology and topology, of the topological relationships among wholes, parts, and the boundaries between parts. ... Mereological nihilism (also called compositional nihilism, or what some philosophers just call nihilism) is the position that objects with parts do not exist (not only objects in space, but also objects existing in time do not have any temporal parts, and thus only exist in the present moment), and only...

References

• Bunt, Harry, 1985. Mass terms and model-theoretic semantics. Cambridge Uni. Press.
• Burkhardt, H., and Dufour, C.A., 1991, "Part/Whole I: History" in Burkhardt, H., and Smith, B., eds., Handbook of Metaphysics and Ontology. Muenchen: Philosophia Verlag.
• Casati, R., and Varzi, A., 1999. Parts and Places: the structures of spatial representation. MIT Press.
• Nelson Goodman, 1977 (1951). The Structure of Appearance. Kluwer.
• Edmund Husserl, 1970. Logical Investigations, Vol. 2. Findlay, J.N., trans. Routledge.
• David K. Lewis, 1991. Parts of Classes. Blackwell.
• Leonard, H.S., and Nelson Goodman, 1940, "The calculus of individuals and its uses," Journal of Symbolic Logic 5: 45-55.
• Stanislaw Lesniewski, 1992. Collected Works. Surma, S.J., Srzednicki, J.T., Barnett, D.I., and Rickey, F.V., eds. and trans. Kluwer.
• Simons, Peter, 1987. Parts: A Study in Ontology. Oxford Uni. Press.
• Alfred Tarski, 1984 (1956), "Foundations of the Geometry of Solids" in his Logic, Semantics, Metamathematics: Papers 1923-38. Woodger, J., and Corcoran, J., eds. and trans. Hackett.

Nelson Goodman (7 August 1906, Somerville, Maryland – 25 November 1998) was an American philosopher, known for his work on counterfactuals, mereology, the problem of induction, and aesthetics. ... Edmund Husserl Edmund Gustav Albrecht Husserl (April 8, 1859 - April 26, 1938, Freiburg) was a German philosopher, known as the father of phenomenology. ... David K. Lewis David Kellogg Lewis (September 28, 1941 - October 14, 2001) is considered by many to have been the leading Analytic philosopher of the latter half of the 20th century. ... Nelson Goodman (7 August 1906, Somerville, Maryland – 25 November 1998) was an American philosopher, known for his work on counterfactuals, mereology, the problem of induction, and aesthetics. ... Stanisław Leśniewski (March 30, 1886–May 13, 1939) was a Polish mathematician, philosopher and logician. ... Alfred Tarski (January 14, 1901, Warsaw Poland – October 26, 1983, Berkeley California) was a logician and mathematician of considerable philosophical importance. ...

External link

• Stanford Encyclopedia of Philosophy: Mereology -- Achille Varzi.
• Varzi, Achille C., 2006, "Spatial Reasoning and Ontology: Parts, Wholes, and Locations."