Mereotopology

In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts.

Mereotopology begins in philosophy with theories articulated by A. N. Whitehead in several books and articles he published between 1916 and 1929, drawing in part on the mereogeometry of De Laguna (1922). The first to have proposed the idea of a point-free definition of the concept of topological space in mathematics was Karl Menger in his book Dimensionstheorie (1928) -- see also his (1940). The early historical background of mereotopology is documented in Bélanger and Marquis (2013) and Whitehead's early work is discussed in Kneebone (1963: chpt. 13.5) and Simons (1987: 2.9.1). The theory of Whitehead's 1929 Process and Reality augmented the part-whole relation with topological notions such as contiguity and connection. Despite Whitehead's acumen as a mathematician, his theories were insufficiently formal, even flawed. By showing how Whitehead's theories could be fully formalized and repaired, Clarke (1981, 1985) founded contemporary mereotopology. The theories of Clarke and Whitehead are discussed in Simons (1987: 2.10.2), and Lucas (2000: chpt. 10). The entry Whitehead's point-free geometry includes two contemporary treatments of Whitehead's theories, due to Giangiacomo Gerla, each different from the theory set out in the next section.

Although mereotopology is a mathematical theory, we owe its subsequent development to logicians and theoretical computer scientists. Lucas (2000: chpt. 10) and Casati and Varzi (1999: chpts. 4,5) are introductions to mereotopology that can be read by anyone having done a course in first-order logic. More advanced treatments of mereotopology include Cohn and Varzi (2003) and, for the mathematically sophisticated, Roeper (1997). For a mathematical treatment of point-free geometry, see Gerla (1995). Lattice-theoretic (algebraic) treatments of mereotopology as contact algebras have been applied to separate the topological from the mereological structure, see Stell (2000), Düntsch and Winter (2004).

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