Mereotopology

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Mereotopology is a formal theory, combining mereology and topology, of the topological relationships among wholes, parts, and the boundaries between parts. Although the theory is essentially mathematical, it has been entirely developed by logicians and theoretical computer scientists.

Smith (1996) and others have shown that mereotopology can be useful in ontology (Smith 1996), by formalizing relations such as contact, connection, boundaries, interiors, holes, and so on.

Beginning in 1916, and culminating in his 1929 Process and Reality, A. N. Whitehead developed theories of the part-whole relation which also included notions of contiguity and connection (for a gentle introduction, see Kneebone 2001: chpt. 12). Despite Whitehead's proven acumen as a mathematician, these theories were flawed and insufficiently formal, but Bowman Clarke (1981, 1985) showed how Whitehead's theories could be repaired and fully formalized. Clarke's work is the foundation of contemporary mereotopology. We can consider mereotopology also as a chapter of point-free geometry. For an advanced treatment revealing its mathematical power, see Roeper (1997). Casati and Varzi (1999) is the university text. For a general presentation of point-free geometry see Gerla (1995).

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Casati, R., and Varzi, A. C., 1999. Parts and places: the structures of spatial representation. MIT Press.
Clarke, Bowman, 1981, "A calculus of individuals based on 'connection'," Notre Dame Journal of Formal Logic 22: 204-18.
------, 1985, "Individuals and Points," Notre Dame Journal of Formal Logic 26: 61-75.
Gerla, G., 1995, "Pointless Geometries," in Buekenhout, F., Kantor, W. (eds), Handbook of incidence geometry: buildings and foundations. North-Holland: 1015-31.
Kneebone, Geoffrey, 2001 (1963). Mathematical Logic and the Foundation of Mathematics. Dover.
Roeper, Peter, 1997, "Region-Based Topology," Journal of Philosophical Logic 26: 251-309.
Smith, Barry, 1996, "Mereotopology: A Theory of Parts and Boundaries," Data and Knowledge Engineering 20 : 287-303.
------, 1997, "Boundaries: An Essay in Mereotopology" in Hahn, L., ed., The Philosophy of Roderick Chisholm. Open Court: 534-61.
Varzi, A. C., 1996, "Parts, wholes, and part-whole relations: the prospects of mereotopology," Data and Knowledge Engineering, 20: 259-286.
------, 1998, "Basic Problems of Mereotopology," in Guarino, N., ed., Formal Ontology in Information Systems. Amsterdam: IOS Press, 29-38.
------, 2006, "Spatial Reasoning and Ontology: Parts, Wholes, and Locations." by Achille Varzi