Abstract
Mereotopology is that branch of the theory of regions concerned with topological properties such as connectedness. It is usually developed by considering the parthood relation that characterizes the, perhaps non-classical, mereology of Space (or Spacetime, or a substance filling Space or Spacetime) and then considering an extra primitive relation. My preferred choice of mereotopological primitive is interior parthood. This choice will have the advantage that filters may be defined with respect to it, constructing “points”, as Peter Roeper has done (“Region-based topology”, Journal of Philosophical Logic, 26 (1997), 25–309). This paper generalizes Roeper’s result, relying only on mereotopological axioms, not requiring an underlying classical mereology, and not assuming the Axiom of Choice. I call the resulting mathematical system an approximate lattice, because although meets and joins are not assumed they are approximated. Theorems are proven establishing the existence and uniqueness of representations of approximate lattices, in which their members, the regions, are represented by sets of “points” in a topological “space”.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arntzenius, F. (2008). ‘Gunk, topology and measure’, Oxford studies in metaphysics, vol 4. Oxford: Oxford University Press.
Casati, R., Smith, B., & Varzi, A. (1998). Ontological tools for geographic representation. In N. Guarino (Ed.), Formal ontology in information systems (pp. 77–85). Amsterdam: IOS.
Clarke, B. L. (1981). Calculus of individuals based on “Connection”. Notre Dame Journal of Formal Logic, 22, 204–18.
Clarke, B. L. (1985). Individuals and points. Notre Dame Journal of Formal Logic, 26, 61–75.
De Laguna, T. (1922). Point, line and surface as sets of solids. The Journal of Philosophy, 19, 449–461.
Dimov, G., & Vakarelov, D. (2007). Contact algebras and region-based theory of space: a proximity approach—I. Fundamenta Informaticae, 74, 209–249.
Dimov, G., & Vakarelov, D. (2007). Contact algebras and region–based theory of space: a proximity approach –II. Fundamenta Informaticae, 74, 251–282.
Düntsch, I., Orlowska, E., & Wang, H. (2001). Algebras of approximating regions. Fundamenta Informaticae, 46, 71–82.
Field, H. H. (1980). Science without numbers: A defence of nominalism. Princeton: Princeton University Press.
Forrest, P. (2007). Mereological summation and the question of unique fusion. Analysis, 67, 237–42.
Hudson, H. (2005). The metaphysics of hyperspace. Oxford: Clarendon.
Kuratowski, K. (1961). Introduction to set theory and topology, translated from the revised polish edition by Leo F. Boron. Oxford: Pergamon.
Leonard, H. S., & Goodman, N. (1940). The calculus of individuals and its uses. Journal of Symbolic Logic, 5, 45–55.
Leśniewski, S. (1916). Podstawy ogólnej teoryi mnogosci. I, Moskow: Prace Polskiego Kola Naukowego w Moskwie, Sekcya matematyczno-przyrodnicza
Leśniewski, S. (1992). ‘Foundations of the General Theory of Sets. I’, Eng. trans. of Leśniewski, S. (1916) by D. I. Barnett, in S. Leśniewski, Collected Works, (ed. S. J. Surma, J. Srzednicki, D. I. Barnett, and F. V. Rickey) Dordrecht: Kluwer, vol. 1: 129–173.
Lewis, D. (1991). Parts of classes. Oxford: Basil Blackwell.
Mormann, T. (2000). Topological representability of mereological systems. In J. Faye (Ed.), Things, facts and events. Atlanta: Rodopi.
Roeper, P. (1997). Region-based topology. Journal of Philosophical Logic, 26, 25–309.
Simons, P. (1987). Parts. A study in ontology. Oxford: Clarendon.
Stell, J. (2000). Boolean connection algebras: a new approach to the region connection calculus. Artificial Intelligence, 122, 111–136.
Vakarelov, D., Dimov, G., Düntsch, I., & Bennett, B. (2002). A proximity approach to some region-based theories of space. Journal of Applied Non-Classical Logics, 12, 527–529.
Whitehead, A. N. (1929). Process and reality. An essay in cosmology. New York: Macmillan.
Author information
Authors and Affiliations
Corresponding author
Additional information
I am indebted to Thomas Mormann with whom I have often discussed mereotopology, to Andrew Percy who checked all the mathematics and whose valuable comments helped me make the paper clearer, to the editor for his patience and especially to the referees for their helpful, detailed comments that helped me complete what was, I confess, a premature submission.
Rights and permissions
About this article
Cite this article
Forrest, P. Mereotopology without Mereology. J Philos Logic 39, 229–254 (2010). https://doi.org/10.1007/s10992-010-9130-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-010-9130-x