Continuous lattices and Whiteheadian theory of space

Thomas Mormann



In this paper a solution ofWhitehead’s problem is presented: Starting with a purely mereological system of regions a topological space is constructed such that the class of regions is isomorphic to the Boolean lattice of regular open sets of that space. This construction may be considered as a generalized completion in analogy to the well-known Dedekind completion of the rational numbers Qyielding the real numbers R. The argument of the paper relies on the theories of continuous lattices and “pointless” topology.

Full Text:



Banaschewsky, B, Hofmann, R.-E., 1981, Continuous Lattices (eds.), Lecture Notes in Mathematics 871, Berlin, Heidelberg, New York, Springer.

Biacino, L., Gerla, G., 1991, “Connection Structures”, Notre Dame Journal of Formal Logic 32, 242–247.

Biacino, L., Gerla, G., 1996, “Connection structures: Grzegorczyk’s and Whitehead’s definitions of point”, Notre Dame Journal of Formal Logic 37, 431–439.

Clarke, B.L., 1981, “A calculus of individuals based on ‘connection’ ”, Notre Dame Journal of Formal Logic 22, 204–218.

Clarke, B.L., 1985, “Individuals and points”, Notre Dame Journal of Formal Logic 26, 61–75.

Davey, B.A., Priestley, H. A., 1990, Introduction to Lattices and Order, Cambridge, Cambridge University Press.

Forrest, P., 1996, “How innocent is mereology?”, Analysis 56, 127–131.

Gerla, G., Tortora, R., 1992, “La relazione di connessione in A. N. Whitehead: Aspetti matematici”, Epistemologia 15, 351–364.

Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S., 1980, A Compendium of Continuous Lattices, Berlin, Heidelberg, New York, Springer.

Gierz, G., Keimel, K., 1981, “Continuous ideal completions and compactifications”, in B. Banaschewsky and R.-E. Hofmann (eds.), 97–124.

Grzegorczyk, A., 1960, “Axiomatizability of geometry without points”, Synthese 12, 228–235.

Johnstone, P. T., 1982, Stone Spaces, Cambridge, Cambridge University Press.

Mac Lane, S., Moerdijk, I., 1992, Sheaves in Geometry and Logic. A First Introduction to Topos Theory, New York, Springer.

Mormann, T., 1998, “Neither mereology nor Whiteheadian account of space yet convicted” (to appear in Analysis).

Mormann 1998a, “Topological representations of mereological systems” (to appear in Poznan Studies in the Philosophy of the Sciences and the Humanities, Amsterdam, Rodopi).

Piazza, M., 1995, “ ‘One must always topologize’: Il teorema di Stone, la ‘topologia influente’ e l’epistemologia matematica”, Rivista di storia della scienza (ser. II), 3, 1–24.

Roeper, P., 1997, “Region-based topology”, Journal of Philosophical Logic 26, 251–309.

Stone, M., 1936, “The theory of representations for Boolean algebras”, Transactions of the American Mathematical Society 40, 37–111.

Vickers, S., 1989, Topology via Logic, Cambridge, Cambridge University Press.

Whitehead, A. N., 1929, Process and Reality. An Essay in Cosmology, New York, Macmillan.

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

Partnerzy platformy czasopism