In this paper a solution of Whitehead’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 yielding the real numbers . The argument of the paper relies on the theories of continuous lattices and “pointless” (...) topology.

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An overview of contemporary part-whole theories, with reference to both their axiomatic developments and their philosophical underpinnings.

Mereological principles are often controversial; perhaps the most stark contrast is between those who claim that Weak Supplementation is analytic—constitutive of our notion of proper parthood—and those who argue that the principle is simply false, and subject to many counterexamples. The aim of this paper is to diagnose the source of this dispute. I’ll suggest that the dispute has arisen by participants failing to be sensitive to two different conceptions of proper parthood: the outstripping conception and the non-identity conception. I’ll (...) argue that the outstripping conception, can deliver the analyticity of Weak Supplementation on at least one sense of ‘analyticity’. I’ll also suggest that the non-identity conception cannot do so independently of considerations to do with mereological extensionality. (shrink)

This article examines the use of mathematical concepts in philosophy, focusing on topology, which may be viewed as a modern supplement to geometry. We show that Plato and Parmenides were already employing geometric ideas in their research, and discuss three examples of the application of topology to philosophical problems: the first concerns the analysis of the Cartesian distinction between res extensa and res cogitans, the second the ontology of possible worlds of Wittgenstein's Tractatus, and the third Leibniz's monadology. We also (...) consider the role of topology in mathematical explanations of the sort found in science, arguing that it can perform a role in philosophy that is of comparable importance. (shrink)

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”. (shrink)