Topological Foundations of Cognitive Science

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Ki

Mereology and uncertainty

Mereology as an art of composing complex concepts out of simpler parts is suited well to the task of reasoning under uncertainty: whereas it is most often difficult to ascertain whether a given thing is an element of a concept, it is possible to decide with belief degree close to certainty that the class of things is an ingredient of an other class, which is sufficient for carrying out the reasoning whose conclusions are taken as true under given conditions. We present in this work a scheme for reasoning based on mereology in which mereology in the classical sense is fuzzified in analogy to the concept fuzzification in the sense of L. A. Zadeh. In this process, mereology becomes rough mereology

Inconsistent boundaries

Research on this paper was supported by a grant from the Marsden Fund, Royal Society of New Zealand.Mereotopology is a theory of connected parts. The existence of boundaries, as parts of everyday objects, is basic to any such theory; but in classical mereotopology, there is a problem: if boundaries exist, then either distinct entities cannot be in contact, or else space is not topologically connected (Varzi in Noûs 31:26–58, 1997). In this paper we urge that this problem can be met with a paraconsistent mereotopology, and sketch the details of one such approach. The resulting theory focuses attention on the role of empty parts, in delivering a balanced and bounded metaphysics of naive space.PostprintPeer reviewe

Stanislaw Lesniewski: rethinking the philosophy of mathematics

Near the end of the nineteenth century, a part of mathematical research was focused on unification: the goal was to find 'one sort of thing' that mathematics is (or could be taken to be) about. Quite quickly sets became the main candidate for this position. While the enterprise hit a rough patch with Frege's failure and set-theoretic paradoxes, by the 1920s mathematicians (roughly speaking) settled on a promising axiomatization of set theory and considered it foundational. In parallel to this development was the work of Stanislaw Lesniewski (1886-1939), a Polish logician who did not accept the existence of abstract (aspatial, atemporal and acausal) objects such as sets. Lesniewski attempted to find a nominalistically acceptable replacement for set theory in the foundations of mathematics. His candidate was Mereology - a theory which, instead of sets and elements, spoke of wholes and parts. The goal of this paper will be to present Mereology in this context, to evaluate the feasibility of Lesniewski's project and to briefly comment on its contemporary relevance

Service integration in distributed control systems: an approach based on fusion of mereologies

International audienceThe objective of the present paper is to provide a methodology where the functional characteristics of a distributed engineering system can be obtained by merging domaindependent knowledge at run-time. We focus on distributed control systems where computing nodes are related to the physical environment in which they operate via sensors/actuators. The knowledge representation is formally expressed with a mereological approach where a structural mereology describes the physical environment and a functional mereology identifies available engineering goals for each computing node. During the design step, a mechanism based on Formal Concept Analysis (FCA) allows to generate the resulting goal mereology. The concept of goal is refined with sub-concepts in the multilevel structure. Because computing nodes depend on each other for goals to be achieved, an agent-based method is proposed to establish dynamically the dependencies among distributed nodes. This method is centered on a fusion mechanism involving the functional mereologies of appropriate nodes. We use an example from an open-channel hydraulic system controlling the water level to motivate and illustrate the model. Although it is limited to the engineering systems, this approach can be reused in related domains where the goal representation can be expressed as a triple including an action, a role and a physical entity

Mereology then and now

This paper offers a critical reconstruction of the motivations that led to the development of mereology as we know it today, along with a brief description of some questions that define current research in the field