To Bill Halchin's question on mereology: Mereology is a subject which traces back to Husserl's Logical Investigations (1900). As a formalized theory, it was developed mainly by the Polish logician and philosopher C. Lesniewski. As for a critical comparison between mereology and category theory, let me remind that, since mereology is no less based on the unanalized notion of sets, common theories of mereology are defective: insofar as semantic entities are the correlate of conceptual constructions, they are not to be viewed as sets, be they punctate or pointless. This remark would itself be pointless, were mereology intended as a theory of parts and wholes independent of the phenomenological aspects of semantics. But then its interest as a ³formal ontology² would be reduced, and lack of consideration for the process aspects (the ³maps²) would mean it served to describe only a static world. As for the relationships between mereology and topology : simple constructions such as the quotient space, the compactification, the connected sum of manifolds, etc., should serve as a reminder that topology is more than a taxonomy of different kinds of spaces (or different regions in space). It is also a theory of how different kinds of continuous variation affect the properties of any given space (and its regions) and how these kinds of variation are algebraically expressible. It is precisely because of this, that topology is more relevant to applied semantics than mereology. So, rather than some weak form of topology, a richer theory has to be sought for, in which different notions of space can be compared. Mereology is also based on the idea of ³objects in general². But one can doubt the existence of ³objects in general², as intended referents of a formal ontology, independent of any material domain. True, a similar doubt can also apply to abstract sets and arbitrary categories: for this very reason it is suitable to see the foundations of semantics and the foundations of mathematics as intimately linked. Typing is relevant here, yet it does not solve the philosophical problem concerning the way any notion of object is possible, as rooted in features (not point-like but still nomological) of the ³common-sense² world. For instance, in natural language even the notion of ³set² is expressed in a typed manner, one sensitive not only to the ³homogeneity² of the entities collected, but also to the mereological modality of their grouping (team, sheaf, fleet, swarm, series, bunch, cluster, etc.). The claim is that set theory results from a (usually implicit) abstraction from the type and the modality whithin which entities are collected modalities are not simply order types. That contemporary mathematics succeds in recovering such modalities (e. g. consider the topos G-Sh(X) of sheaves over a space X, with a group G acting on them) is a fact of crucial philosophical importance. It also suggests that mathematics is directly involved with structures of the material world and does not need logic or language as intermediacy agencies though, evidently, mathematics needs language and logic in order to be communicated and organised in a stable way. A theory investigating the relationships of part and whole remains inadequate until the stability of the objects involved with respect to (actual or imagined) actions is taken into account. There have been attempts at axiomatising mereology in (classical) first order logic, but they either presuppose (classical) set theory or occur in a semantic vacuum. Phenomenological wholes are rich in structure, so they have rather to be investigated by starting with a category of G-spaces (e. g. the category of differentiable manifolds with the action of a Lie group), then passing to sheaves over such spaces. Likewise, the problem of constraining the formation of unions of parts (of a given whole) also has a direct solution: parts and their unions are different from arbitrary subsets, being closed with respect to G-action. (In order to be applied in cognitive science, such a model will have to be greatly refined, taking into account edges, junctures, etc.) Of course, the difficulty lies in dealing with singularities. Let me suggest you the reading of what I wrote on the intrinsic limits of mereology, both from a mathematical point of view and from the point of view of its use as a tool to be applied in the semantic theory of (fragments) of natural language, in the following papers: Constraints on Universals, in R. Casati, B. Smith (eds.), Philosophy and the Cognitive Sciences, Hölder-Pichler-Tempsky, Vienna 1994, pp. 357-370. Action of Structures, Structure of Actions, Axiomathes, 7 (1996), pp. 325-348. An Essay on the Notion of Schema, in L. Albertazzi (ed.), Shapes of Form, Kluwer, Amsterdam 1999, pp. 191-243. The Geometric Roots of Semantics, in L. Albertazzi (ed.) Meaning and Cognition, John Benjamins, Amsterdam, 2000, pp. 169-201. On 20-10-2002 2:29, "Galchin Vasili" <vngalchin@yahoo.com> wrote:
Hi Cat community,
In Jeremy Buffterfield's paper (Section 1.3) below he says "a formal generalization of the notion of part as studied in mereology, so as to allow for 'structured parts'."
http://philsci-archive.pitt.edu/documents/disk0/00/00/01/92/
What is meteology? (I did look on the Web but I didn't understand anything).
Thanks and regards, Bill Halchin
Alberto Peruzzi Dipartimento di Filosofia Via Bolognese 52 50139 Firenze Italia 21-Oct-2002 22:24:25 -0300,3794;000000000000-00000000