Alberto Peruzzi

The issue as to whether an atomistic or holistic viewof knowledge and meaning is correct relies on the way part/whole relationships is analysed,exactly as the issue as to whether a constructive or realistic view of knowledge and meaningis correct relies on the way internal/external relationships is analysed. Both theprinciple of compositionality and the context principle depend on how finely the constituents,the nature and the size of the context are identified; both the notion of meaning andthe notion of truth depend on the resources of internalisation/externalisation. Thus thespectrum of semantic and epistemological theories varies from (global) atomismto (global) holism, and from minimal to maximal internalisation. Are compositionaltheories necessarily extensional? Does formal semantics necessarily rely on set theory?Does the domain-specific character of the notions of element, part and whole prevent anygeneral, non-trivial account? The aim of the present paper is to provide a negative answerto these questions by exploring some of the features a theory covering the phenomenologyof part and whole should have. This phenomenology will only be sketched through a fewparadigmatic examples, showing how the reference of notions of part and wholevaries and which are the constraints inherentin such variation. Categorytheory provides the tools for fashioning this framework, since it allows describing any coherentcollection of objects (with actions defined over them) and (action preserving) maps betweenthe objects, as well as the variation of such collections in terms of suitablefunctors, coding the ways parts and wholes undergo co-variation. The main thesis is thatthere are interference patterns between the two pairs Local/Global and Internal/External,only in terms of which the above phenomenology can be properly described.Se mai tu diventeraimetá di te stesso, e te l''auguro, ragazzo, capirai cose al di lá della comune intelligenza dei cervelli interi.Avrai perso metá di te e del mondo, ma la realtá rimasta sará mille volte piú profonda e preziosa. E tupure vorrai che tutto sia dimezzato e straziato a tua immagine, perchè bellezza e sapienza e giustizia ci sono soloin ció che é fatto a brani.

The main motivation of the paper lies in an argument which shows the relevance of the topological notion of lifting for semantic theory. After a brief examination of aspects of knowledge which are described by means of concepts of algebraic geometry, the development of a functorial connection between topology and group theory is related to aspects of logical analysis. In categorical terms, lifting is, with extension, a form of division. As such it is investigated here, starting from simple examples in general topology up to fiber spaces and topoi, in order to appreciate the recurrent pattern, which is finally used to explain the understanding of sentences about abstract domains.