Representing points as classes of mereotopologically structured basic entities

Laurenz Hudetz (University of Salzburg)

Representing points as classes of mereotopologically structured basic entities

Abstract

It has been suggested by a number of authors (most prominently Whitehead and Russell) that spacetime points should be identified with classes of mereotopologically structured basic entities. These suggestions are mainly motivated by either of the two following views: (a) the empiricist or relationist view that statements about spacetime points should be reduced to statements describing relations between epistemologically or metaphysically preferable entities such as processes and events; (b) the view that talk about spacetime points should be meaningful even in the case that the mereological structure of spacetime regions is atomless (given substantivalism about spacetime regions).

In order to evaluate the feasibility of such suggestions, two main questions need to be answered: (Q1) Under which conditions is a point representation method-i.e. a method of identifying points with classes of mereotopologically structured basic entities—generally adequate? (Q2) Are there any adequate point representation methods in that sense? My talk addresses exactly these two questions.
If we want to treat question Q1 in a systematic and rigorous way, we first of all need a unified formal framework for analysing and evaluating different point representation methods. I propose a unified framework, in which I explicate a general notion of point representations and the notion of general adequacy of point representations. Thereby, we obtain an answer to Q1 and transform the informal question Q2 into a precise, mathematical question.

I then examine important point representation methods within the proposed framework and present the main results I have achieved so far. It can be proven in a rigorous manner that the method which identifies points with limited maximal round filters-as suggested by Roeper (1997) and Mormann (2010)-is generally adequate. So question Q2 has a positive answer. Other salient methods such as the method employing ultrafilters as points (along the lines of Stone’s representation theorem) and the method using completely prime filters (as usual in point-free topology) can be proven to be not generally adequate and we can pinpoint the reason for their inadequacy.