The distinction between object-based and field-based conceptions of geographical reality has become well-known in recent years. It replicates at a conceptual level the distinction between vector-based and raster-based implementation methods for GIS. In this paper I endeavour to lay the groundwork for a careful, mathematically rigorous development of the relevant ideas at the conceptual level. The notions of object and field are given precise mathematical definitions, and an appropriate formal apparatus is constructed by which to address such issues as the interconvertibility of the object-based and field-based paradigms, and their relative adequacy for different representational problems.