Prolegomena to an ontology of shape

Abstract

Influenced by the four-category ontology of Aristotle, many modern ontologies treat shapes as accidental particulars which (a) are specifically dependent on the substantial particulars which act as their bearers, and (b) instantiate accidental universals which are exemplified by those bearers. It is also common to distinguish between, on the one hand, these physical shapes which form part of the empirical world and, on the other, ideal geometrical shapes which belong to the abstract realm of mathematics. Shapes of the former kind are often said to approximate, but never to exactly instantiate, shapes of the latter kind. Following a suggestion of Frege, ideal mathematical shapes can be given precise definitions as equivalence classes under the relation of geometrical similarity. One might, analogously, attempt to define physical shape universals as equivalence classes under a relation of physical similarity, but this fails because physical similarity is not an equivalence relation. In this talk I will examine the implications of this for the ontology of shape and in particular for the relationship between mathematical shapes and the shapes we attribute to physical objects.