Open texture, rigor, and proof

Open texture is a kind of semantic indeterminacy first systematically studied by Waismann. In this paper, extant definitions of open texture will be compared and contrasted, with a view towards the consequences of open-textured concepts in mathematics. It has been suggested that these would threaten the traditional virtues of proof, primarily the certainty bestowed by proof-possession, and this suggestion will be critically investigated using recent work on informal proof. It will be argued that informal proofs have virtues that mitigate the danger posed by open texture. Moreover, it will be argued that while rigor in the guise of formalisation and axiomatisation might banish open texture from mathematical theories through implicit definition, it can do so only at the cost of restricting the tamed concepts in certain ways.

Automated summary

This paper compares and contrasts different definitions of open texture and explores the consequences of open-textured concepts in mathematics. It argues that informal proofs have advantages in mitigating the dangers posed by open texture, and points out that eliminating open texture through formalization and axiomatization comes at the cost of restricting concepts.