Open texture clarified
Joost Jacob Vecht
Abstract
‘Open texture’ is the property of concepts or terms that they are not fully defined with regard to unexpected questions. Even though it is a ubiquitous phenomenon, there is no consensus on which concepts are subject to open texture. I present three equivalent definitions of open texture, connecting the concept to the notions of analyticity and algebraicity. I show that the claim that a concept is open-textured is equivalent to the claim that the class of analytic sentences using it can be changed, which in turn is equivalent to the claim that its definition is not algebraic. While this equivalence is interesting on its own, with it I can show the extent of open texture: all concepts that are not algebraically defined mathematical concepts are open textured. Finally, I show that this has unfortunate consequences for attempts to provide a foundation of mathematics.