Semantic indeterminacy, concept sharpening, and set theories

Abstract

Friedrich Waismann once suggested that mathematical concepts are not subject to open-texture; they are “closed”. This is not quite right, as there are some traditional mathematical notions that were, at least at one time, open-textured. One of them is the notion of “polyhedron” following the history sketched in Imre Lakatos’s Proofs and refutations. Another is “computability”, which has now been sharpened into an arguably closed notion, via the Church-Turing thesis.
There are also some mathematical notions that have longstanding, intuitive principles underlying them, principles that later proved to be inconsistent with each other, sometimes when the notion is applied to cases not considered previously (in which case it is perhaps an instance of open-texture). One example is “same size”, which is or was governed by the part-whole principle (one of Euclid’s Common Notions) and the one-one principle, now called “Hume’s Principle". Another is the notion of continuity.
The purpose of this paper is to explore the notion of “set” and other related notions like “class”, “totality”, and the like. I tentatively put forward a thesis that this notion, too, is or at least was subject to open-texture (or something like it) and has been sharpened in various ways.
This raises some questions concerning what the purposes of a (sharpened) theory of sets are to be. And questions about how one goes about trying to give non-ad-hoc explanations or answers to various questions.