Talk (Work in Progress): Andreas Frenzel (MCMP)

Title:

The Problem of Open Texture in Mathematics: An Attempt at a Solution

Abstract:

In this talk, I will examine the problem of open texture in mathematics and consider a possible solution based on different levels of formality.
A concept exhibits open texture, when no definition of it can ensure that every future case can be decided as either falling or not falling under the concept. If concepts in mathematics exhibited open texture, this would seemingly threaten its claim to certain knowledge by rigorous, exhaustive, and conclusive proofs.
I will first illustrate how open texture could arise in mathematics by discussing the problem of the definition of polyhedron. I will show how this concept seemingly resists a precise definition, making it difficult to ascertain whether a particular theorem about it holds.
Afterwards, I will present an account by Lakatos that differentiates between three levels of formality of proofs. According to Lakatos, preformal and postformal proofs employ an informal notion of concepts that exhibits open texture. Formal proofs on the other hand are completely precise and closed textured, due to the rigorous, often set-theoretical definitions they rely on.
This formality comes with a tradeoff, namely the loss of the intuitive access to the subject matter of the proof. It is not entirely clear what kind of objects the proof is about, since the definitions are stipulative and not based on intuitions.
I will argue that this formal notion of proof is nevertheless useful, since it restores the desirable properties of mathematics that open texture had threatened. It does so by closing a concept off against intuitively appealing counterexamples. Since the stipulative definition does not capture any preformal concept, it is final and unrevisable, thus allowing for exhaustive and conclusive proofs.