Are We Still Babylonians? The Structure of the Foundations of Mathematics from a Wimsattian Perspective

Abstract

We will investigate the usefulness of Wimsatt’s concept of robustness for mathematics. In the experimental sciences, there are no demonstrations in the strict sense but only ‘confirmations’ of various types of the propositions one believes in. Wimsatt stressed that our conviction of a proposition grows with the number of independent and convergent confirmations. In the chapter, we shall investigate whether and to which degree wimsattian robustness is at issue in the practice of mathematicians, namely in the discussion of propositions considered as very useful, desirable, likely etc. but found to be logically independent of usual well-established bases of deduction. We will study two examples: the proposition asserting the consistency of set theory, and the proposition asserting the relative consistency of a certain large cardinal axiom. In these cases, one finds utterings from practitioners (including Bourbaki and Grothendieck) which suggest that confidence in the reliability of the propositions is based on the (alleged) presence of non-necessary but multiple, independent and convergent confirmations. These claims are submitted to thorough scrutiny. The second example might seem quite technical, but its discussion is useful for judging the relevance of such a concept of robustness for mathematics because it concerns a mathematical discipline (category theory) which is very important but which can’t claim so far to have reached a state of development rendering unlikely the future discovery of contradictions.