A little while ago a wrote a piece on agnotology (the study of ignorance) and set-theoretic pluralism (journal, academia.edu, preprint).
The rough idea was that the ways in which we might be ignorant of set-theoretic claims might lead us to accept a methodological pluralism concerning set theory, even if we think there’s just one universe.
What do I mean by methodological pluralism concerning some discipline X? Simply that we should investigate a wide range of diverse theories pertaining to X. So in the case of set theory, proponents of the one universe view, even those with a preferred foundational theory, should still want people investigating other `competing’ theories.
Why so? Well, one obvious reason is that technical insight obtained by researching a particular theory can often be transferred across to ones own foundational theory. However, more philosophically, the point (made in detail in the paper) is that research into competing theories (thereby completing the theoretical picture they provide) provides evidence that we are not in a case of ignorance concerning our own foundational theory. In particular, it is only through the research of other programmes that we can be convinced that we are not in a state of unconscious ignorance (i.e. a state where we don’t know that we don’t know something) with respect to a particular mathematical question. So, we should actually want research into theories we think are false, in order to bolster the case for our own preferred theory actually being true.
A substantial question left open here is whether these ideas can be fed into a wider pluralistic framework. For instance, there’s some interesting technical facts about how category theory can be used set-theoretically (see Bagaria and Brooke-Taylor, `On Colimits and Elementary Embeddings’), but can this be fed into a wider epistemological/agnotological story elucidating the two kinds of foundation?
Here’s the abstract for anyone interested:
Much of the discussion of set-theoretic independence, and whether or not we could legitimately expand our foundational theory, concerns how we could possibly come to know the truth value of independent sentences. This paper pursues a slightly different tack, examining how we are ignorant of issues surrounding their truth. We argue that a study of how we are ignorant reveals a need for an understanding of set-theoretic explanation and motivates a pluralism concerning the adoption of foundational theory.