I just realized it’s been almost 2 months since I’ve posted here! You may have noticed trouble leaving comments in the last month or so – apparently my host site updated something in their system, and only today did I find the change I needed to make to make it work again.
Anyway, after finishing up at the Canada/USA Mathcamp, I visited some friends in Bellingham and Vancouver, and then had the beginning of the semester to deal with, which all distracted me from blogging for a while. Last week I was in Berlin for a workshop, Towards a New Epistemology of Mathematics, attached to the Gesellschaft für Analytische Philosophie’s large conference. An overview of the workshop by David Corfield is here.
A few talks caught my attention that he didn’t mention, so I’ll briefly mention those here. Tatiana Arrigoni presented some discussion of the candidate set-theoretic axiom V=L, mentioning that although many philosophers of set theory argue that it should be rejected, there are some (like Ronald Jensen) that argue in its favor. Her idea seemed to be that there might be at least two different ideas of set-theoretic intuition that lead to different sets of axioms.
Curtis Franks suggested that although Hilbert is traditionally regarded as a mathematical formalist, some of his early writings suggest that his program was motivated by a sort of naturalism, perhaps in Maddy’s vein. He objected to the intuitionists and others by saying that standard mathematical practice is just obviously justified, because it’s been so successful and hasn’t led to any problems – in a sense, their philosophical arguments are no better than those of the skeptic. However, Hilbert wanted to reformulate the consistency of mathematics as a mathematical (rather than philosophical) question – Gödel just showed that this was impossible.
And in the main conference, Øystein Linnebo suggested that John Burgess’ system inspired by Bernays and Boolos, although it very elegantly derives all of ZFC (and large cardinals up to indescribables) just by means of a fairly straightforward plural quantification system together with something like Cantor’s limitation of size principle, doesn’t provide a much stronger justification than Bernays’ original system. The particular plural logic used here does matter, and a seemingly similarly justified limitation of size principle leads to Russell’s paradox. I’m not convinced that Linnebo undermines Burgess’ system terribly much, but it’s definitely interesting to see how these systems develop.
Anyway, now I should return to more regular posting.