A new class of philosophically-minded mathematician that I just learned from the logician Paul Levy: smallist.
CH is a statement of third-order arithmetic. It doesn’t quantify over the universe of sets. GCH, on the other hand, does. For smallists, who take a platonic view of PPN (powerset of powerset of the naturals) but not of the universe of sets, this is a big difference. (http://www.cs.nyu.edu/pipermail/fom/2016-October/020149.html)
I guess it means a mathematician who doesn’t necessarily want in their axiomatic system arbitrary powersets, rather just the few that are needed for ‘ordinary’ mathematics (say up to PPPN, which is plenty to deal with differential geometry, differential equations, functional analysis, number theory, algebraic geometry over number fields or rings of integers therein etc). I think he just invented the word 🙂 but I like it. For a categorically-minded person like me, this means I could work in a pretopos with just a few powersets posited.
(Originally posted to Google+ on 11 November 2016)