Hamkins on the Multiverse

Ever since the rise of non-standard models and the proliferation of the independence results there have been two conflicting positions in the foundations of mathematics. The first position—which we shall call pluralism—maintains that certain statements of mathematics do not have determinate truth-values. On this view, although it is admitted that there are are practical reasons that one might give in favour of one set of axioms over another—say, that the one set of axioms is more useful than the other, with respect to a given task—, there are no theoretical reasons that can be given since, on this view, we are dealing with statements that lack determinate truth-values. The second position—which we shall call non-pluralism—maintains that statements of mathematics do have determinate truth-values. On this view, the matter of selecting one set of axioms over another, incompatible set of axioms is more than one of mere practical expedience—it is a substantive matter, one where the there is hope of giving theoretical reasons for one over the other since, on this view, the statements in question do have determinate truth-values. I say that there is a hope of finding theoretical reasons and not that there are theoretical reasons since I want to leave it open—at least as far as the characterization of the position is concerned—for the non-pluralist to admit that there are absolutely undecidable statements, that is, statements for which one cannot give any (convincing) theoretical reasons.