Reassessing the Explanatory Indispensability Argument: A Bayesian Defence of Nominalism

Advocates of the explanatory indispensability argument for platonism say two things. First, we should believe in the parts of our best scientific theories that are explanatory. Second, mathematical objects play an explanatory role within those theories. I give a two-part response. I start by using a Bayesian framework to argue that the standards many have proposed must be met to show that mathematical objects are dispensable are too demanding. In particular, nominalistic theories may be more probable than platonistic ones even if they are extremely complicated by comparison. This is true even if there are genuine cases of mathematical explanation in science. The point made here is a matter of principle, holding regardless of how one assesses nominalistic theories already on offer. I then examine my recent nominalization of second-order impure set theory in light of the correct, laxer standards. I make a tentative case that my nominalistic theory meets those standards, which would undermine the explanatory indispensability argument. While this case is provisional, I aim to bring attention to my nominalization and others in light of the revised standards for demonstrating dispensability.