One of Penelope Maddy’s objections to the indispensability argument as the justification for mathematical practice is that it seems to make set theory a hostage to quantum gravity. That is, if set theorists are realists, and are justified in their position because their work is an indispensable part of fully specifying the mathematical theories that are an inherent part of physics, then they should be eagerly awaiting the results of physical research to find out just what things need to be included in order to support the physics.
A potential response is to notice that the form of this argument is just the same as the argument against Quine’s web of belief that points out that “2+2=4” is not going anywhere. I think Maddy is right to notice that higher set theory is at least more vulnerable to this sort of attack than basic number theory. I would be hard-pressed to imagine a version of science that doesn’t apply basic number theory, while the motivations for set theory stem from deep reasoning about the continuum and about various other transfinite objects.
Mark Colyvan responds by suggesting that in fact, some mathematics may not be applied. Such mathematics carries with it no ontological commitments, and he calls it “recreational mathematics”. (I don’t recall if Maddy used this term as well.) He suggests that set theorists may be free to investigate set theory however they want, because part of it will be applied, and part will be purely recreational.
However, this doesn’t seem right to me. Set theory seems to have a fairly unified methodology (ignoring the fact that Californians work on extending large cardinal axioms and sloving CH, while east coasters and Israelis do something else), and this applied/recreational divide would cut across this work in some totally unexpected way. It doesn’t seem plausible that this divide between the applied and the recreational could be important enough to base ontology on, but unimportant enough that practitioners don’t even notice it.
I think that it is far more likely that either all set theory falls on the applied side, or almost all of it falls on the recreational side. (I say “almost all”, because I could imagine the countable being seen as applied while the uncountable is recreational.) It seems that once one adopts ZFC, there are important reasons to adopt large cardinal axioms. Maddy gives these explanations quite well (see her “Believing the Axioms” parts I and II in the 1988 Journal of Symbolic Logic) and I think both she and Colyvan are unnecessarily worried that proper scientific methodology might only pick out some of this. The real worry about indispensability will come much lower down, and I think the argument may well be found wanting at that level. But rather than considering the rest of mathematics to be “recreational”, a Fieldian fictionalist position will be the right attitude to take. In practice, this should be no different from the realism that Colyvan supports, or the agnosticism that Maddy seems to endorse now.