If Quine-Putnam's argument is (following the SEP):
- (P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
- (P2) Mathematical entities are indispensable to our best scientific theories.
- (C) We ought to have ontological commitment to mathematical entities.
I am reading this as a a transfer argument, where the supposed existence of the entities indispensable to our scientific theories is transferred to the mathematical objects used in those scientific theories.
Assuming this is tenable, is there a possible attack on premise (P1) along the lines that our best scientific theories do not necessarily involve any ontological commitment, so that no ontological commitment would be required of mathematics even if they were indispensable to our best scientific theories?
To restrict the discussion to physics, physics could be argued to propose only descriptive models that "work" more or less well in some range of parameters but do not necessarily carry a commitment to the existence of what they model. For example, "elementary particles" as point particles of matter are now superseded by "fields", just because the fields picture has more explanatory power. But if physicists needed to abandon the fields picture in favor of another model that would be a more accurate description, they would presumably do so, rather than cling to "particles" or "fields" just because they would have made some sort of "ontological commitment" through these models (it is actually more complicated, Newtonian gravity is useful in some range of parameters, while Einstein's SR and GR are useful in other ranges of parameters, there is a "classical limit" to QM, etc etc.)
So, if physics itself was seen to not actually make any kind of "ontological commitment" but instead be concerned with the descriptive power of models only, there would be no "ontological commitment" in physics to transfer to mathematics, even if mathematics were somehow deemed "indispensable" to physics.
As this kind of attack on (P1) been advanced by any philosopher?
Addendum: a variant on this attack could possibly be conceived by asking what happened to the ontological commitment to obsolete objects. If physics used some mathematical object X in theory A that was then superseded by theory B which no longer used mathematical object X, would the ontological commitment conferred to mathematical object X by its indispensability in physical theory A have to be retracted?