Notes on Field, "math realism and modality"

p.14 "An indispensability argument is an argument that we should believe a certain claim (for instance, a claim asserting the existence of a certain kind of eneity) because doing so is indispensable for certain purposes (which the argument then details). In this section I will focus on one special kind of ind argument: one involving indispensability for explnations. (There is some discussion of other kinds of indispensability argument in several of the essays in this volume, especially essays 3 and 7). To rely on this special kind of ind argument is to rely on a principle of inference to the best explanation. Some such principle seems to underlie much of our knowledge of the physical world.

It seems to me that most of us accept the principle of "inference to the best explanation" in the sense that this principle (or something pretty close to it) governs our ordinary inductive methodology.

The principle of inference to the best explanation makes no discrimination among these three cases: if a belief plays an ineliminable role in explanations of our observations, then other things being equal we should believe it, regardless of whether that beleif is itself observational, and regardless of whether the entitires it is about are observable. That I think is the metholdogly we (nealry) all employ and I think it would be unwise to change it. The fact tat the principle doesn't discriinate over whether the explanation is obesrvational (or whether it postulates unobservable entities) stands up well to reflection: intuitively, the observational nature of the explanation shoul dmake no difference in an inference to the best explanation. After all, in any case where we rely on inference to the best explanation our belief goes beyond what we have observed; the fact that one belief could be fairly directly tested by observation while the other couldn't seems to have no relevance to their evidential status when such an independent test has not been made. (When the independent test has been made--when th eleak behind the wall has been directly observed--then we need no longer rely on inference to the best explanation. When we do rely on inference to the best explanation our belief s go beyond th observations we have made, and my point is that the difference with respect to possible observations that haven't been made is irrlevant to our actual evidential situation.

I think that this sort of argument for the existence of mathematical entities (the Q-P argument, I'll call it) is an extremely powerful one, at least prima facie. It should be noted that the 1p argument is not merely that just as there are good explanations in which the posulation of unobservables is essential, so too are there good explanations in which the posulation of math entitie siessneital, so that if inference to the best explanation licnesess one is licenses the other. The argument is stronger in that it says that the very same explanations in whichthe posulation of unobseravbles is essential are explanations in which the posultaton of math enties is essential: math enters essentially into our theory of (say) electrons.

"At present of course we do not know in detail how to eliminate mathematical entities from every scientific explanation we accept; consequently I think that our inductive methodology does at present give us some justification for believing in math entities. But this brings me to my second point, which is that justification is not an all or nothing affair. The belief in math entities raises some problems which I and many other s believe to be fairly serious (I will briefly discuss two of these problems in the next section ...It seems to me that the most satisfying explanations are usually intrinsic ones that don't invoke entities that are causally irrelevant to what is being explained. Extrinsic explanations ar e acceptable but it is natural to think that for any good extrinsic explanation there is an intrinsic explanation that underlies it. This principle seems plausible indpenddently of anti-platonist scrulples...