I’ve been reading Mike Resnik’s book Mathematics as a Science of Patterns and have found a lot of stuff I like in it. He makes the point that if we use the indispensability argument to show that mathematical entities exist, then they shouldn’t be that different from the entities postulated by theoretical physics. I didn’t know much about the particular examples he gives from physics, but I think I would take the argument a little bit in the other direction – if mathematical entities really are indispensable to physical theory, then we might as well take them to be concrete physical objects that just happen to lack a lot of the causal and spatiotemporal properties that other physical objects have.
In addition, the indispensability argument meets the epistemological challenge, because we get epistemic access to the objects by confirming the whole theory to which they are indispensable. I’m not sure if Hartry Field discusses this point much in Science without Numbers, but Burgess and Rosen (in A Subject with no Object) seem to be puzzled by the fact that Field both gives an epistemological challenge and argues for the dispensability of mathematics. They think either one alone should be enough, if successful, and therefore the fact that he has to give both calls each into question. But if the point of the epistemological challenge is merely to show that there is no direct epistemic access to these objects, then we need to establish dispensability to show that the indispensability argument doesn’t give us indirect epistemic access. So both arguments are needed.
However, on page 109, Resnik criticizes Field, saying “whether space-time points are mathematical or physical, abstract or concrete, there will be no real gain in using them to dispense with (other) mathematical objects unless they are more epistemically accessible than the objects they replace.” This sort of objection is related to the ones that say that space-time points really are mathematical objects, and therefore Field hasn’t succeeded in nominalizing anything.
However, I think all of this misses an important point. Although Field says he’s a nominalist, it seems to me that a more important point is that he’s trying to give an internal explanation of everything in physics rather than an external one. (Resnik compares this to the contrast between synthetic geometry (where we only talk about points and lines and such) and analytic geometry (where we refer to real-number coordinates as well) and thus talks about “synthesizing” physics rather than “nominalizing” it.) Whether space-time points are concrete, physical objects or abstract, mathematical ones, and whether we have good epistemic access to them or not, they are somehow much more intrinsic to the physical system than real numbers seem to be. Real numbers are applied in measuring distances, calculating probabilities, stating temperatures, and many other things. These seem to be many different areas of the natural world, and using real numbers to explain all of them seems to involve some sort of “spooky action at a distance” as I discussed several months ago. Field’s reconstruction of Newtonian mechanics is certainly an advance on this front, whether or not it has any metaphysical, epistemological, or nominalistic gains. Thus, Resnik is wrong when he says there is no real gain in using space-time points instead of real numbers.