Rayo aims to defend the stability or coherence of a form of noncommittalism according to which the use of mathematical language does not commit one to the existence of any abstract mathematical objects. An important negative point that Rayo makes early on is that the noncommittalist is unable to accomplish this task by “translating each arithmetical sentence into a sentence that wears its ontological innocence on its sleeze” (385). Still, the positive program is to set out a method of specifying associated truth conditions for these sentences so that these conditions do not involve abstract objects, but only the world being a certain way. The key innovation, if I am not misunderstanding the paper, is to employ the full range of semantic machinery in explaining how the world has to be for a given sentence to be correct. Rayo carefully explores the issue of whether this is legitimate, and concludes that if one begins as a noncommittalist, then one will feel entitled to appeal to numbers and functions in one’s semantic theory. As a result, the noncommittalist can claim an internally coherent or stable package of views.
While much of the interest of the paper is in the details, the following frank admission by Rayo struck me as worth highlighting:
Whatever its plausibility as an explanation of how the standard arithmetical axioms might be rendered meaningful in such a way that their truth is knowable a priori, it should be clear that the arithmetical stipulation is not very plausible as an explanation of how the axioms were actually rendered meaningful or how it is that we actually acquire a priori knowledge of their truth (431-432).A committalist seems well within her rights to wonder, then, what the point is of carving out a stable position like Rayo’s if it has no bearing on our actual mathematical knowledge or on the actual contribution that mathematics makes in applied contexts.