The Status of Mathematical Proofs and the Enhanced Indispensability Argument

One of the most important arguments for mathematical realism is Quine’s indispensability argument. In recent years, the original argument has been modified, and taken on the form of the Enhanced Indispensability Argument (EIA), where the explanatory virtues of mathematics are of primary importance. In particular, the notion of mathematical explanation (in contrast to causal or mechanistic explanation) in science is crucial for the debate. The issue of the explanatory role of mathematics in science is closely related to the problem of the explanatory role of mathematical proofs within mathematics. It is reasonable to claim, that the explanatory role of a mathematical theorem in physics depends on the fact how it is proven (that is, in particular, on the intra-mathematical explanatory role of the proof). But here we have to take into account the problem of the relationship between informal proofs known from practice and their formal counterparts. From the point of view of logical analysis (allowing, to identify the logically necessary ontological commitments, for instance using the tools from Reverse Mathematics), the formal version is important. From the point of view of the EIA, the "real", explanatory version is crucial. This leads to a tension between these versions of the argument.