In this talk I explore the possibility of understanding mathematical beliefs as structured in a web of interconnected hyperintensional beliefs. The goal is to make sense of explanation in mathematics, a domain that is traditionally seen as lacking modal distinctions. The view is inspired by Quine’s web of belief, which is modified to make sense of “thick” explanatory relations between beliefs. Each belief that is held by a mathematician is characterized by a set of exact truthmakers, i.e. the mathematical facts (in a possible mathematical universe) that would make just those statements true. Beliefs are connected by relations of inference and support, which are functions of the truthmakers of the connected beliefs. The explanatory power of (i) statements, (ii) entailment relations between statements and (iii) proofs of statements is then determined by their capacity to provide a fruitful or efficient rewiring of a part of the web of beliefs.
The upshot of all this is that a model is provided for some aspects of pragmatics in mathematics. The Gricean principle of quantity requires, for example, that when mathematicians say that their Theorem A follows from Theorem B and Lemma C, they mean something stronger than merely that the A is a logical consequence of some background theory together with B and C (by the monotonicity of logical consequence, C could, with that weak interpretation, have absolutely nothing to do with A). They mean that B and C really contribute to proving A. In our model the pragmatic meaning of “A follows from B and C” in a mathematical paper is the fact that belief A is connected to belief B and C in the web of beliefs held by the author of the paper (and presumably shared with the reader of the paper).