One of the central questions in the philosophy of mathematics concerns the nature of mathematical knowledge. The version of this question familiar from [Benacerraf, 1973] asks how knowledge of any mathematical proposition could be consistent with any picture of the semantics of mathematical language (and in particular with the apparently abstract and acausal nature of mathematical objects). However, there is a further question even granting existing knowledge of mathematical propositions, one may wonder what exactly it takes for a mathematician to come to know yet more propositions. To begin to address this question, I note that there is some extremely close connection in mathematics between knowledge and proof. Mathematicians often say that a claim is not known until a proof has been given, and an account somewhat like this is presupposed in some naturalistic discussions of mathematical knowledge (see [Horsten, 2001, pp. 186-9], where he concedes that other means may provide knowledge of mathematical propositions, but suggests that proof must underlie a notion of “mathematical knowledge”). However, for any account like this to work, it must be clear what counts as a proof. Although this question is in the end a normative one, about which proofs are right to play a role in understanding mathematical knowledge, in this paper I will only start to address it, by focusing on the descriptive question of which proofs mathematicians actually accept in practice. Although this is a descriptive question, it is still a question about norms in particular, it’s about isolating the norms that seem to underlie the practice of accepting and rejecting mathematical arguments. This question has a certain similarity to the question of grammaticality in a language linguists are interested in what norms seem to be actually at work in the practices of a certain language community, and I am interested in what norms seem to be actually at work in the practices of an epistemic community. The main difference is that in the epistemic case there is room for a difference between the norms that are followed by a community and the norms that ought to be followed. The proofs mathematicians accept are not complete formal proofs of the sort studied in proof theory, but are rather some sort of informal approximation