Modality in mathematics

In this paper I argue that there are some quite basic questions that we can’t yet answer, about how we write and read mathematics. The questions themselves are straightforward enough to state, provided that we don’t allow ourselves to be distracted by irrelevances. I formulate them in terms of the use of modal notions in mathematical writing, but I think it will become clear that these formulations are special cases of much larger questions. How far the answers depend on general facts about language, and how far on peculiar features of mathematics, is one of the things we don’t yet know. Readers who want background information on English modals can find a readable treatment in Palmer [7]. I am in debt to various audiences and correspondents. But let me particularly thank the organisers and contributors of the Amsterdam meeting on ‘Practice-based philosophy of Logic and Mathematics’ in August and September 2009, and especially Catarina Dutilh who designed and led the whole enterprise.