It is a commonplace that statements of pure mathematics are necessarily true if true at all. But why should we think this? A cursory investigation of the practice of mathematics itself presents something of a puzzle here. Mathematicians do not appear to make use of the language of metaphysical necessity and possibility in their own investigations. Of course they do use the modal idioms ‘might’ and ‘must’ and their cognates. However, their use of these idioms does not provide much evidence that metaphysical modality is in play in any serious way. On the one hand, many of their uses seem to be metaphorical. As Wilfrid Hodges points out, when a mathematician says, for example, that one system ‘can be embedded’ in another, this is little more than a colorful way of saying that there is an embedding of one into the other. What the modal ‘can’ adds is a certain human colouring, by suggesting that part of the mathematics is carried out by a human being. This adds nothing to the mathematical content, but somehow it helps the readability (Hodges 2013: 6).
On the other hand, many uses of modals in mathematics express epistemic modality. For example, when mathematicians say at some point in their investigations, ‘Various answers might be correct’, they are not giving voice to a perceived metaphysical contingency in mathematical reality, but signaling that which answer is correct is an open question at the relevant stage in the process of mathematical discovery. And similarly, when they say, ‘Only one answer can be correct’, they are talking about what has been established at the relevant stage, not about what is metaphysically necessary: if it turns out that two answers are epistemically live at the time of speaking or writing, then the ‘must’ claim will be reckoned false. Also similarly, when a mathematician says that ‘Given that
A, it must be that B’, it is arguable that the ‘must’ again expresses a kind of epistemic modality.1 (One of us has explored elsewhere the behavior of epistemic modals embedded in logically complex sentences.2) And it is far from clear that there is any modality left once we set aside the metaphorical and epistemic occurrences of modals in mathematical texts.
