 The problem with paraphrase nominalist views is very simple: they involve empirical hypotheses about the meanings of ordinary mathematical utterances that are extremely implausible. For instance, in connection with if-thenism, it’s just really hard to believe that the best interpretation of what ordinary speakers of mathematical discourse (ordinary mathematicians and ordinary folk) are saying when they utter, e.g., ‘3 is prime’ is that if there were numbers then 3 would be prime. This just seems to get wrong what people actually mean when they utter sentences like this. Indeed, it seems that a more general point can be made here. There is a good interpretive principle that says something like this: we should interpret people’s utterances at face value unless there’s evidence that they have positive intentions to be interpreted nonliterally. Given this, and given (what seems obvious) that ordinary people don’t have positive intentions for their mathematical utterances to be interpreted nonliterally—e.g., as expressing conditional propositions—it seems to follow that we ought to interpret our mathematical utterances at face value. But this means that we ought to accept premise (1) and reject paraphrase nominalism.