 If the arguments given so far are correct, then the only remaining views—the only philosophies of mathematics that haven’t been ruled out—are platonism and fictionalism. Thus, to complete their argument, fictionalists need merely to provide an argument for premise (5); in other words, they just need to argue against platonism. But this turns out to be a lot harder than arguing against the various non-fictionalistic versions of anti-platonism considered above. As we’ve seen, fictionalists can argue against those views by simply motivating a series of empirical hypotheses about ordinary mathematical discourse and the ordinary meaning of the word ‘true’. More specifically, fictionalists can argue against these views by arguing that (a) ordinary mathematical utterances are best interpreted at face value, and (b) these utterances cannot plausibly be interpreted as being about physical or mental objects, and (c) sentences of the form ‘The object a is an F’ cannot be true, in the ordinary sense of the term, unless there really is such a thing as a. But fictionalists cannot argue against platonism in anything like this way, because fictionalists and platonists are in agreement about the meanings of ordinary mathematical utterances (and the word ‘true’). Indeed, platonists and fictionalists don’t disagree about any semantic theses. Their disagreement is about an ontological thesis: platonists believe in abstract objects, whereas fictionalists do not. Thus, if fictionalists are going to argue against platonism, they’re going to have to use a different kind of argument.